Physics Book - User contributions [en]

Lenses

2015-11-30T00:11:25Z

Aallaire3: /* References */

Lenses are found in a huge array of optical instruments, ranging from a simple magnifying glass to the eye to a camera’s zoom
lens. Law of refraction is used to explore the properties of lenses and how they form images.

==The Main Idea==
[[File:Convex.PNG|200px|thumb|right|Convex]] [[File:Concave.PNG|200px|thumb|right|Concave]]
Index of refraction depends on the wavelength. Thus, light of different wavelengths is bent, or deflected, by different amounts as it passes through a lens. The shape of a lens, either concave or convex, also plays a role in the deflection pattern of light. The images above show that how these two shapes determines the behavior of the light rays. A lens where the middle is thicker than the two ends is called a "convex" lens, through which incoming light rays converge towards the center axis of the lens. A lens where the middle is thinner than the two ends is called a "concave" lens the prisms represent a "diverging" lens, through which incoming light rays diverge away from the center axis. The angle at which light rays converge or diverge is called the deflection angle. Deflection angles for thin lenses will be modeled mathematically in the following section. Thin lenses are lenses where the y position of a light ray does not change very much as the light ray travels through it. In other words, the lens is thick enough to refract light rays, but does not allow dispersion or aberrations.

===A Mathematical Model===

==== Law of Refraction ====
Refraction occurs when light travels through an area of space that has a changing index of refraction. The simplest case of refraction involves a uniform medium with index of refraction <math>n_1</math> and another medium with index of refraction <math>n_2</math>. The following equation describes the resulting deflection of the light ray:

:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>

==== Thin Lens Equation and Magnification ====
[[File:lens3b.svg|200px|thumb|A ray tracing diagram for a converging lens.]]

Thin lenses produce focal points on either side that can be modeled using the lensmaker's equation. Thin lenses follow a simple equation that determines the location of the images given a particular focal length (<math>f</math>) and object distance (<math>S_1</math>):

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

===A Computational Model===
[http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/23-lens Lens Simulation]

==Examples==

===Determine the Index of Refraction from Refraction Data===
[[File:Figure1.PNG|200px|thumb|right|Figure1]]
Find the index of refraction for medium 2 in Figure1 (a), assuming medium 1 is air and given the incident angle is 30.0º and the angle of refraction is 22.0º.
==== Strategy ====
The index of refraction for air is taken to be 1 in most cases (and up to four significant figures, it is 1.000). Thus <math>n_1 = 1.00 </math> here. From the given information, <math>\theta_1 = 30.0º </math> and <math>\theta_2 = 22.0º </math>. With this information, the only unknown in the lens equation is <math>n_2</math>, so that it can be used to find this unknown.
====Solution====
:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>
Entering known values,<math>n_2 = 1.33</math>.

===Fining the Image of a Light Bulb Filament by Ray Tracing and by the Thin Lens Equations===
[[File:Figure2.png|200px|thumb|right|Figure2]]
A clear glass light bulb is placed 0.750 m from a convex lens having a 0.500 m focal length, as shown in Figure2. Use ray tracing to get an approximate location for the image. Then use the thin lens equations to calculate (a) the location of the image and (b) its magnification. Verify that ray tracing and the thin lens equations produce consistent results.

====Solution====
The ray tracing to scale shows two rays from a point on the bulb’s filament crossing about 1.50 m on the farside of the lens. Thus the image <math>d_i</math> distance is about 1.50 m. Similarly, the image height based on ray tracing is greater than the object height by about a factor of 2, and the image is inverted. Thus <math>m</math> is about –2. The minus sign indicates that the image is inverted.

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

Solving for <math>d_i</math> from the given known values gives <math>d_i = 1.50m</math>.

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

Solving for <math>m</math> gives <math>m = -2.00 </math>

===Image Produced by a Concave Lens===
Suppose an object such as a book page is held 7.50 cm from a concave lens of focal length –10.0 cm. Such a lens could be used in eyeglasses to correct pronounced nearsightedness. What magnification is produced?

====Solution====
To find the magnification, we must first find <math>s_1</math> using the thin lens equation.

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

Using known values, we can solve for <math>s_1</math> which gives <math>s_1 = -4.29 cm</math>.
Now using the equation for magnification <math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>, <math> m = 0.571 </math>.

==Connectedness==
Optics is part of everyday life. Optics plays a central role in visual systems in biology. An industry of optical instruments allows many people to benefit from eyeglasses or contact lenses, and even camera lenses.

==History==
[[File:Ibn Sahl manuscript.jpg|thumb|right|upright|Reproduction of a page of Ibn Sahl's manuscript showing his knowledge of the law of refraction, now known as Snell's law]]
Optics began with the development of lenses by the ancient Egyptians and Mesopotamians. The ancient Romans and Ancient Greece filled glass spheres with water to make lenses. These practical developments were followed by the development of theories of light and vision by ancient philosophers and the development of geometrical optics.

Euclid wrote a treatise entitled ''Optics'' where he linked vision to geometry, creating ''geometrical optics''. Ptolemy summarized much of Euclid and went on to describe a way to measure the angle of refraction, though he failed to notice the empirical relationship between it and the angle of incidence.

In 984, the Persian mathematician Ibn Sahl wrote the treatise "On burning mirrors and lenses", correctly describing a law of refraction equivalent to Snell's law.He used this law to compute optimum shapes for lenses.

== See also ==

===Further reading===
*Alhacen. ''Book of Optics''.

*Isaac Newton. ''Opticks or, a Treatise of the reflexions, refractions, inflexions and colours of light. Also two treatises of the species and magnitude of curvilinear figures''.

*Fresnel, Augustin. ''The Wave Theory of Light''.

*William Rowan Hamilton.''The Mathematical Papers of Sir William Rowan Hamilton, Volume I: Geometrical Optics''.

===External links===
*[https://en.wikipedia.org/wiki/Index_of_optics_articles Index of Optics Articles]

==References==
*OpenStax algebra based intro physics textbook [https://openstaxcollege.org/textbooks/college-physics College Physics]

*An online concept map of intro physics [http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics]

* A physics resource written by experts for an expert audience [https://en.wikipedia.org/wiki/Portal:Physics Physics Portal]

Lenses

2015-11-30T00:09:25Z

Aallaire3: /* See also */

Lenses

2015-11-30T00:08:57Z

Aallaire3: /* References */

Lenses are found in a huge array of optical instruments, ranging from a simple magnifying glass to the eye to a camera’s zoom
lens. Law of refraction is used to explore the properties of lenses and how they form images.

==The Main Idea==
[[File:Convex.PNG|200px|thumb|right|Convex]] [[File:Concave.PNG|200px|thumb|right|Concave]]
Index of refraction depends on the wavelength. Thus, light of different wavelengths is bent, or deflected, by different amounts as it passes through a lens. The shape of a lens, either concave or convex, also plays a role in the deflection pattern of light. The images above show that how these two shapes determines the behavior of the light rays. A lens where the middle is thicker than the two ends is called a "convex" lens, through which incoming light rays converge towards the center axis of the lens. A lens where the middle is thinner than the two ends is called a "concave" lens the prisms represent a "diverging" lens, through which incoming light rays diverge away from the center axis. The angle at which light rays converge or diverge is called the deflection angle. Deflection angles for thin lenses will be modeled mathematically in the following section. Thin lenses are lenses where the y position of a light ray does not change very much as the light ray travels through it. In other words, the lens is thick enough to refract light rays, but does not allow dispersion or aberrations.

===A Mathematical Model===

==== Law of Refraction ====
Refraction occurs when light travels through an area of space that has a changing index of refraction. The simplest case of refraction involves a uniform medium with index of refraction <math>n_1</math> and another medium with index of refraction <math>n_2</math>. The following equation describes the resulting deflection of the light ray:

:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>

==== Thin Lens Equation and Magnification ====
[[File:lens3b.svg|200px|thumb|A ray tracing diagram for a converging lens.]]

Thin lenses produce focal points on either side that can be modeled using the lensmaker's equation. Thin lenses follow a simple equation that determines the location of the images given a particular focal length (<math>f</math>) and object distance (<math>S_1</math>):

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

===A Computational Model===
[http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/23-lens Lens Simulation]

==Examples==

===Determine the Index of Refraction from Refraction Data===
[[File:Figure1.PNG|200px|thumb|right|Figure1]]
Find the index of refraction for medium 2 in Figure1 (a), assuming medium 1 is air and given the incident angle is 30.0º and the angle of refraction is 22.0º.
==== Strategy ====
The index of refraction for air is taken to be 1 in most cases (and up to four significant figures, it is 1.000). Thus <math>n_1 = 1.00 </math> here. From the given information, <math>\theta_1 = 30.0º </math> and <math>\theta_2 = 22.0º </math>. With this information, the only unknown in the lens equation is <math>n_2</math>, so that it can be used to find this unknown.
====Solution====
:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>
Entering known values,<math>n_2 = 1.33</math>.

===Fining the Image of a Light Bulb Filament by Ray Tracing and by the Thin Lens Equations===
[[File:Figure2.png|200px|thumb|right|Figure2]]
A clear glass light bulb is placed 0.750 m from a convex lens having a 0.500 m focal length, as shown in Figure2. Use ray tracing to get an approximate location for the image. Then use the thin lens equations to calculate (a) the location of the image and (b) its magnification. Verify that ray tracing and the thin lens equations produce consistent results.

====Solution====
The ray tracing to scale shows two rays from a point on the bulb’s filament crossing about 1.50 m on the farside of the lens. Thus the image <math>d_i</math> distance is about 1.50 m. Similarly, the image height based on ray tracing is greater than the object height by about a factor of 2, and the image is inverted. Thus <math>m</math> is about –2. The minus sign indicates that the image is inverted.

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

Solving for <math>d_i</math> from the given known values gives <math>d_i = 1.50m</math>.

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

Solving for <math>m</math> gives <math>m = -2.00 </math>

===Image Produced by a Concave Lens===
Suppose an object such as a book page is held 7.50 cm from a concave lens of focal length –10.0 cm. Such a lens could be used in eyeglasses to correct pronounced nearsightedness. What magnification is produced?

====Solution====
To find the magnification, we must first find <math>s_1</math> using the thin lens equation.

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

Using known values, we can solve for <math>s_1</math> which gives <math>s_1 = -4.29 cm</math>.
Now using the equation for magnification <math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>, <math> m = 0.571 </math>.

==Connectedness==
Optics is part of everyday life. Optics plays a central role in visual systems in biology. An industry of optical instruments allows many people to benefit from eyeglasses or contact lenses, and even camera lenses.

==History==
[[File:Ibn Sahl manuscript.jpg|thumb|right|upright|Reproduction of a page of Ibn Sahl's manuscript showing his knowledge of the law of refraction, now known as Snell's law]]
Optics began with the development of lenses by the ancient Egyptians and Mesopotamians. The ancient Romans and Ancient Greece filled glass spheres with water to make lenses. These practical developments were followed by the development of theories of light and vision by ancient philosophers and the development of geometrical optics.

Euclid wrote a treatise entitled ''Optics'' where he linked vision to geometry, creating ''geometrical optics''. Ptolemy summarized much of Euclid and went on to describe a way to measure the angle of refraction, though he failed to notice the empirical relationship between it and the angle of incidence.

In 984, the Persian mathematician Ibn Sahl wrote the treatise "On burning mirrors and lenses", correctly describing a law of refraction equivalent to Snell's law.He used this law to compute optimum shapes for lenses.

== See also ==

===Further reading===
Alhacen. ''Book of Optics''.

Isaac Newton. ''Opticks or, a Treatise of the reflexions, refractions, inflexions and colours of light. Also two treatises of the species and magnitude of curvilinear figures''.

Fresnel, Augustin. ''The Wave Theory of Light''.

William Rowan Hamilton.''The Mathematical Papers of Sir William Rowan Hamilton, Volume I: Geometrical Optics''.

===External links===
[https://en.wikipedia.org/wiki/Index_of_optics_articles Index of Optics Articles]

==References==
*OpenStax algebra based intro physics textbook [https://openstaxcollege.org/textbooks/college-physics College Physics]

*An online concept map of intro physics [http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics]

Lenses

2015-11-30T00:07:47Z

Aallaire3: /* References */

Lenses are found in a huge array of optical instruments, ranging from a simple magnifying glass to the eye to a camera’s zoom
lens. Law of refraction is used to explore the properties of lenses and how they form images.

==The Main Idea==
[[File:Convex.PNG|200px|thumb|right|Convex]] [[File:Concave.PNG|200px|thumb|right|Concave]]
Index of refraction depends on the wavelength. Thus, light of different wavelengths is bent, or deflected, by different amounts as it passes through a lens. The shape of a lens, either concave or convex, also plays a role in the deflection pattern of light. The images above show that how these two shapes determines the behavior of the light rays. A lens where the middle is thicker than the two ends is called a "convex" lens, through which incoming light rays converge towards the center axis of the lens. A lens where the middle is thinner than the two ends is called a "concave" lens the prisms represent a "diverging" lens, through which incoming light rays diverge away from the center axis. The angle at which light rays converge or diverge is called the deflection angle. Deflection angles for thin lenses will be modeled mathematically in the following section. Thin lenses are lenses where the y position of a light ray does not change very much as the light ray travels through it. In other words, the lens is thick enough to refract light rays, but does not allow dispersion or aberrations.

===A Mathematical Model===

==== Law of Refraction ====
Refraction occurs when light travels through an area of space that has a changing index of refraction. The simplest case of refraction involves a uniform medium with index of refraction <math>n_1</math> and another medium with index of refraction <math>n_2</math>. The following equation describes the resulting deflection of the light ray:

:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>

==== Thin Lens Equation and Magnification ====
[[File:lens3b.svg|200px|thumb|A ray tracing diagram for a converging lens.]]

Thin lenses produce focal points on either side that can be modeled using the lensmaker's equation. Thin lenses follow a simple equation that determines the location of the images given a particular focal length (<math>f</math>) and object distance (<math>S_1</math>):

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

===A Computational Model===
[http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/23-lens Lens Simulation]

==Examples==

===Determine the Index of Refraction from Refraction Data===
[[File:Figure1.PNG|200px|thumb|right|Figure1]]
Find the index of refraction for medium 2 in Figure1 (a), assuming medium 1 is air and given the incident angle is 30.0º and the angle of refraction is 22.0º.
==== Strategy ====
The index of refraction for air is taken to be 1 in most cases (and up to four significant figures, it is 1.000). Thus <math>n_1 = 1.00 </math> here. From the given information, <math>\theta_1 = 30.0º </math> and <math>\theta_2 = 22.0º </math>. With this information, the only unknown in the lens equation is <math>n_2</math>, so that it can be used to find this unknown.
====Solution====
:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>
Entering known values,<math>n_2 = 1.33</math>.

===Fining the Image of a Light Bulb Filament by Ray Tracing and by the Thin Lens Equations===
[[File:Figure2.png|200px|thumb|right|Figure2]]
A clear glass light bulb is placed 0.750 m from a convex lens having a 0.500 m focal length, as shown in Figure2. Use ray tracing to get an approximate location for the image. Then use the thin lens equations to calculate (a) the location of the image and (b) its magnification. Verify that ray tracing and the thin lens equations produce consistent results.

====Solution====
The ray tracing to scale shows two rays from a point on the bulb’s filament crossing about 1.50 m on the farside of the lens. Thus the image <math>d_i</math> distance is about 1.50 m. Similarly, the image height based on ray tracing is greater than the object height by about a factor of 2, and the image is inverted. Thus <math>m</math> is about –2. The minus sign indicates that the image is inverted.

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

Solving for <math>d_i</math> from the given known values gives <math>d_i = 1.50m</math>.

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

Solving for <math>m</math> gives <math>m = -2.00 </math>

===Image Produced by a Concave Lens===
Suppose an object such as a book page is held 7.50 cm from a concave lens of focal length –10.0 cm. Such a lens could be used in eyeglasses to correct pronounced nearsightedness. What magnification is produced?

====Solution====
To find the magnification, we must first find <math>s_1</math> using the thin lens equation.

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

Using known values, we can solve for <math>s_1</math> which gives <math>s_1 = -4.29 cm</math>.
Now using the equation for magnification <math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>, <math> m = 0.571 </math>.

==Connectedness==
Optics is part of everyday life. Optics plays a central role in visual systems in biology. An industry of optical instruments allows many people to benefit from eyeglasses or contact lenses, and even camera lenses.

==History==
[[File:Ibn Sahl manuscript.jpg|thumb|right|upright|Reproduction of a page of Ibn Sahl's manuscript showing his knowledge of the law of refraction, now known as Snell's law]]
Optics began with the development of lenses by the ancient Egyptians and Mesopotamians. The ancient Romans and Ancient Greece filled glass spheres with water to make lenses. These practical developments were followed by the development of theories of light and vision by ancient philosophers and the development of geometrical optics.

Euclid wrote a treatise entitled ''Optics'' where he linked vision to geometry, creating ''geometrical optics''. Ptolemy summarized much of Euclid and went on to describe a way to measure the angle of refraction, though he failed to notice the empirical relationship between it and the angle of incidence.

In 984, the Persian mathematician Ibn Sahl wrote the treatise "On burning mirrors and lenses", correctly describing a law of refraction equivalent to Snell's law.He used this law to compute optimum shapes for lenses.

== See also ==

===Further reading===
Alhacen. ''Book of Optics''.

Isaac Newton. ''Opticks or, a Treatise of the reflexions, refractions, inflexions and colours of light. Also two treatises of the species and magnitude of curvilinear figures''.

Fresnel, Augustin. ''The Wave Theory of Light''.

William Rowan Hamilton.''The Mathematical Papers of Sir William Rowan Hamilton, Volume I: Geometrical Optics''.

===External links===
[https://en.wikipedia.org/wiki/Index_of_optics_articles Index of Optics Articles]

==References==
OpenStax algebra based intro physics textbook [https://openstaxcollege.org/textbooks/college-physics College Physics]

Lenses

2015-11-30T00:05:22Z

Aallaire3: /* See also */

Lenses are found in a huge array of optical instruments, ranging from a simple magnifying glass to the eye to a camera’s zoom
lens. Law of refraction is used to explore the properties of lenses and how they form images.

==The Main Idea==
[[File:Convex.PNG|200px|thumb|right|Convex]] [[File:Concave.PNG|200px|thumb|right|Concave]]
Index of refraction depends on the wavelength. Thus, light of different wavelengths is bent, or deflected, by different amounts as it passes through a lens. The shape of a lens, either concave or convex, also plays a role in the deflection pattern of light. The images above show that how these two shapes determines the behavior of the light rays. A lens where the middle is thicker than the two ends is called a "convex" lens, through which incoming light rays converge towards the center axis of the lens. A lens where the middle is thinner than the two ends is called a "concave" lens the prisms represent a "diverging" lens, through which incoming light rays diverge away from the center axis. The angle at which light rays converge or diverge is called the deflection angle. Deflection angles for thin lenses will be modeled mathematically in the following section. Thin lenses are lenses where the y position of a light ray does not change very much as the light ray travels through it. In other words, the lens is thick enough to refract light rays, but does not allow dispersion or aberrations.

===A Mathematical Model===

==== Law of Refraction ====
Refraction occurs when light travels through an area of space that has a changing index of refraction. The simplest case of refraction involves a uniform medium with index of refraction <math>n_1</math> and another medium with index of refraction <math>n_2</math>. The following equation describes the resulting deflection of the light ray:

:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>

==== Thin Lens Equation and Magnification ====
[[File:lens3b.svg|200px|thumb|A ray tracing diagram for a converging lens.]]

Thin lenses produce focal points on either side that can be modeled using the lensmaker's equation. Thin lenses follow a simple equation that determines the location of the images given a particular focal length (<math>f</math>) and object distance (<math>S_1</math>):

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

===A Computational Model===
[http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/23-lens Lens Simulation]

==Examples==

===Determine the Index of Refraction from Refraction Data===
[[File:Figure1.PNG|200px|thumb|right|Figure1]]
Find the index of refraction for medium 2 in Figure1 (a), assuming medium 1 is air and given the incident angle is 30.0º and the angle of refraction is 22.0º.
==== Strategy ====
The index of refraction for air is taken to be 1 in most cases (and up to four significant figures, it is 1.000). Thus <math>n_1 = 1.00 </math> here. From the given information, <math>\theta_1 = 30.0º </math> and <math>\theta_2 = 22.0º </math>. With this information, the only unknown in the lens equation is <math>n_2</math>, so that it can be used to find this unknown.
====Solution====
:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>
Entering known values,<math>n_2 = 1.33</math>.

===Fining the Image of a Light Bulb Filament by Ray Tracing and by the Thin Lens Equations===
[[File:Figure2.png|200px|thumb|right|Figure2]]
A clear glass light bulb is placed 0.750 m from a convex lens having a 0.500 m focal length, as shown in Figure2. Use ray tracing to get an approximate location for the image. Then use the thin lens equations to calculate (a) the location of the image and (b) its magnification. Verify that ray tracing and the thin lens equations produce consistent results.

====Solution====
The ray tracing to scale shows two rays from a point on the bulb’s filament crossing about 1.50 m on the farside of the lens. Thus the image <math>d_i</math> distance is about 1.50 m. Similarly, the image height based on ray tracing is greater than the object height by about a factor of 2, and the image is inverted. Thus <math>m</math> is about –2. The minus sign indicates that the image is inverted.

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

Solving for <math>d_i</math> from the given known values gives <math>d_i = 1.50m</math>.

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

Solving for <math>m</math> gives <math>m = -2.00 </math>

===Image Produced by a Concave Lens===
Suppose an object such as a book page is held 7.50 cm from a concave lens of focal length –10.0 cm. Such a lens could be used in eyeglasses to correct pronounced nearsightedness. What magnification is produced?

====Solution====
To find the magnification, we must first find <math>s_1</math> using the thin lens equation.

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

Using known values, we can solve for <math>s_1</math> which gives <math>s_1 = -4.29 cm</math>.
Now using the equation for magnification <math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>, <math> m = 0.571 </math>.

==Connectedness==
Optics is part of everyday life. Optics plays a central role in visual systems in biology. An industry of optical instruments allows many people to benefit from eyeglasses or contact lenses, and even camera lenses.

==History==
[[File:Ibn Sahl manuscript.jpg|thumb|right|upright|Reproduction of a page of Ibn Sahl's manuscript showing his knowledge of the law of refraction, now known as Snell's law]]
Optics began with the development of lenses by the ancient Egyptians and Mesopotamians. The ancient Romans and Ancient Greece filled glass spheres with water to make lenses. These practical developments were followed by the development of theories of light and vision by ancient philosophers and the development of geometrical optics.

Euclid wrote a treatise entitled ''Optics'' where he linked vision to geometry, creating ''geometrical optics''. Ptolemy summarized much of Euclid and went on to describe a way to measure the angle of refraction, though he failed to notice the empirical relationship between it and the angle of incidence.

In 984, the Persian mathematician Ibn Sahl wrote the treatise "On burning mirrors and lenses", correctly describing a law of refraction equivalent to Snell's law.He used this law to compute optimum shapes for lenses.

== See also ==

===Further reading===
Alhacen. ''Book of Optics''.

Isaac Newton. ''Opticks or, a Treatise of the reflexions, refractions, inflexions and colours of light. Also two treatises of the species and magnitude of curvilinear figures''.

Fresnel, Augustin. ''The Wave Theory of Light''.

William Rowan Hamilton.''The Mathematical Papers of Sir William Rowan Hamilton, Volume I: Geometrical Optics''.

===External links===
[https://en.wikipedia.org/wiki/Index_of_optics_articles Index of Optics Articles]

==References==

Lenses

2015-11-30T00:04:56Z

Aallaire3: /* External links */

Lenses are found in a huge array of optical instruments, ranging from a simple magnifying glass to the eye to a camera’s zoom
lens. Law of refraction is used to explore the properties of lenses and how they form images.

==The Main Idea==
[[File:Convex.PNG|200px|thumb|right|Convex]] [[File:Concave.PNG|200px|thumb|right|Concave]]
Index of refraction depends on the wavelength. Thus, light of different wavelengths is bent, or deflected, by different amounts as it passes through a lens. The shape of a lens, either concave or convex, also plays a role in the deflection pattern of light. The images above show that how these two shapes determines the behavior of the light rays. A lens where the middle is thicker than the two ends is called a "convex" lens, through which incoming light rays converge towards the center axis of the lens. A lens where the middle is thinner than the two ends is called a "concave" lens the prisms represent a "diverging" lens, through which incoming light rays diverge away from the center axis. The angle at which light rays converge or diverge is called the deflection angle. Deflection angles for thin lenses will be modeled mathematically in the following section. Thin lenses are lenses where the y position of a light ray does not change very much as the light ray travels through it. In other words, the lens is thick enough to refract light rays, but does not allow dispersion or aberrations.

===A Mathematical Model===

==== Law of Refraction ====
Refraction occurs when light travels through an area of space that has a changing index of refraction. The simplest case of refraction involves a uniform medium with index of refraction <math>n_1</math> and another medium with index of refraction <math>n_2</math>. The following equation describes the resulting deflection of the light ray:

:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>

==== Thin Lens Equation and Magnification ====
[[File:lens3b.svg|200px|thumb|A ray tracing diagram for a converging lens.]]

Thin lenses produce focal points on either side that can be modeled using the lensmaker's equation. Thin lenses follow a simple equation that determines the location of the images given a particular focal length (<math>f</math>) and object distance (<math>S_1</math>):

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

===A Computational Model===
[http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/23-lens Lens Simulation]

==Examples==

===Determine the Index of Refraction from Refraction Data===
[[File:Figure1.PNG|200px|thumb|right|Figure1]]
Find the index of refraction for medium 2 in Figure1 (a), assuming medium 1 is air and given the incident angle is 30.0º and the angle of refraction is 22.0º.
==== Strategy ====
The index of refraction for air is taken to be 1 in most cases (and up to four significant figures, it is 1.000). Thus <math>n_1 = 1.00 </math> here. From the given information, <math>\theta_1 = 30.0º </math> and <math>\theta_2 = 22.0º </math>. With this information, the only unknown in the lens equation is <math>n_2</math>, so that it can be used to find this unknown.
====Solution====
:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>
Entering known values,<math>n_2 = 1.33</math>.

===Fining the Image of a Light Bulb Filament by Ray Tracing and by the Thin Lens Equations===
[[File:Figure2.png|200px|thumb|right|Figure2]]
A clear glass light bulb is placed 0.750 m from a convex lens having a 0.500 m focal length, as shown in Figure2. Use ray tracing to get an approximate location for the image. Then use the thin lens equations to calculate (a) the location of the image and (b) its magnification. Verify that ray tracing and the thin lens equations produce consistent results.

====Solution====
The ray tracing to scale shows two rays from a point on the bulb’s filament crossing about 1.50 m on the farside of the lens. Thus the image <math>d_i</math> distance is about 1.50 m. Similarly, the image height based on ray tracing is greater than the object height by about a factor of 2, and the image is inverted. Thus <math>m</math> is about –2. The minus sign indicates that the image is inverted.

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

Solving for <math>d_i</math> from the given known values gives <math>d_i = 1.50m</math>.

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

Solving for <math>m</math> gives <math>m = -2.00 </math>

===Image Produced by a Concave Lens===
Suppose an object such as a book page is held 7.50 cm from a concave lens of focal length –10.0 cm. Such a lens could be used in eyeglasses to correct pronounced nearsightedness. What magnification is produced?

====Solution====
To find the magnification, we must first find <math>s_1</math> using the thin lens equation.

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

Using known values, we can solve for <math>s_1</math> which gives <math>s_1 = -4.29 cm</math>.
Now using the equation for magnification <math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>, <math> m = 0.571 </math>.

==Connectedness==
Optics is part of everyday life. Optics plays a central role in visual systems in biology. An industry of optical instruments allows many people to benefit from eyeglasses or contact lenses, and even camera lenses.

==History==
[[File:Ibn Sahl manuscript.jpg|thumb|right|upright|Reproduction of a page of Ibn Sahl's manuscript showing his knowledge of the law of refraction, now known as Snell's law]]
Optics began with the development of lenses by the ancient Egyptians and Mesopotamians. The ancient Romans and Ancient Greece filled glass spheres with water to make lenses. These practical developments were followed by the development of theories of light and vision by ancient philosophers and the development of geometrical optics.

Euclid wrote a treatise entitled ''Optics'' where he linked vision to geometry, creating ''geometrical optics''. Ptolemy summarized much of Euclid and went on to describe a way to measure the angle of refraction, though he failed to notice the empirical relationship between it and the angle of incidence.

In 984, the Persian mathematician Ibn Sahl wrote the treatise "On burning mirrors and lenses", correctly describing a law of refraction equivalent to Snell's law.He used this law to compute optimum shapes for lenses.

== See also ==

===Further reading===
Alhacen. ''Book of Optics''.

Isaac Newton. ''Opticks or, a Treatise of the reflexions, refractions, inflexions and colours of light. Also two treatises of the species and magnitude of curvilinear figures''.

Fresnel, Augustin. ''The Wave Theory of Light''.

William Rowan Hamilton.''The Mathematical Papers of Sir William Rowan Hamilton, Volume I: Geometrical Optics''.

===External links===
[[https://en.wikipedia.org/wiki/Index_of_optics_articles Index of Optics Articles]]

==References==

Lenses

2015-11-30T00:03:47Z

Aallaire3: /* See also */

Lenses are found in a huge array of optical instruments, ranging from a simple magnifying glass to the eye to a camera’s zoom
lens. Law of refraction is used to explore the properties of lenses and how they form images.

==The Main Idea==
[[File:Convex.PNG|200px|thumb|right|Convex]] [[File:Concave.PNG|200px|thumb|right|Concave]]
Index of refraction depends on the wavelength. Thus, light of different wavelengths is bent, or deflected, by different amounts as it passes through a lens. The shape of a lens, either concave or convex, also plays a role in the deflection pattern of light. The images above show that how these two shapes determines the behavior of the light rays. A lens where the middle is thicker than the two ends is called a "convex" lens, through which incoming light rays converge towards the center axis of the lens. A lens where the middle is thinner than the two ends is called a "concave" lens the prisms represent a "diverging" lens, through which incoming light rays diverge away from the center axis. The angle at which light rays converge or diverge is called the deflection angle. Deflection angles for thin lenses will be modeled mathematically in the following section. Thin lenses are lenses where the y position of a light ray does not change very much as the light ray travels through it. In other words, the lens is thick enough to refract light rays, but does not allow dispersion or aberrations.

===A Mathematical Model===

==== Law of Refraction ====
Refraction occurs when light travels through an area of space that has a changing index of refraction. The simplest case of refraction involves a uniform medium with index of refraction <math>n_1</math> and another medium with index of refraction <math>n_2</math>. The following equation describes the resulting deflection of the light ray:

:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>

==== Thin Lens Equation and Magnification ====
[[File:lens3b.svg|200px|thumb|A ray tracing diagram for a converging lens.]]

Thin lenses produce focal points on either side that can be modeled using the lensmaker's equation. Thin lenses follow a simple equation that determines the location of the images given a particular focal length (<math>f</math>) and object distance (<math>S_1</math>):

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

===A Computational Model===
[http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/23-lens Lens Simulation]

==Examples==

===Determine the Index of Refraction from Refraction Data===
[[File:Figure1.PNG|200px|thumb|right|Figure1]]
Find the index of refraction for medium 2 in Figure1 (a), assuming medium 1 is air and given the incident angle is 30.0º and the angle of refraction is 22.0º.
==== Strategy ====
The index of refraction for air is taken to be 1 in most cases (and up to four significant figures, it is 1.000). Thus <math>n_1 = 1.00 </math> here. From the given information, <math>\theta_1 = 30.0º </math> and <math>\theta_2 = 22.0º </math>. With this information, the only unknown in the lens equation is <math>n_2</math>, so that it can be used to find this unknown.
====Solution====
:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>
Entering known values,<math>n_2 = 1.33</math>.

===Fining the Image of a Light Bulb Filament by Ray Tracing and by the Thin Lens Equations===
[[File:Figure2.png|200px|thumb|right|Figure2]]
A clear glass light bulb is placed 0.750 m from a convex lens having a 0.500 m focal length, as shown in Figure2. Use ray tracing to get an approximate location for the image. Then use the thin lens equations to calculate (a) the location of the image and (b) its magnification. Verify that ray tracing and the thin lens equations produce consistent results.

====Solution====
The ray tracing to scale shows two rays from a point on the bulb’s filament crossing about 1.50 m on the farside of the lens. Thus the image <math>d_i</math> distance is about 1.50 m. Similarly, the image height based on ray tracing is greater than the object height by about a factor of 2, and the image is inverted. Thus <math>m</math> is about –2. The minus sign indicates that the image is inverted.

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

Solving for <math>d_i</math> from the given known values gives <math>d_i = 1.50m</math>.

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

Solving for <math>m</math> gives <math>m = -2.00 </math>

===Image Produced by a Concave Lens===
Suppose an object such as a book page is held 7.50 cm from a concave lens of focal length –10.0 cm. Such a lens could be used in eyeglasses to correct pronounced nearsightedness. What magnification is produced?

====Solution====
To find the magnification, we must first find <math>s_1</math> using the thin lens equation.

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

Using known values, we can solve for <math>s_1</math> which gives <math>s_1 = -4.29 cm</math>.
Now using the equation for magnification <math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>, <math> m = 0.571 </math>.

==Connectedness==
Optics is part of everyday life. Optics plays a central role in visual systems in biology. An industry of optical instruments allows many people to benefit from eyeglasses or contact lenses, and even camera lenses.

==History==
[[File:Ibn Sahl manuscript.jpg|thumb|right|upright|Reproduction of a page of Ibn Sahl's manuscript showing his knowledge of the law of refraction, now known as Snell's law]]
Optics began with the development of lenses by the ancient Egyptians and Mesopotamians. The ancient Romans and Ancient Greece filled glass spheres with water to make lenses. These practical developments were followed by the development of theories of light and vision by ancient philosophers and the development of geometrical optics.

Euclid wrote a treatise entitled ''Optics'' where he linked vision to geometry, creating ''geometrical optics''. Ptolemy summarized much of Euclid and went on to describe a way to measure the angle of refraction, though he failed to notice the empirical relationship between it and the angle of incidence.

In 984, the Persian mathematician Ibn Sahl wrote the treatise "On burning mirrors and lenses", correctly describing a law of refraction equivalent to Snell's law.He used this law to compute optimum shapes for lenses.

== See also ==

===Further reading===
Alhacen. ''Book of Optics''.

Isaac Newton. ''Opticks or, a Treatise of the reflexions, refractions, inflexions and colours of light. Also two treatises of the species and magnitude of curvilinear figures''.

Fresnel, Augustin. ''The Wave Theory of Light''.

William Rowan Hamilton.''The Mathematical Papers of Sir William Rowan Hamilton, Volume I: Geometrical Optics''.

===External links===

==References==

Lenses

2015-11-30T00:03:23Z

Aallaire3: /* Further reading */

Lenses are found in a huge array of optical instruments, ranging from a simple magnifying glass to the eye to a camera’s zoom
lens. Law of refraction is used to explore the properties of lenses and how they form images.

==The Main Idea==
[[File:Convex.PNG|200px|thumb|right|Convex]] [[File:Concave.PNG|200px|thumb|right|Concave]]
Index of refraction depends on the wavelength. Thus, light of different wavelengths is bent, or deflected, by different amounts as it passes through a lens. The shape of a lens, either concave or convex, also plays a role in the deflection pattern of light. The images above show that how these two shapes determines the behavior of the light rays. A lens where the middle is thicker than the two ends is called a "convex" lens, through which incoming light rays converge towards the center axis of the lens. A lens where the middle is thinner than the two ends is called a "concave" lens the prisms represent a "diverging" lens, through which incoming light rays diverge away from the center axis. The angle at which light rays converge or diverge is called the deflection angle. Deflection angles for thin lenses will be modeled mathematically in the following section. Thin lenses are lenses where the y position of a light ray does not change very much as the light ray travels through it. In other words, the lens is thick enough to refract light rays, but does not allow dispersion or aberrations.

===A Mathematical Model===

==== Law of Refraction ====
Refraction occurs when light travels through an area of space that has a changing index of refraction. The simplest case of refraction involves a uniform medium with index of refraction <math>n_1</math> and another medium with index of refraction <math>n_2</math>. The following equation describes the resulting deflection of the light ray:

:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>

==== Thin Lens Equation and Magnification ====
[[File:lens3b.svg|200px|thumb|A ray tracing diagram for a converging lens.]]

Thin lenses produce focal points on either side that can be modeled using the lensmaker's equation. Thin lenses follow a simple equation that determines the location of the images given a particular focal length (<math>f</math>) and object distance (<math>S_1</math>):

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

===A Computational Model===
[http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/23-lens Lens Simulation]

==Examples==

===Determine the Index of Refraction from Refraction Data===
[[File:Figure1.PNG|200px|thumb|right|Figure1]]
Find the index of refraction for medium 2 in Figure1 (a), assuming medium 1 is air and given the incident angle is 30.0º and the angle of refraction is 22.0º.
==== Strategy ====
The index of refraction for air is taken to be 1 in most cases (and up to four significant figures, it is 1.000). Thus <math>n_1 = 1.00 </math> here. From the given information, <math>\theta_1 = 30.0º </math> and <math>\theta_2 = 22.0º </math>. With this information, the only unknown in the lens equation is <math>n_2</math>, so that it can be used to find this unknown.
====Solution====
:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>
Entering known values,<math>n_2 = 1.33</math>.

===Fining the Image of a Light Bulb Filament by Ray Tracing and by the Thin Lens Equations===
[[File:Figure2.png|200px|thumb|right|Figure2]]
A clear glass light bulb is placed 0.750 m from a convex lens having a 0.500 m focal length, as shown in Figure2. Use ray tracing to get an approximate location for the image. Then use the thin lens equations to calculate (a) the location of the image and (b) its magnification. Verify that ray tracing and the thin lens equations produce consistent results.

====Solution====
The ray tracing to scale shows two rays from a point on the bulb’s filament crossing about 1.50 m on the farside of the lens. Thus the image <math>d_i</math> distance is about 1.50 m. Similarly, the image height based on ray tracing is greater than the object height by about a factor of 2, and the image is inverted. Thus <math>m</math> is about –2. The minus sign indicates that the image is inverted.

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

Solving for <math>d_i</math> from the given known values gives <math>d_i = 1.50m</math>.

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

Solving for <math>m</math> gives <math>m = -2.00 </math>

===Image Produced by a Concave Lens===
Suppose an object such as a book page is held 7.50 cm from a concave lens of focal length –10.0 cm. Such a lens could be used in eyeglasses to correct pronounced nearsightedness. What magnification is produced?

====Solution====
To find the magnification, we must first find <math>s_1</math> using the thin lens equation.

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

Using known values, we can solve for <math>s_1</math> which gives <math>s_1 = -4.29 cm</math>.
Now using the equation for magnification <math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>, <math> m = 0.571 </math>.

==Connectedness==
Optics is part of everyday life. Optics plays a central role in visual systems in biology. An industry of optical instruments allows many people to benefit from eyeglasses or contact lenses, and even camera lenses.

==History==
[[File:Ibn Sahl manuscript.jpg|thumb|right|upright|Reproduction of a page of Ibn Sahl's manuscript showing his knowledge of the law of refraction, now known as Snell's law]]
Optics began with the development of lenses by the ancient Egyptians and Mesopotamians. The ancient Romans and Ancient Greece filled glass spheres with water to make lenses. These practical developments were followed by the development of theories of light and vision by ancient philosophers and the development of geometrical optics.

Euclid wrote a treatise entitled ''Optics'' where he linked vision to geometry, creating ''geometrical optics''. Ptolemy summarized much of Euclid and went on to describe a way to measure the angle of refraction, though he failed to notice the empirical relationship between it and the angle of incidence.

In 984, the Persian mathematician Ibn Sahl wrote the treatise "On burning mirrors and lenses", correctly describing a law of refraction equivalent to Snell's law.He used this law to compute optimum shapes for lenses.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===
Alhacen. ''Book of Optics''.

Isaac Newton. ''Opticks or, a Treatise of the reflexions, refractions, inflexions and colours of light. Also two treatises of the species and magnitude of curvilinear figures''.

Fresnel, Augustin. ''The Wave Theory of Light''.

William Rowan Hamilton.''The Mathematical Papers of Sir William Rowan Hamilton, Volume I: Geometrical Optics''.

===External links===

==References==

Lenses

2015-11-30T00:00:11Z

Aallaire3: /* See also */

Lenses are found in a huge array of optical instruments, ranging from a simple magnifying glass to the eye to a camera’s zoom
lens. Law of refraction is used to explore the properties of lenses and how they form images.

==The Main Idea==
[[File:Convex.PNG|200px|thumb|right|Convex]] [[File:Concave.PNG|200px|thumb|right|Concave]]
Index of refraction depends on the wavelength. Thus, light of different wavelengths is bent, or deflected, by different amounts as it passes through a lens. The shape of a lens, either concave or convex, also plays a role in the deflection pattern of light. The images above show that how these two shapes determines the behavior of the light rays. A lens where the middle is thicker than the two ends is called a "convex" lens, through which incoming light rays converge towards the center axis of the lens. A lens where the middle is thinner than the two ends is called a "concave" lens the prisms represent a "diverging" lens, through which incoming light rays diverge away from the center axis. The angle at which light rays converge or diverge is called the deflection angle. Deflection angles for thin lenses will be modeled mathematically in the following section. Thin lenses are lenses where the y position of a light ray does not change very much as the light ray travels through it. In other words, the lens is thick enough to refract light rays, but does not allow dispersion or aberrations.

===A Mathematical Model===

==== Law of Refraction ====
Refraction occurs when light travels through an area of space that has a changing index of refraction. The simplest case of refraction involves a uniform medium with index of refraction <math>n_1</math> and another medium with index of refraction <math>n_2</math>. The following equation describes the resulting deflection of the light ray:

:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>

==== Thin Lens Equation and Magnification ====
[[File:lens3b.svg|200px|thumb|A ray tracing diagram for a converging lens.]]

Thin lenses produce focal points on either side that can be modeled using the lensmaker's equation. Thin lenses follow a simple equation that determines the location of the images given a particular focal length (<math>f</math>) and object distance (<math>S_1</math>):

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

===A Computational Model===
[http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/23-lens Lens Simulation]

==Examples==

===Determine the Index of Refraction from Refraction Data===
[[File:Figure1.PNG|200px|thumb|right|Figure1]]
Find the index of refraction for medium 2 in Figure1 (a), assuming medium 1 is air and given the incident angle is 30.0º and the angle of refraction is 22.0º.
==== Strategy ====
The index of refraction for air is taken to be 1 in most cases (and up to four significant figures, it is 1.000). Thus <math>n_1 = 1.00 </math> here. From the given information, <math>\theta_1 = 30.0º </math> and <math>\theta_2 = 22.0º </math>. With this information, the only unknown in the lens equation is <math>n_2</math>, so that it can be used to find this unknown.
====Solution====
:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>
Entering known values,<math>n_2 = 1.33</math>.

===Fining the Image of a Light Bulb Filament by Ray Tracing and by the Thin Lens Equations===
[[File:Figure2.png|200px|thumb|right|Figure2]]
A clear glass light bulb is placed 0.750 m from a convex lens having a 0.500 m focal length, as shown in Figure2. Use ray tracing to get an approximate location for the image. Then use the thin lens equations to calculate (a) the location of the image and (b) its magnification. Verify that ray tracing and the thin lens equations produce consistent results.

====Solution====
The ray tracing to scale shows two rays from a point on the bulb’s filament crossing about 1.50 m on the farside of the lens. Thus the image <math>d_i</math> distance is about 1.50 m. Similarly, the image height based on ray tracing is greater than the object height by about a factor of 2, and the image is inverted. Thus <math>m</math> is about –2. The minus sign indicates that the image is inverted.

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

Solving for <math>d_i</math> from the given known values gives <math>d_i = 1.50m</math>.

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

Solving for <math>m</math> gives <math>m = -2.00 </math>

===Image Produced by a Concave Lens===
Suppose an object such as a book page is held 7.50 cm from a concave lens of focal length –10.0 cm. Such a lens could be used in eyeglasses to correct pronounced nearsightedness. What magnification is produced?

====Solution====
To find the magnification, we must first find <math>s_1</math> using the thin lens equation.

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

Using known values, we can solve for <math>s_1</math> which gives <math>s_1 = -4.29 cm</math>.
Now using the equation for magnification <math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>, <math> m = 0.571 </math>.

==Connectedness==
Optics is part of everyday life. Optics plays a central role in visual systems in biology. An industry of optical instruments allows many people to benefit from eyeglasses or contact lenses, and even camera lenses.

==History==
[[File:Ibn Sahl manuscript.jpg|thumb|right|upright|Reproduction of a page of Ibn Sahl's manuscript showing his knowledge of the law of refraction, now known as Snell's law]]
Optics began with the development of lenses by the ancient Egyptians and Mesopotamians. The ancient Romans and Ancient Greece filled glass spheres with water to make lenses. These practical developments were followed by the development of theories of light and vision by ancient philosophers and the development of geometrical optics.

Euclid wrote a treatise entitled ''Optics'' where he linked vision to geometry, creating ''geometrical optics''. Ptolemy summarized much of Euclid and went on to describe a way to measure the angle of refraction, though he failed to notice the empirical relationship between it and the angle of incidence.

In 984, the Persian mathematician Ibn Sahl wrote the treatise "On burning mirrors and lenses", correctly describing a law of refraction equivalent to Snell's law.He used this law to compute optimum shapes for lenses.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

{{cite book |last=Alhacen |author-link=Alhazen |title=Book of Optics |year=1021}}

{{cite book |first=Isaac |last=Newton |author-link=Isaac Newton |title=Opticks|Opticks or, a Treatise of the reflexions, refractions, inflexions and colours of light. Also two treatises of the species and magnitude of curvilinear figures |location=London |publisher=printed for Sam. Smith. and Benj. Walford |year=1704 }}

{{Cite book |last=Fresnel |first=Augustin |author-link=Augustin Fresnel |year=1819 |chapter=Memoir on the Diffraction of Light |title=The Wave Theory of Light – Memoirs by Huygens, Young and Fresnel |publisher=American Book Company |pages=79–145 |url=http://books.google.com/?id=_0hWAAAAMAAJ&dq=memoir%20of%20fresnel&pg=PA79#v=onepage&q&f=false}}

{{cite book |first=William Rowan |last=Hamilton |url=http://books.google.com/?id=ekPnMgEACAAJ |title=The Mathematical Papers of Sir William Rowan Hamilton, Volume I: Geometrical Optics |others=Edited for the Royal Irish Academy by A. W. Conway and J. L. Synge |publisher=Cambridge University Press |year=1931 |accessdate=2013-07-13}}

===External links===

==References==

Lenses

2015-11-29T23:52:41Z

Aallaire3: /* Connectedness */

Lenses are found in a huge array of optical instruments, ranging from a simple magnifying glass to the eye to a camera’s zoom
lens. Law of refraction is used to explore the properties of lenses and how they form images.

==The Main Idea==
[[File:Convex.PNG|200px|thumb|right|Convex]] [[File:Concave.PNG|200px|thumb|right|Concave]]
Index of refraction depends on the wavelength. Thus, light of different wavelengths is bent, or deflected, by different amounts as it passes through a lens. The shape of a lens, either concave or convex, also plays a role in the deflection pattern of light. The images above show that how these two shapes determines the behavior of the light rays. A lens where the middle is thicker than the two ends is called a "convex" lens, through which incoming light rays converge towards the center axis of the lens. A lens where the middle is thinner than the two ends is called a "concave" lens the prisms represent a "diverging" lens, through which incoming light rays diverge away from the center axis. The angle at which light rays converge or diverge is called the deflection angle. Deflection angles for thin lenses will be modeled mathematically in the following section. Thin lenses are lenses where the y position of a light ray does not change very much as the light ray travels through it. In other words, the lens is thick enough to refract light rays, but does not allow dispersion or aberrations.

===A Mathematical Model===

==== Law of Refraction ====
Refraction occurs when light travels through an area of space that has a changing index of refraction. The simplest case of refraction involves a uniform medium with index of refraction <math>n_1</math> and another medium with index of refraction <math>n_2</math>. The following equation describes the resulting deflection of the light ray:

:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>

==== Thin Lens Equation and Magnification ====
[[File:lens3b.svg|200px|thumb|A ray tracing diagram for a converging lens.]]

Thin lenses produce focal points on either side that can be modeled using the lensmaker's equation. Thin lenses follow a simple equation that determines the location of the images given a particular focal length (<math>f</math>) and object distance (<math>S_1</math>):

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

===A Computational Model===
[http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/23-lens Lens Simulation]

==Examples==

===Determine the Index of Refraction from Refraction Data===
[[File:Figure1.PNG|200px|thumb|right|Figure1]]
Find the index of refraction for medium 2 in Figure1 (a), assuming medium 1 is air and given the incident angle is 30.0º and the angle of refraction is 22.0º.
==== Strategy ====
The index of refraction for air is taken to be 1 in most cases (and up to four significant figures, it is 1.000). Thus <math>n_1 = 1.00 </math> here. From the given information, <math>\theta_1 = 30.0º </math> and <math>\theta_2 = 22.0º </math>. With this information, the only unknown in the lens equation is <math>n_2</math>, so that it can be used to find this unknown.
====Solution====
:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>
Entering known values,<math>n_2 = 1.33</math>.

===Fining the Image of a Light Bulb Filament by Ray Tracing and by the Thin Lens Equations===
[[File:Figure2.png|200px|thumb|right|Figure2]]
A clear glass light bulb is placed 0.750 m from a convex lens having a 0.500 m focal length, as shown in Figure2. Use ray tracing to get an approximate location for the image. Then use the thin lens equations to calculate (a) the location of the image and (b) its magnification. Verify that ray tracing and the thin lens equations produce consistent results.

====Solution====
The ray tracing to scale shows two rays from a point on the bulb’s filament crossing about 1.50 m on the farside of the lens. Thus the image <math>d_i</math> distance is about 1.50 m. Similarly, the image height based on ray tracing is greater than the object height by about a factor of 2, and the image is inverted. Thus <math>m</math> is about –2. The minus sign indicates that the image is inverted.

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

Solving for <math>d_i</math> from the given known values gives <math>d_i = 1.50m</math>.

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

Solving for <math>m</math> gives <math>m = -2.00 </math>

===Image Produced by a Concave Lens===
Suppose an object such as a book page is held 7.50 cm from a concave lens of focal length –10.0 cm. Such a lens could be used in eyeglasses to correct pronounced nearsightedness. What magnification is produced?

====Solution====
To find the magnification, we must first find <math>s_1</math> using the thin lens equation.

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

Using known values, we can solve for <math>s_1</math> which gives <math>s_1 = -4.29 cm</math>.
Now using the equation for magnification <math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>, <math> m = 0.571 </math>.

==Connectedness==
Optics is part of everyday life. Optics plays a central role in visual systems in biology. An industry of optical instruments allows many people to benefit from eyeglasses or contact lenses, and even camera lenses.

==History==
[[File:Ibn Sahl manuscript.jpg|thumb|right|upright|Reproduction of a page of Ibn Sahl's manuscript showing his knowledge of the law of refraction, now known as Snell's law]]
Optics began with the development of lenses by the ancient Egyptians and Mesopotamians. The ancient Romans and Ancient Greece filled glass spheres with water to make lenses. These practical developments were followed by the development of theories of light and vision by ancient philosophers and the development of geometrical optics.

Euclid wrote a treatise entitled ''Optics'' where he linked vision to geometry, creating ''geometrical optics''. Ptolemy summarized much of Euclid and went on to describe a way to measure the angle of refraction, though he failed to notice the empirical relationship between it and the angle of incidence.

In 984, the Persian mathematician Ibn Sahl wrote the treatise "On burning mirrors and lenses", correctly describing a law of refraction equivalent to Snell's law.He used this law to compute optimum shapes for lenses.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

Lenses

2015-11-29T23:48:59Z

Aallaire3:

Lenses are found in a huge array of optical instruments, ranging from a simple magnifying glass to the eye to a camera’s zoom
lens. Law of refraction is used to explore the properties of lenses and how they form images.

==The Main Idea==
[[File:Convex.PNG|200px|thumb|right|Convex]] [[File:Concave.PNG|200px|thumb|right|Concave]]
Index of refraction depends on the wavelength. Thus, light of different wavelengths is bent, or deflected, by different amounts as it passes through a lens. The shape of a lens, either concave or convex, also plays a role in the deflection pattern of light. The images above show that how these two shapes determines the behavior of the light rays. A lens where the middle is thicker than the two ends is called a "convex" lens, through which incoming light rays converge towards the center axis of the lens. A lens where the middle is thinner than the two ends is called a "concave" lens the prisms represent a "diverging" lens, through which incoming light rays diverge away from the center axis. The angle at which light rays converge or diverge is called the deflection angle. Deflection angles for thin lenses will be modeled mathematically in the following section. Thin lenses are lenses where the y position of a light ray does not change very much as the light ray travels through it. In other words, the lens is thick enough to refract light rays, but does not allow dispersion or aberrations.

===A Mathematical Model===

==== Law of Refraction ====
Refraction occurs when light travels through an area of space that has a changing index of refraction. The simplest case of refraction involves a uniform medium with index of refraction <math>n_1</math> and another medium with index of refraction <math>n_2</math>. The following equation describes the resulting deflection of the light ray:

:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>

==== Thin Lens Equation and Magnification ====
[[File:lens3b.svg|200px|thumb|A ray tracing diagram for a converging lens.]]

Thin lenses produce focal points on either side that can be modeled using the lensmaker's equation. Thin lenses follow a simple equation that determines the location of the images given a particular focal length (<math>f</math>) and object distance (<math>S_1</math>):

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

===A Computational Model===
[http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/23-lens Lens Simulation]

==Examples==

===Determine the Index of Refraction from Refraction Data===
[[File:Figure1.PNG|200px|thumb|right|Figure1]]
Find the index of refraction for medium 2 in Figure1 (a), assuming medium 1 is air and given the incident angle is 30.0º and the angle of refraction is 22.0º.
==== Strategy ====
The index of refraction for air is taken to be 1 in most cases (and up to four significant figures, it is 1.000). Thus <math>n_1 = 1.00 </math> here. From the given information, <math>\theta_1 = 30.0º </math> and <math>\theta_2 = 22.0º </math>. With this information, the only unknown in the lens equation is <math>n_2</math>, so that it can be used to find this unknown.
====Solution====
:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>
Entering known values,<math>n_2 = 1.33</math>.

===Fining the Image of a Light Bulb Filament by Ray Tracing and by the Thin Lens Equations===
[[File:Figure2.png|200px|thumb|right|Figure2]]
A clear glass light bulb is placed 0.750 m from a convex lens having a 0.500 m focal length, as shown in Figure2. Use ray tracing to get an approximate location for the image. Then use the thin lens equations to calculate (a) the location of the image and (b) its magnification. Verify that ray tracing and the thin lens equations produce consistent results.

====Solution====
The ray tracing to scale shows two rays from a point on the bulb’s filament crossing about 1.50 m on the farside of the lens. Thus the image <math>d_i</math> distance is about 1.50 m. Similarly, the image height based on ray tracing is greater than the object height by about a factor of 2, and the image is inverted. Thus <math>m</math> is about –2. The minus sign indicates that the image is inverted.

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

Solving for <math>d_i</math> from the given known values gives <math>d_i = 1.50m</math>.

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

Solving for <math>m</math> gives <math>m = -2.00 </math>

===Image Produced by a Concave Lens===
Suppose an object such as a book page is held 7.50 cm from a concave lens of focal length –10.0 cm. Such a lens could be used in eyeglasses to correct pronounced nearsightedness. What magnification is produced?

====Solution====
To find the magnification, we must first find <math>s_1</math> using the thin lens equation.

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

Using known values, we can solve for <math>s_1</math> which gives <math>s_1 = -4.29 cm</math>.
Now using the equation for magnification <math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>, <math> m = 0.571 </math>.

==Connectedness==
#How is this topic connected to something that you are interested in?
#How is it connected to your major?
#Is there an interesting industrial application?

==History==
[[File:Ibn Sahl manuscript.jpg|thumb|right|upright|Reproduction of a page of Ibn Sahl's manuscript showing his knowledge of the law of refraction, now known as Snell's law]]
Optics began with the development of lenses by the ancient Egyptians and Mesopotamians. The ancient Romans and Ancient Greece filled glass spheres with water to make lenses. These practical developments were followed by the development of theories of light and vision by ancient philosophers and the development of geometrical optics.

Euclid wrote a treatise entitled ''Optics'' where he linked vision to geometry, creating ''geometrical optics''. Ptolemy summarized much of Euclid and went on to describe a way to measure the angle of refraction, though he failed to notice the empirical relationship between it and the angle of incidence.

In 984, the Persian mathematician Ibn Sahl wrote the treatise "On burning mirrors and lenses", correctly describing a law of refraction equivalent to Snell's law.He used this law to compute optimum shapes for lenses.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

Lenses

2015-11-29T23:46:05Z

Aallaire3: /* Image Produced by a Concave Lens */

Lenses are found in a huge array of optical instruments, ranging from a simple magnifying glass to the eye to a camera’s zoom
lens. Law of refraction is used to explore the properties of lenses and how they form images.

==The Main Idea==
[[File:Convex.PNG|200px|thumb|right|Convex]] [[File:Concave.PNG|200px|thumb|right|Concave]]
Index of refraction depends on the wavelength. Thus, light of different wavelengths is bent, or deflected, by different amounts as it passes through a lens. The shape of a lens, either concave or convex, also plays a role in the deflection pattern of light. The images above show that how these two shapes determines the behavior of the light rays. A lens where the middle is thicker than the two ends is called a "convex" lens, through which incoming light rays converge towards the center axis of the lens. A lens where the middle is thinner than the two ends is called a "concave" lens the prisms represent a "diverging" lens, through which incoming light rays diverge away from the center axis. The angle at which light rays converge or diverge is called the deflection angle. Deflection angles for thin lenses will be modeled mathematically in the following section. Thin lenses are lenses where the y position of a light ray does not change very much as the light ray travels through it. In other words, the lens is thick enough to refract light rays, but does not allow dispersion or aberrations.

===A Mathematical Model===

==== Law of Refraction ====
Refraction occurs when light travels through an area of space that has a changing index of refraction. The simplest case of refraction involves a uniform medium with index of refraction <math>n_1</math> and another medium with index of refraction <math>n_2</math>. The following equation describes the resulting deflection of the light ray:

:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>

==== Thin Lens Equation and Magnification ====
[[File:lens3b.svg|200px|thumb|A ray tracing diagram for a converging lens.]]

Thin lenses produce focal points on either side that can be modeled using the lensmaker's equation. Thin lenses follow a simple equation that determines the location of the images given a particular focal length (<math>f</math>) and object distance (<math>S_1</math>):

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

===A Computational Model===
[http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/23-lens Lens Simulation]

==Examples==

===Determine the Index of Refraction from Refraction Data===
[[File:Figure1.PNG|200px|thumb|right|Figure1]]
Find the index of refraction for medium 2 in Figure1 (a), assuming medium 1 is air and given the incident angle is 30.0º and the angle of refraction is 22.0º.
==== Strategy ====
The index of refraction for air is taken to be 1 in most cases (and up to four significant figures, it is 1.000). Thus <math>n_1 = 1.00 </math> here. From the given information, <math>\theta_1 = 30.0º </math> and <math>\theta_2 = 22.0º </math>. With this information, the only unknown in the lens equation is <math>n_2</math>, so that it can be used to find this unknown.
====Solution====
:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>
Entering known values,<math>n_2 = 1.33</math>.

===Fining the Image of a Light Bulb Filament by Ray Tracing and by the Thin Lens Equations===
[[File:Figure2.png|200px|thumb|right|Figure2]]
A clear glass light bulb is placed 0.750 m from a convex lens having a 0.500 m focal length, as shown in Figure2. Use ray tracing to get an approximate location for the image. Then use the thin lens equations to calculate (a) the location of the image and (b) its magnification. Verify that ray tracing and the thin lens equations produce consistent results.

====Solution====
The ray tracing to scale shows two rays from a point on the bulb’s filament crossing about 1.50 m on the farside of the lens. Thus the image <math>d_i</math> distance is about 1.50 m. Similarly, the image height based on ray tracing is greater than the object height by about a factor of 2, and the image is inverted. Thus <math>m</math> is about –2. The minus sign indicates that the image is inverted.

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

Solving for <math>d_i</math> from the given known values gives <math>d_i = 1.50m</math>.

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

Solving for <math>m</math> gives <math>m = -2.00 </math>

===Image Produced by a Concave Lens===
Suppose an object such as a book page is held 7.50 cm from a concave lens of focal length –10.0 cm. Such a lens could be used in eyeglasses to correct pronounced nearsightedness. What magnification is produced?

====Solution====
To find the magnification, we must first find <math>s_1</math> using the thin lens equation.

==Connectedness==
#How is this topic connected to something that you are interested in?
#How is it connected to your major?
#Is there an interesting industrial application?

==History==
[[File:Ibn Sahl manuscript.jpg|thumb|right|upright|Reproduction of a page of Ibn Sahl's manuscript showing his knowledge of the law of refraction, now known as Snell's law]]
Optics began with the development of lenses by the ancient Egyptians and Mesopotamians. The ancient Romans and Ancient Greece filled glass spheres with water to make lenses. These practical developments were followed by the development of theories of light and vision by ancient philosophers and the development of geometrical optics.

Euclid wrote a treatise entitled ''Optics'' where he linked vision to geometry, creating ''geometrical optics''. Ptolemy summarized much of Euclid and went on to describe a way to measure the angle of refraction, though he failed to notice the empirical relationship between it and the angle of incidence.

In 984, the Persian mathematician Ibn Sahl wrote the treatise "On burning mirrors and lenses", correctly describing a law of refraction equivalent to Snell's law.He used this law to compute optimum shapes for lenses.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

Lenses

2015-11-29T23:42:12Z

Aallaire3: /* History */

Lenses are found in a huge array of optical instruments, ranging from a simple magnifying glass to the eye to a camera’s zoom
lens. Law of refraction is used to explore the properties of lenses and how they form images.

==The Main Idea==
[[File:Convex.PNG|200px|thumb|right|Convex]] [[File:Concave.PNG|200px|thumb|right|Concave]]
Index of refraction depends on the wavelength. Thus, light of different wavelengths is bent, or deflected, by different amounts as it passes through a lens. The shape of a lens, either concave or convex, also plays a role in the deflection pattern of light. The images above show that how these two shapes determines the behavior of the light rays. A lens where the middle is thicker than the two ends is called a "convex" lens, through which incoming light rays converge towards the center axis of the lens. A lens where the middle is thinner than the two ends is called a "concave" lens the prisms represent a "diverging" lens, through which incoming light rays diverge away from the center axis. The angle at which light rays converge or diverge is called the deflection angle. Deflection angles for thin lenses will be modeled mathematically in the following section. Thin lenses are lenses where the y position of a light ray does not change very much as the light ray travels through it. In other words, the lens is thick enough to refract light rays, but does not allow dispersion or aberrations.

===A Mathematical Model===

==== Law of Refraction ====
Refraction occurs when light travels through an area of space that has a changing index of refraction. The simplest case of refraction involves a uniform medium with index of refraction <math>n_1</math> and another medium with index of refraction <math>n_2</math>. The following equation describes the resulting deflection of the light ray:

:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>

==== Thin Lens Equation and Magnification ====
[[File:lens3b.svg|200px|thumb|A ray tracing diagram for a converging lens.]]

Thin lenses produce focal points on either side that can be modeled using the lensmaker's equation. Thin lenses follow a simple equation that determines the location of the images given a particular focal length (<math>f</math>) and object distance (<math>S_1</math>):

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

===A Computational Model===
[http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/23-lens Lens Simulation]

==Examples==

===Determine the Index of Refraction from Refraction Data===
[[File:Figure1.PNG|200px|thumb|right|Figure1]]
Find the index of refraction for medium 2 in Figure1 (a), assuming medium 1 is air and given the incident angle is 30.0º and the angle of refraction is 22.0º.
==== Strategy ====
The index of refraction for air is taken to be 1 in most cases (and up to four significant figures, it is 1.000). Thus <math>n_1 = 1.00 </math> here. From the given information, <math>\theta_1 = 30.0º </math> and <math>\theta_2 = 22.0º </math>. With this information, the only unknown in the lens equation is <math>n_2</math>, so that it can be used to find this unknown.
====Solution====
:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>
Entering known values,<math>n_2 = 1.33</math>.

===Fining the Image of a Light Bulb Filament by Ray Tracing and by the Thin Lens Equations===
[[File:Figure2.png|200px|thumb|right|Figure2]]
A clear glass light bulb is placed 0.750 m from a convex lens having a 0.500 m focal length, as shown in Figure2. Use ray tracing to get an approximate location for the image. Then use the thin lens equations to calculate (a) the location of the image and (b) its magnification. Verify that ray tracing and the thin lens equations produce consistent results.

====Solution====
The ray tracing to scale shows two rays from a point on the bulb’s filament crossing about 1.50 m on the farside of the lens. Thus the image <math>d_i</math> distance is about 1.50 m. Similarly, the image height based on ray tracing is greater than the object height by about a factor of 2, and the image is inverted. Thus <math>m</math> is about –2. The minus sign indicates that the image is inverted.

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

Solving for <math>d_i</math> from the given known values gives <math>d_i = 1.50m</math>.

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

Solving for <math>m</math> gives <math>m = -2.00 </math>

===Image Produced by a Concave Lens===
Suppose an object such as a book page is held 7.50 cm from a concave lens of focal length –10.0 cm. Such a lens could be used in eyeglasses to correct pronounced nearsightedness. What magnification is produced?

==Connectedness==
#How is this topic connected to something that you are interested in?
#How is it connected to your major?
#Is there an interesting industrial application?

==History==
[[File:Ibn Sahl manuscript.jpg|thumb|right|upright|Reproduction of a page of Ibn Sahl's manuscript showing his knowledge of the law of refraction, now known as Snell's law]]
Optics began with the development of lenses by the ancient Egyptians and Mesopotamians. The ancient Romans and Ancient Greece filled glass spheres with water to make lenses. These practical developments were followed by the development of theories of light and vision by ancient philosophers and the development of geometrical optics.

Euclid wrote a treatise entitled ''Optics'' where he linked vision to geometry, creating ''geometrical optics''. Ptolemy summarized much of Euclid and went on to describe a way to measure the angle of refraction, though he failed to notice the empirical relationship between it and the angle of incidence.

In 984, the Persian mathematician Ibn Sahl wrote the treatise "On burning mirrors and lenses", correctly describing a law of refraction equivalent to Snell's law.He used this law to compute optimum shapes for lenses.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

Lenses

2015-11-29T23:41:18Z

Aallaire3: /* History */

Lenses are found in a huge array of optical instruments, ranging from a simple magnifying glass to the eye to a camera’s zoom
lens. Law of refraction is used to explore the properties of lenses and how they form images.

==The Main Idea==
[[File:Convex.PNG|200px|thumb|right|Convex]] [[File:Concave.PNG|200px|thumb|right|Concave]]
Index of refraction depends on the wavelength. Thus, light of different wavelengths is bent, or deflected, by different amounts as it passes through a lens. The shape of a lens, either concave or convex, also plays a role in the deflection pattern of light. The images above show that how these two shapes determines the behavior of the light rays. A lens where the middle is thicker than the two ends is called a "convex" lens, through which incoming light rays converge towards the center axis of the lens. A lens where the middle is thinner than the two ends is called a "concave" lens the prisms represent a "diverging" lens, through which incoming light rays diverge away from the center axis. The angle at which light rays converge or diverge is called the deflection angle. Deflection angles for thin lenses will be modeled mathematically in the following section. Thin lenses are lenses where the y position of a light ray does not change very much as the light ray travels through it. In other words, the lens is thick enough to refract light rays, but does not allow dispersion or aberrations.

===A Mathematical Model===

==== Law of Refraction ====
Refraction occurs when light travels through an area of space that has a changing index of refraction. The simplest case of refraction involves a uniform medium with index of refraction <math>n_1</math> and another medium with index of refraction <math>n_2</math>. The following equation describes the resulting deflection of the light ray:

:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>

==== Thin Lens Equation and Magnification ====
[[File:lens3b.svg|200px|thumb|A ray tracing diagram for a converging lens.]]

Thin lenses produce focal points on either side that can be modeled using the lensmaker's equation. Thin lenses follow a simple equation that determines the location of the images given a particular focal length (<math>f</math>) and object distance (<math>S_1</math>):

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

===A Computational Model===
[http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/23-lens Lens Simulation]

==Examples==

===Determine the Index of Refraction from Refraction Data===
[[File:Figure1.PNG|200px|thumb|right|Figure1]]
Find the index of refraction for medium 2 in Figure1 (a), assuming medium 1 is air and given the incident angle is 30.0º and the angle of refraction is 22.0º.
==== Strategy ====
The index of refraction for air is taken to be 1 in most cases (and up to four significant figures, it is 1.000). Thus <math>n_1 = 1.00 </math> here. From the given information, <math>\theta_1 = 30.0º </math> and <math>\theta_2 = 22.0º </math>. With this information, the only unknown in the lens equation is <math>n_2</math>, so that it can be used to find this unknown.
====Solution====
:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>
Entering known values,<math>n_2 = 1.33</math>.

===Fining the Image of a Light Bulb Filament by Ray Tracing and by the Thin Lens Equations===
[[File:Figure2.png|200px|thumb|right|Figure2]]
A clear glass light bulb is placed 0.750 m from a convex lens having a 0.500 m focal length, as shown in Figure2. Use ray tracing to get an approximate location for the image. Then use the thin lens equations to calculate (a) the location of the image and (b) its magnification. Verify that ray tracing and the thin lens equations produce consistent results.

====Solution====
The ray tracing to scale shows two rays from a point on the bulb’s filament crossing about 1.50 m on the farside of the lens. Thus the image <math>d_i</math> distance is about 1.50 m. Similarly, the image height based on ray tracing is greater than the object height by about a factor of 2, and the image is inverted. Thus <math>m</math> is about –2. The minus sign indicates that the image is inverted.

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

Solving for <math>d_i</math> from the given known values gives <math>d_i = 1.50m</math>.

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

Solving for <math>m</math> gives <math>m = -2.00 </math>

===Image Produced by a Concave Lens===
Suppose an object such as a book page is held 7.50 cm from a concave lens of focal length –10.0 cm. Such a lens could be used in eyeglasses to correct pronounced nearsightedness. What magnification is produced?

==Connectedness==
#How is this topic connected to something that you are interested in?
#How is it connected to your major?
#Is there an interesting industrial application?

==History==
[[File:Ibn Sahl manuscript.jpg|thumb|right|upright|Reproduction of a page of [[Ibn Sahl]]'s manuscript showing his knowledge of the law of refraction, now known as [[Snell's law]]]]
Optics began with the development of lenses by the ancient Egyptians and Mesopotamians. The ancient Romans and Ancient Greece filled glass spheres with water to make lenses. These practical developments were followed by the development of theories of light and vision by ancient philosophers and the development of geometrical optics.

Euclid wrote a treatise entitled ''[[Euclid#Other works|Optics]]'' where he linked vision to geometry, creating ''geometrical optics''. Ptolemy summarized much of Euclid and went on to describe a way to measure the angle of refraction, though he failed to notice the empirical relationship between it and the angle of incidence.

In 984, the Persian mathematician Ibn Sahl wrote the treatise "On burning mirrors and lenses", correctly describing a law of refraction equivalent to Snell's law.He used this law to compute optimum shapes for lenses.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

Lenses

2015-11-29T23:37:17Z

Aallaire3: /* History */

Lenses are found in a huge array of optical instruments, ranging from a simple magnifying glass to the eye to a camera’s zoom
lens. Law of refraction is used to explore the properties of lenses and how they form images.

==The Main Idea==
[[File:Convex.PNG|200px|thumb|right|Convex]] [[File:Concave.PNG|200px|thumb|right|Concave]]
Index of refraction depends on the wavelength. Thus, light of different wavelengths is bent, or deflected, by different amounts as it passes through a lens. The shape of a lens, either concave or convex, also plays a role in the deflection pattern of light. The images above show that how these two shapes determines the behavior of the light rays. A lens where the middle is thicker than the two ends is called a "convex" lens, through which incoming light rays converge towards the center axis of the lens. A lens where the middle is thinner than the two ends is called a "concave" lens the prisms represent a "diverging" lens, through which incoming light rays diverge away from the center axis. The angle at which light rays converge or diverge is called the deflection angle. Deflection angles for thin lenses will be modeled mathematically in the following section. Thin lenses are lenses where the y position of a light ray does not change very much as the light ray travels through it. In other words, the lens is thick enough to refract light rays, but does not allow dispersion or aberrations.

===A Mathematical Model===

==== Law of Refraction ====
Refraction occurs when light travels through an area of space that has a changing index of refraction. The simplest case of refraction involves a uniform medium with index of refraction <math>n_1</math> and another medium with index of refraction <math>n_2</math>. The following equation describes the resulting deflection of the light ray:

:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>

==== Thin Lens Equation and Magnification ====
[[File:lens3b.svg|200px|thumb|A ray tracing diagram for a converging lens.]]

Thin lenses produce focal points on either side that can be modeled using the lensmaker's equation. Thin lenses follow a simple equation that determines the location of the images given a particular focal length (<math>f</math>) and object distance (<math>S_1</math>):

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

===A Computational Model===
[http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/23-lens Lens Simulation]

==Examples==

===Determine the Index of Refraction from Refraction Data===
[[File:Figure1.PNG|200px|thumb|right|Figure1]]
Find the index of refraction for medium 2 in Figure1 (a), assuming medium 1 is air and given the incident angle is 30.0º and the angle of refraction is 22.0º.
==== Strategy ====
The index of refraction for air is taken to be 1 in most cases (and up to four significant figures, it is 1.000). Thus <math>n_1 = 1.00 </math> here. From the given information, <math>\theta_1 = 30.0º </math> and <math>\theta_2 = 22.0º </math>. With this information, the only unknown in the lens equation is <math>n_2</math>, so that it can be used to find this unknown.
====Solution====
:<math>n_1\sin\theta_1 = n_2\sin\theta_2\ </math>
Entering known values,<math>n_2 = 1.33</math>.

===Fining the Image of a Light Bulb Filament by Ray Tracing and by the Thin Lens Equations===
[[File:Figure2.png|200px|thumb|right|Figure2]]
A clear glass light bulb is placed 0.750 m from a convex lens having a 0.500 m focal length, as shown in Figure2. Use ray tracing to get an approximate location for the image. Then use the thin lens equations to calculate (a) the location of the image and (b) its magnification. Verify that ray tracing and the thin lens equations produce consistent results.

====Solution====
The ray tracing to scale shows two rays from a point on the bulb’s filament crossing about 1.50 m on the farside of the lens. Thus the image <math>d_i</math> distance is about 1.50 m. Similarly, the image height based on ray tracing is greater than the object height by about a factor of 2, and the image is inverted. Thus <math>m</math> is about –2. The minus sign indicates that the image is inverted.

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

Solving for <math>d_i</math> from the given known values gives <math>d_i = 1.50m</math>.

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

Solving for <math>m</math> gives <math>m = -2.00 </math>

===Image Produced by a Concave Lens===
Suppose an object such as a book page is held 7.50 cm from a concave lens of focal length –10.0 cm. Such a lens could be used in eyeglasses to correct pronounced nearsightedness. What magnification is produced?

==Connectedness==
#How is this topic connected to something that you are interested in?
#How is it connected to your major?
#Is there an interesting industrial application?

==History==
Optics began with the development of lenses by the [[ancient Egypt]]ians and [[Mesopotamia]]ns. The earliest known lenses, made from polished crystal, often [[quartz]], date from as early as 700 BC for [[Assyria]]n lenses such as the Layard/[[Nimrud lens]].<ref>{{cite news |url=http://news.bbc.co.uk/1/hi/sci/tech/380186.stm |title=World's oldest telescope? |publisher=BBC News |date=July 1, 1999 |accessdate=Jan 3, 2010}}</ref> The [[ancient Roman]]s and [[Ancient Greece|Greeks]] filled glass spheres with water to make lenses. These practical developments were followed by the development of theories of light and vision by ancient [[Greek philosophy|Greek]] and [[Indian philosophy|Indian]] philosophers, and the development of [[geometrical optics]] in the [[Greco-Roman world]]. The word ''optics'' comes from the [[ancient Greek]] word {{lang|grc|ὀπτική}}, meaning "appearance, look".<ref>{{cite book|title=The Concise Oxford Dictionary of English Etymology|year=1996|author=T. F. Hoad|isbn=0-19-283098-8}}</ref>

[[Euclid]] wrote a treatise entitled ''[[Euclid#Other works|Optics]]'' where he linked vision to [[geometry]], creating ''geometrical optics''. [[Ptolemy]] summarised much of Euclid and went on to describe a way to measure the [[angle of refraction]], though he failed to notice the empirical relationship between it and the angle of incidence.

[[File:Alhazen, the Persian.gif|thumb|upright|[[Alhazen]] (Ibn al-Haytham), "the father of Optics" <ref>{{Citation | first = RL | last = Verma | year = 1969 | title = Al-Hazen: father of modern optics}}</ref>]]
[[File:Ibn Sahl manuscript.jpg|thumb|right|upright|Reproduction of a page of [[Ibn Sahl]]'s manuscript showing his knowledge of the law of refraction, now known as [[Snell's law]]]]
In 984, the [[Persia]]n mathematician [[Ibn Sahl]] wrote the treatise "On burning mirrors and lenses", correctly describing a law of refraction equivalent to [[Snell's law]].He used this law to compute optimum shapes for lenses and [[curved mirror]]s. In the early 11th century, [[Alhazen]] (Ibn al-Haytham) wrote the ''[[Book of Optics]]'' (''Kitab al-manazir'') in which he explored reflection and refraction and proposed a new system for explaining vision and light based on observation and experiment.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

Lenses

2015-11-29T23:30:46Z

Aallaire3: /* Image Produced by a Concave Lens */

Lenses

2015-11-29T23:21:21Z

Aallaire3: /* Difficult */

Lenses

2015-11-29T23:19:17Z

Aallaire3: /* Fining the Image of a Light Bulb Filament by Ray Tracing and by the Thin Lens Equations */

Lenses

2015-11-29T23:15:53Z

Aallaire3: /* Fining the Image of a Light Bulb Filament by Ray Tracing and by the Thin Lens Equations */

Lenses

2015-11-29T23:10:40Z

Aallaire3: /* Fining the Image of a Light Bulb Filament by Ray Tracing and by the Thin Lens Equations */

File:Figure2.png

2015-11-29T23:09:52Z

Aallaire3: Aallaire3 uploaded a new version of "File:Figure2.png"

File:Figure2.png

2015-11-29T23:09:25Z

Aallaire3:

Lenses

2015-11-29T23:08:58Z

Aallaire3: /* Examples */

Lenses

2015-11-29T23:07:36Z

Aallaire3: /* Middling */

Lenses

2015-11-29T23:02:26Z

Aallaire3: /* Determine the Index of Refraction from Refraction Data */

Lenses

2015-11-29T23:01:07Z

Aallaire3: /* Thin Lens Equation and Magnification */

Lenses

2015-11-29T22:57:28Z

Aallaire3: /* Determine the Index of Refraction from Refraction Data */

Lenses

2015-11-29T22:54:56Z

Aallaire3: /* Determine the Index of Refraction from Refraction Data */

File:Figure1.PNG

2015-11-29T22:53:51Z

Aallaire3: Aallaire3 uploaded a new version of "File:Figure1.PNG"

Lenses

2015-11-29T22:53:31Z

Aallaire3: /* Determine the Index of Refraction from Refraction Data */

File:Figure1.PNG

2015-11-29T22:45:11Z

Aallaire3:

Lenses

2015-11-29T22:44:35Z

Aallaire3: /* Examples */

Lenses

2015-11-29T22:37:19Z

Aallaire3: /* Law of Refraction */

Lenses

2015-11-29T22:36:56Z

Aallaire3: /* Law of Refraction */

Lenses

2015-11-29T22:29:35Z

Aallaire3: /* The Main Idea */

Lenses

2015-11-29T22:28:04Z

Aallaire3: /* References */

Lenses are found in a huge array of optical instruments, ranging from a simple magnifying glass to the eye to a camera’s zoom
lens. Law of refraction is used to explore the properties of lenses and how they form images.

==The Main Idea==
Index of refraction depends on the wavelength. Thus, light of different wavelengths is bent, or deflected, by different amounts as it passes through a lens. The shape of a lens, either concave or convex, also plays a role in the deflection pattern of light. [[File:Convex.PNG|200px|thumb|center|Convex]] [[File:Concave.PNG|200px|thumb|center|Concave]] The images above show that how these two shapes determines the behavior of the light rays. A lens where the middle is thicker than the two ends is called a "convex" lens, through which incoming light rays converge towards the center axis of the lens. A lens where the middle is thinner than the two ends is called a "concave" lens the prisms represent a "diverging" lens, through which incoming light rays diverge away from the center axis. The angle at which light rays converge or diverge is called the deflection angle. Deflection angles for thin lenses will be modeled mathematically in the following section. Thin lenses are lenses where the y position of a light ray does not change very much as the light ray travels through it. In other words, the lens is thick enough to refract light rays, but does not allow dispersion or aberrations.

===A Mathematical Model===

==== Law of Refraction ====
:The law of reflection says that the reflected ray lies in the plane of incidence, and the angle of reflection equals the angle of incidence.

:The law of refraction says that the refracted ray lies in the plane of incidence, and the sine of the angle of refraction divided by the sine of the angle of incidence is a constant.

::<math>\frac {\sin {\theta_1}}{\sin {\theta_2}} = n</math>

:where <math>n</math> is a constant for any two materials and a given color of light. It is known as the refractive index.

==== Thin Lens Equation and Magnification ====
[[File:lens3b.svg|360px|thumb|A ray tracing diagram for a converging lens.]]

Thin lenses produce focal points on either side that can be modeled using the lensmaker's equation. Thin lenses follow a simple equation that determines the location of the images given a particular focal length (<math>f</math>) and object distance (<math>S_1</math>):

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

===A Computational Model===
[http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/23-lens Lens Simulation]

==Examples==

Be sure to show all steps in your solution and include diagrams whenever possible

===Simple===
===Middling===
===Difficult===

==Connectedness==
#How is this topic connected to something that you are interested in?
#How is it connected to your major?
#Is there an interesting industrial application?

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

Lenses

2015-11-29T22:25:48Z

Aallaire3: /* References */

Lenses are found in a huge array of optical instruments, ranging from a simple magnifying glass to the eye to a camera’s zoom
lens. Law of refraction is used to explore the properties of lenses and how they form images.

==The Main Idea==
Index of refraction depends on the wavelength. Thus, light of different wavelengths is bent, or deflected, by different amounts as it passes through a lens. The shape of a lens, either concave or convex, also plays a role in the deflection pattern of light. [[File:Convex.PNG|200px|thumb|center|Convex]] [[File:Concave.PNG|200px|thumb|center|Concave]] The images above show that how these two shapes determines the behavior of the light rays. A lens where the middle is thicker than the two ends is called a "convex" lens, through which incoming light rays converge towards the center axis of the lens. A lens where the middle is thinner than the two ends is called a "concave" lens the prisms represent a "diverging" lens, through which incoming light rays diverge away from the center axis. The angle at which light rays converge or diverge is called the deflection angle. Deflection angles for thin lenses will be modeled mathematically in the following section. Thin lenses are lenses where the y position of a light ray does not change very much as the light ray travels through it. In other words, the lens is thick enough to refract light rays, but does not allow dispersion or aberrations.

===A Mathematical Model===

==== Law of Refraction ====
:The law of reflection says that the reflected ray lies in the plane of incidence, and the angle of reflection equals the angle of incidence.

:The law of refraction says that the refracted ray lies in the plane of incidence, and the sine of the angle of refraction divided by the sine of the angle of incidence is a constant.

::<math>\frac {\sin {\theta_1}}{\sin {\theta_2}} = n</math>

:where <math>n</math> is a constant for any two materials and a given color of light. It is known as the refractive index.

==== Thin Lens Equation and Magnification ====
[[File:lens3b.svg|360px|thumb|A ray tracing diagram for a converging lens.]]

Thin lenses produce focal points on either side that can be modeled using the lensmaker's equation. Thin lenses follow a simple equation that determines the location of the images given a particular focal length (<math>f</math>) and object distance (<math>S_1</math>):

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

===A Computational Model===
[http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/23-lens Lens Simulation]

==Examples==

Be sure to show all steps in your solution and include diagrams whenever possible

===Simple===
===Middling===
===Difficult===

==Connectedness==
#How is this topic connected to something that you are interested in?
#How is it connected to your major?
#Is there an interesting industrial application?

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

This section contains the the references you used while writing this page

[[Category:Radiation]]

Lenses

2015-11-29T22:25:04Z

Aallaire3: /* Thin Lens Equation and Magnification */

Lenses are found in a huge array of optical instruments, ranging from a simple magnifying glass to the eye to a camera’s zoom
lens. Law of refraction is used to explore the properties of lenses and how they form images.

==The Main Idea==
Index of refraction depends on the wavelength. Thus, light of different wavelengths is bent, or deflected, by different amounts as it passes through a lens. The shape of a lens, either concave or convex, also plays a role in the deflection pattern of light. [[File:Convex.PNG|200px|thumb|center|Convex]] [[File:Concave.PNG|200px|thumb|center|Concave]] The images above show that how these two shapes determines the behavior of the light rays. A lens where the middle is thicker than the two ends is called a "convex" lens, through which incoming light rays converge towards the center axis of the lens. A lens where the middle is thinner than the two ends is called a "concave" lens the prisms represent a "diverging" lens, through which incoming light rays diverge away from the center axis. The angle at which light rays converge or diverge is called the deflection angle. Deflection angles for thin lenses will be modeled mathematically in the following section. Thin lenses are lenses where the y position of a light ray does not change very much as the light ray travels through it. In other words, the lens is thick enough to refract light rays, but does not allow dispersion or aberrations.

===A Mathematical Model===

==== Law of Refraction ====
:The law of reflection says that the reflected ray lies in the plane of incidence, and the angle of reflection equals the angle of incidence.

:The law of refraction says that the refracted ray lies in the plane of incidence, and the sine of the angle of refraction divided by the sine of the angle of incidence is a constant.

::<math>\frac {\sin {\theta_1}}{\sin {\theta_2}} = n</math>

:where <math>n</math> is a constant for any two materials and a given color of light. It is known as the refractive index.

==== Thin Lens Equation and Magnification ====
[[File:lens3b.svg|360px|thumb|A ray tracing diagram for a converging lens.]]

Thin lenses produce focal points on either side that can be modeled using the lensmaker's equation. Thin lenses follow a simple equation that determines the location of the images given a particular focal length (<math>f</math>) and object distance (<math>S_1</math>):

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

===A Computational Model===
[http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/23-lens Lens Simulation]

==Examples==

Be sure to show all steps in your solution and include diagrams whenever possible

===Simple===
===Middling===
===Difficult===

==Connectedness==
#How is this topic connected to something that you are interested in?
#How is it connected to your major?
#Is there an interesting industrial application?

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]

Lenses

2015-11-29T22:24:47Z

Aallaire3: /* Law of Refraction */

Lenses are found in a huge array of optical instruments, ranging from a simple magnifying glass to the eye to a camera’s zoom
lens. Law of refraction is used to explore the properties of lenses and how they form images.

==The Main Idea==
Index of refraction depends on the wavelength. Thus, light of different wavelengths is bent, or deflected, by different amounts as it passes through a lens. The shape of a lens, either concave or convex, also plays a role in the deflection pattern of light. [[File:Convex.PNG|200px|thumb|center|Convex]] [[File:Concave.PNG|200px|thumb|center|Concave]] The images above show that how these two shapes determines the behavior of the light rays. A lens where the middle is thicker than the two ends is called a "convex" lens, through which incoming light rays converge towards the center axis of the lens. A lens where the middle is thinner than the two ends is called a "concave" lens the prisms represent a "diverging" lens, through which incoming light rays diverge away from the center axis. The angle at which light rays converge or diverge is called the deflection angle. Deflection angles for thin lenses will be modeled mathematically in the following section. Thin lenses are lenses where the y position of a light ray does not change very much as the light ray travels through it. In other words, the lens is thick enough to refract light rays, but does not allow dispersion or aberrations.

===A Mathematical Model===

==== Law of Refraction ====
:The law of reflection says that the reflected ray lies in the plane of incidence, and the angle of reflection equals the angle of incidence.

:The law of refraction says that the refracted ray lies in the plane of incidence, and the sine of the angle of refraction divided by the sine of the angle of incidence is a constant.

::<math>\frac {\sin {\theta_1}}{\sin {\theta_2}} = n</math>

:where <math>n</math> is a constant for any two materials and a given color of light. It is known as the refractive index.

== Thin Lens Equation and Magnification ==
[[File:lens3b.svg|360px|thumb|A ray tracing diagram for a converging lens.]]

Thin lenses produce focal points on either side that can be modeled using the lensmaker's equation. Thin lenses follow a simple equation that determines the location of the images given a particular focal length (<math>f</math>) and object distance (<math>S_1</math>):

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

===A Computational Model===
[http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/23-lens Lens Simulation]

==Examples==

Be sure to show all steps in your solution and include diagrams whenever possible

===Simple===
===Middling===
===Difficult===

==Connectedness==
#How is this topic connected to something that you are interested in?
#How is it connected to your major?
#Is there an interesting industrial application?

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]

Lenses

2015-11-29T22:23:04Z

Aallaire3: /* Thin Lens Equation and Magnification */

Lenses are found in a huge array of optical instruments, ranging from a simple magnifying glass to the eye to a camera’s zoom
lens. Law of refraction is used to explore the properties of lenses and how they form images.

==The Main Idea==
Index of refraction depends on the wavelength. Thus, light of different wavelengths is bent, or deflected, by different amounts as it passes through a lens. The shape of a lens, either concave or convex, also plays a role in the deflection pattern of light. [[File:Convex.PNG|200px|thumb|center|Convex]] [[File:Concave.PNG|200px|thumb|center|Concave]] The images above show that how these two shapes determines the behavior of the light rays. A lens where the middle is thicker than the two ends is called a "convex" lens, through which incoming light rays converge towards the center axis of the lens. A lens where the middle is thinner than the two ends is called a "concave" lens the prisms represent a "diverging" lens, through which incoming light rays diverge away from the center axis. The angle at which light rays converge or diverge is called the deflection angle. Deflection angles for thin lenses will be modeled mathematically in the following section. Thin lenses are lenses where the y position of a light ray does not change very much as the light ray travels through it. In other words, the lens is thick enough to refract light rays, but does not allow dispersion or aberrations.

===A Mathematical Model===

== Law of Refraction ==
:The law of reflection says that the reflected ray lies in the plane of incidence, and the angle of reflection equals the angle of incidence.

:The law of refraction says that the refracted ray lies in the plane of incidence, and the sine of the angle of refraction divided by the sine of the angle of incidence is a constant.

::<math>\frac {\sin {\theta_1}}{\sin {\theta_2}} = n</math>

:where <math>n</math> is a constant for any two materials and a given color of light. It is known as the refractive index.

== Thin Lens Equation and Magnification ==
[[File:lens3b.svg|360px|thumb|A ray tracing diagram for a converging lens.]]

Thin lenses produce focal points on either side that can be modeled using the lensmaker's equation. Thin lenses follow a simple equation that determines the location of the images given a particular focal length (<math>f</math>) and object distance (<math>S_1</math>):

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

The magnification of a lens is given by

:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math>

===A Computational Model===
[http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/23-lens Lens Simulation]

==Examples==

Be sure to show all steps in your solution and include diagrams whenever possible

===Simple===
===Middling===
===Difficult===

==Connectedness==
#How is this topic connected to something that you are interested in?
#How is it connected to your major?
#Is there an interesting industrial application?

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]

Lenses

2015-11-29T22:21:21Z

Aallaire3: /* Thin Lens Equation and Magnification */

Lenses are found in a huge array of optical instruments, ranging from a simple magnifying glass to the eye to a camera’s zoom
lens. Law of refraction is used to explore the properties of lenses and how they form images.

==The Main Idea==
Index of refraction depends on the wavelength. Thus, light of different wavelengths is bent, or deflected, by different amounts as it passes through a lens. The shape of a lens, either concave or convex, also plays a role in the deflection pattern of light. [[File:Convex.PNG|200px|thumb|center|Convex]] [[File:Concave.PNG|200px|thumb|center|Concave]] The images above show that how these two shapes determines the behavior of the light rays. A lens where the middle is thicker than the two ends is called a "convex" lens, through which incoming light rays converge towards the center axis of the lens. A lens where the middle is thinner than the two ends is called a "concave" lens the prisms represent a "diverging" lens, through which incoming light rays diverge away from the center axis. The angle at which light rays converge or diverge is called the deflection angle. Deflection angles for thin lenses will be modeled mathematically in the following section. Thin lenses are lenses where the y position of a light ray does not change very much as the light ray travels through it. In other words, the lens is thick enough to refract light rays, but does not allow dispersion or aberrations.

===A Mathematical Model===

== Law of Refraction ==
:The law of reflection says that the reflected ray lies in the plane of incidence, and the angle of reflection equals the angle of incidence.

:The law of refraction says that the refracted ray lies in the plane of incidence, and the sine of the angle of refraction divided by the sine of the angle of incidence is a constant.

::<math>\frac {\sin {\theta_1}}{\sin {\theta_2}} = n</math>

:where <math>n</math> is a constant for any two materials and a given color of light. It is known as the refractive index.

== Thin Lens Equation and Magnification ==
[[File:lens3b.svg|360px|thumb|A ray tracing diagram for a converging lens.]]

Thin lenses produce focal points on either side that can be modeled using the lensmaker's equation. Thin lenses follow a simple equation that determines the location of the images given a particular focal length (<math>f</math>) and object distance (<math>S_1</math>):

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

where <math>S_2</math> is the distance associated with the image and is considered by convention to be negative if on the same side of the lens as the object and positive if on the opposite side of the lens. The focal length f is considered negative for concave lenses.

===A Computational Model===
[http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/23-lens Lens Simulation]

==Examples==

Be sure to show all steps in your solution and include diagrams whenever possible

===Simple===
===Middling===
===Difficult===

==Connectedness==
#How is this topic connected to something that you are interested in?
#How is it connected to your major?
#Is there an interesting industrial application?

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]

Lenses

2015-11-29T22:20:20Z

Aallaire3: /* Law of Refraction */

Lenses are found in a huge array of optical instruments, ranging from a simple magnifying glass to the eye to a camera’s zoom
lens. Law of refraction is used to explore the properties of lenses and how they form images.

==The Main Idea==
Index of refraction depends on the wavelength. Thus, light of different wavelengths is bent, or deflected, by different amounts as it passes through a lens. The shape of a lens, either concave or convex, also plays a role in the deflection pattern of light. [[File:Convex.PNG|200px|thumb|center|Convex]] [[File:Concave.PNG|200px|thumb|center|Concave]] The images above show that how these two shapes determines the behavior of the light rays. A lens where the middle is thicker than the two ends is called a "convex" lens, through which incoming light rays converge towards the center axis of the lens. A lens where the middle is thinner than the two ends is called a "concave" lens the prisms represent a "diverging" lens, through which incoming light rays diverge away from the center axis. The angle at which light rays converge or diverge is called the deflection angle. Deflection angles for thin lenses will be modeled mathematically in the following section. Thin lenses are lenses where the y position of a light ray does not change very much as the light ray travels through it. In other words, the lens is thick enough to refract light rays, but does not allow dispersion or aberrations.

===A Mathematical Model===

== Law of Refraction ==
:The law of reflection says that the reflected ray lies in the plane of incidence, and the angle of reflection equals the angle of incidence.

:The law of refraction says that the refracted ray lies in the plane of incidence, and the sine of the angle of refraction divided by the sine of the angle of incidence is a constant.

::<math>\frac {\sin {\theta_1}}{\sin {\theta_2}} = n</math>

:where <math>n</math> is a constant for any two materials and a given color of light. It is known as the refractive index.

== Thin Lens Equation and Magnification ==
[[File:lens3b.svg|360px|thumb|A ray tracing diagram for a converging lens.]]

Thin lenses produce focal points on either side that can be modeled using the lensmaker's equation.

:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math>

where <math>S_2</math> is the distance associated with the image and is considered by convention to be negative if on the same side of the lens as the object and positive if on the opposite side of the lens. The focal length f is considered negative for concave lenses.

===A Computational Model===
[http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/23-lens Lens Simulation]

==Examples==

Be sure to show all steps in your solution and include diagrams whenever possible

===Simple===
===Middling===
===Difficult===

==Connectedness==
#How is this topic connected to something that you are interested in?
#How is it connected to your major?
#Is there an interesting industrial application?

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]

File:Convex.PNG

2015-11-29T22:14:40Z

Aallaire3: Aallaire3 uploaded a new version of "File:Convex.PNG"

File:Concave.PNG

2015-11-29T22:14:09Z

Aallaire3: Aallaire3 uploaded a new version of "File:Concave.PNG"

File:Convex.PNG

2015-11-29T22:12:43Z

Aallaire3: Aallaire3 uploaded a new version of "File:Convex.PNG"

File:Concave.PNG

2015-11-29T22:12:15Z

Aallaire3: Aallaire3 uploaded a new version of "File:Concave.PNG"

Lenses

2015-11-29T22:10:02Z

Aallaire3: /* A Mathematical Model */

Lenses are found in a huge array of optical instruments, ranging from a simple magnifying glass to the eye to a camera’s zoom
lens. Law of refraction is used to explore the properties of lenses and how they form images.

==The Main Idea==
Index of refraction depends on the wavelength. Thus, light of different wavelengths is bent, or deflected, by different amounts as it passes through a lens. The shape of a lens, either concave or convex, also plays a role in the deflection pattern of light. [[File:Convex.PNG|200px|thumb|center|Convex]] [[File:Concave.PNG|200px|thumb|center|Concave]] The images above show that how these two shapes determines the behavior of the light rays. A lens where the middle is thicker than the two ends is called a "convex" lens, through which incoming light rays converge towards the center axis of the lens. A lens where the middle is thinner than the two ends is called a "concave" lens the prisms represent a "diverging" lens, through which incoming light rays diverge away from the center axis. The angle at which light rays converge or diverge is called the deflection angle. Deflection angles for thin lenses will be modeled mathematically in the following section. Thin lenses are lenses where the y position of a light ray does not change very much as the light ray travels through it. In other words, the lens is thick enough to refract light rays, but does not allow dispersion or aberrations.

===A Mathematical Model===

== Law of Refraction ==
:The law of reflection says that the reflected ray lies in the plane of incidence, and the angle of reflection equals the angle of incidence.

:The law of refraction says that the refracted ray lies in the plane of incidence, and the sine of the angle of refraction divided by the sine of the angle of incidence is a constant.

::<math>\frac {\sin {\theta_1}}{\sin {\theta_2}} = n</math>

:where <math>n</math> is a constant for any two materials and a given color of light. It is known as the refractive index.

===A Computational Model===
[http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/23-lens Lens Simulation]

==Examples==

Be sure to show all steps in your solution and include diagrams whenever possible

===Simple===
===Middling===
===Difficult===

==Connectedness==
#How is this topic connected to something that you are interested in?
#How is it connected to your major?
#Is there an interesting industrial application?

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]

Lenses

2015-11-29T21:54:44Z

Aallaire3:

Lenses are found in a huge array of optical instruments, ranging from a simple magnifying glass to the eye to a camera’s zoom
lens. Law of refraction is used to explore the properties of lenses and how they form images.

==The Main Idea==
Index of refraction depends on the wavelength. Thus, light of different wavelengths is bent, or deflected, by different amounts as it passes through a lens. The shape of a lens, either concave or convex, also plays a role in the deflection pattern of light. [[File:Convex.PNG|200px|thumb|center|Convex]] [[File:Concave.PNG|200px|thumb|center|Concave]] The images above show that how these two shapes determines the behavior of the light rays. A lens where the middle is thicker than the two ends is called a "convex" lens, through which incoming light rays converge towards the center axis of the lens. A lens where the middle is thinner than the two ends is called a "concave" lens the prisms represent a "diverging" lens, through which incoming light rays diverge away from the center axis. The angle at which light rays converge or diverge is called the deflection angle. Deflection angles for thin lenses will be modeled mathematically in the following section. Thin lenses are lenses where the y position of a light ray does not change very much as the light ray travels through it. In other words, the lens is thick enough to refract light rays, but does not allow dispersion or aberrations.

===A Mathematical Model===

What are the mathematical equations that allow us to model this topic. For example <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings.

===A Computational Model===
[http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/23-lens Lens Simulation]

==Examples==

Be sure to show all steps in your solution and include diagrams whenever possible

===Simple===
===Middling===
===Difficult===

==Connectedness==
#How is this topic connected to something that you are interested in?
#How is it connected to your major?
#Is there an interesting industrial application?

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]

File:Concave.PNG

2015-11-29T21:53:14Z

Aallaire3:

File:Convex.PNG

2015-11-29T21:53:00Z

Aallaire3: