Physics Book - User contributions [en]

Lenz's Law

2019-04-14T18:07:49Z

Acasado3:

"Edited by Alex Casado (Spring 2019)"

==The Main Idea==

Lenz's Law is applies the law of conservation of energy to Faraday's Law, which states that any change in the magnetic field will cause an induced current. Lenz's law accounts for the direction of the new current - with the change coming from either the strength of the magnetic field, the direction of the magnetic field, the position of a circuit, the shape of a circuit, or the orientation of a circuit. To keep the magnetic flux of a loop of wires constant, there is an induced magnetic field. Change in magnetic flux results in an equal and opposite change in the loop which is why we use negative <math>{\frac{dB}{dt}}</math>. Below is a basic visualization of Lenz's Law where there is an existing magnetic field.

[[File:lenzlaw1.png]]
(image obtained from https://brilliant.org/wiki/lenzs-law/)

This Law is applied to determine the direction of current and the field it creates. When an induced field is generated by a change in magnetic flux (Faraday's Law), the induced current will flow creating its own magnetic field that opposes the magnetic field that created it. Direction is important when dealing with the law of conservation of energy. It can be seen applied when looking at electromagnetism such as the direction of voltage induced in an inductor.

In summary, whenever there is a change in magnetic flux in a conducting loop, the charge is balanced by the induction of a magnetic field. The integral of B * n remains constant.

==Theory==

===Mathematical Model===

Lenz's Law is mathematically modeled as part of Faraday's Law. The negative sign in the equation represents the opposing induced field.
<math>\epsilon = -{\frac{d\phi}{dt}} </math> where '''<math>\epsilon</math>''' is the emf of the system and '''<math>d\phi</math>''' is the change in the magnetic field. This means the induced electromotive force and rate of change in magnetic flux both have opposite signs. While there is a formula, it is important to note that this is mainly a qualitative law and does not incorporate magnitude, only direction.

Example:
The currents seen within strong magnets can create currents rotating in the opposite direction in metals like cooper or aluminum. If you were to drop a strong magnet through a cooper or aluminum pipe, you would observe that the velocity is decently slower than if you were to drop the magnet outside of the pipe.

=== Right Hand Rule ===

The Right Hand Rule is a heuristic used to determine the direction of angular momentum as a vector. The direction of angular momentum is determined by calculating the cross product of the vectors. To use the Right Hand Rule, place your thumb in the direction of <math> -dB </math> and curl your fingers. The direction of the non-coulombic electric field is the same direction as your fingers curl. An example of how the cross product is performed is below.

[[File:Crossproductcasado.png]]

==Examples==
'''Right Hand Rule'''

In the examples above, the direction of the electric field is the pink arrows. This can be determined by first placing your thumb in the direction of the <math> -dB </math> and then curling your fingers. The direction your fingers curl is the direction of the electric field.

===Example 1: Easy===

A solenoid has current <math> I_1 </math> flowing around it increasing from 0 to 40A. A plain loop of wire is placed around the solenoid, perpendicular to the axis of the solenoid. An emf is produced, therefore producing an <math> I_2 </math>. Are <math> I_1 </math> and <math> I_2 </math> flowing in the same direction or opposite?

'''Solution'''

The original current <math> I_1 </math> produced a magnetic field. In order to maintain the conservation of energy and Newton's Third Law, the magnetic field produced by <math> I_2 </math> must oppose this field. This is in accordance with Lenz's Law. Therefore, <math> I_2 </math> must oppose <math> I_1 </math> in direction.

===Example 2: Medium===

An magnet is moving through a copper tube (velocity drawn). Find the direction of -dB/dt and the direction of the induced current. Remember to use the right hand rule.

[[File:LenzLaw.jpg]]
Graphic created by Nicole Romer

'''Solution'''

A)-y, clockwise B)+y, counterclockwise C)+y, counterclockwise D)-y, clockwise

The magnetic field in this graphic is decreasing at a rate of 5.0mT/s. What is the direction of the current in the circle of wire?

[[File:hm235.jpg]]

'''Solution'''
The current in the circle of wire will produce a magnetic field that needs to supplement the existing diminishing field. This is in accordance with Lenz's Law. Therefore, the magnetic field produced needs to be into the page. Using the right hand rule, to produce a field going into the page the current in the circle of the wire must be in the clockwise direction.

===Example 3: Difficult===
The magnetic field is decreasing at a rate of 5.0mT/s. The radius of the loop of wire is 5.0m, and the resistance is 5 ohms. What is the magnitude and direction of the current?

[[File:hm235.jpg]]

'''Solution'''

To find the magnitude of the current we must first use the formula <math>\epsilon = -{\frac{d\phi}{dt}} </math> to find the <math>\epsilon</math> representing the emf of the system. We know, more specifically, that <math>\epsilon = -NAcos\theta{\frac{d\phi}{dt}} </math>. Therefore, <math>\epsilon = -1*5.0^2*\pi*1*-5*10^{-3} </math> which resolves to <math>\epsilon = .392699</math>. From there, to find the current we know that <math>I = {\frac{\epsilon}{R}}</math>. Plugging in the values we know we find, <math>I = {\frac{.392699}{5.0}} = .07854A</math>. That is the magnitude of the current. To find the direction we must use Lenz's Law. The current in the circle of wire will produce a magnetic field that needs to supplement the existing diminishing field. Therefore, the magnetic field produced needs to be into the page. Using the right hand rule, to produce a field going into the page the current in the circle of the wire must be in the clockwise direction.

==Connectedness==

An application of Lenz's Law is to cause rotation to create energy. In industry, Lenz's Law can be applied to electric generators or motors. When a current is induced in a generator, the direction of the induced current will flow in opposition of the magnetic field that created it. This causes rotation in the generator.

[[File:Electricmotor1.jpeg]]

Another application of Lenz's Law is in electromagnetic braking for vehicles. This process begins with electromagnets inducing eddy currents into the spinning rotor. Magnetic fields that oppose the initial change in magnetic flux are created from these eddy currents. This slows the rotor.

[[File:Electromagbraking.png]]

One last application of Lenz's Law is induction stove tops. These cooktops heat up as a result of changing magnetic fields and eddy currents operating according to Lenz's Law.

[[File:inductionstove.jpeg]]

==History==

Henrich Friefrich Emil Lenz (1804-1865), a Russian physicist of German origin was born in Dorpat, nowadays Tartu, Estonia. Henrich studied chemistry and physics at the University of Dorpat in 1820 after his secondary education. From 1823 to 1826, he traveled with the navigator, Otto von Kotzebue on his third expedition around the world. During this journey he studied climate conditions, and properties of seawater. After his travels, he worked at the University of St. Petersburg, Russia, where he later became the Dean of Mathematics and Physics from 1840 to 1863. In the year of 1831, he started studying electromagnetism, and soon after in 1835, what is known today as Lenz's Law was created. Lenz was also know for carefully checking his work, testing any variable that might effect his results. Lenz died on February 10, 1865, just two days before his 61st birthday, after suffering a stroke, while in Rome.

===Further reading===

Faraday's Law
[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html]
Conservation Laws
[http://hyperphysics.phy-astr.gsu.edu/hbase/conser.html]

===More Videos For Study===

https://www.youtube.com/watch?v=xxZenoBs2Pg This links to a video by Khan Academy that explains how Lenz's Law works through an example.

https://www.youtube.com/watch?v=Vs3afgStVy4 This links to a video by Grand Illusions that displays Lenz's Law using a magnetic falling down a tube.

==References==

https://nationalmaglab.org/education/magnet-academy/history-of-electricity-magnetism/pioneers/heinrich-friedrich-emil-lenz
http://hyperphysics.phy-astr.gsu.edu/hbase/conser.html
http://regentsprep.org/regents/physics/phys08/clenslaw/
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html
http://farside.ph.utexas.edu/teaching/302l/lectures/node85.html
http://www.electrical4u.com/lenz-law-of-electromagnetic-induction/

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

File:Crossproductcasado.png

2019-04-14T18:06:11Z

Acasado3:

Lenz's Law

2019-04-14T18:04:01Z

Acasado3:

"Edited by Alex Casado (Spring 2019)"

==The Main Idea==

Lenz's Law is applies the law of conservation of energy to Faraday's Law, which states that any change in the magnetic field will cause an induced current. Lenz's law accounts for the direction of the new current - with the change coming from either the strength of the magnetic field, the direction of the magnetic field, the position of a circuit, the shape of a circuit, or the orientation of a circuit. To keep the magnetic flux of a loop of wires constant, there is an induced magnetic field. Change in magnetic flux results in an equal and opposite change in the loop which is why we use negative <math>{\frac{dB}{dt}}</math>. Below is a basic visualization of Lenz's Law where there is an existing magnetic field.

[[File:lenzlaw1.png]]
(image obtained from https://brilliant.org/wiki/lenzs-law/)

This Law is applied to determine the direction of current and the field it creates. When an induced field is generated by a change in magnetic flux (Faraday's Law), the induced current will flow creating its own magnetic field that opposes the magnetic field that created it. Direction is important when dealing with the law of conservation of energy. It can be seen applied when looking at electromagnetism such as the direction of voltage induced in an inductor.

In summary, whenever there is a change in magnetic flux in a conducting loop, the charge is balanced by the induction of a magnetic field. The integral of B * n remains constant.

==Theory==

===Mathematical Model===

Lenz's Law is mathematically modeled as part of Faraday's Law. The negative sign in the equation represents the opposing induced field.
<math>\epsilon = -{\frac{d\phi}{dt}} </math> where '''<math>\epsilon</math>''' is the emf of the system and '''<math>d\phi</math>''' is the change in the magnetic field. This means the induced electromotive force and rate of change in magnetic flux both have opposite signs. While there is a formula, it is important to note that this is mainly a qualitative law and does not incorporate magnitude, only direction.

Example:
The currents seen within strong magnets can create currents rotating in the opposite direction in metals like cooper or aluminum. If you were to drop a strong magnet through a cooper or aluminum pipe, you would observe that the velocity is decently slower than if you were to drop the magnet outside of the pipe.

=== Right Hand Rule ===

The Right Hand Rule is a heuristic used to determine the direction of angular momentum as a vector. The direction of angular momentum is determined by calculating the cross product of the vectors. To use the Right Hand Rule, place your thumb in the direction of <math> -dB </math> and curl your fingers. The direction of the non-coulombic electric field is the same direction as your fingers curl. An example of how the cross product is performed is below.

[[File:crossproduct.png]]

==Examples==
'''Right Hand Rule'''

In the examples above, the direction of the electric field is the pink arrows. This can be determined by first placing your thumb in the direction of the <math> -dB </math> and then curling your fingers. The direction your fingers curl is the direction of the electric field.

===Example 1: Easy===

A solenoid has current <math> I_1 </math> flowing around it increasing from 0 to 40A. A plain loop of wire is placed around the solenoid, perpendicular to the axis of the solenoid. An emf is produced, therefore producing an <math> I_2 </math>. Are <math> I_1 </math> and <math> I_2 </math> flowing in the same direction or opposite?

'''Solution'''

The original current <math> I_1 </math> produced a magnetic field. In order to maintain the conservation of energy and Newton's Third Law, the magnetic field produced by <math> I_2 </math> must oppose this field. This is in accordance with Lenz's Law. Therefore, <math> I_2 </math> must oppose <math> I_1 </math> in direction.

===Example 2: Medium===

An magnet is moving through a copper tube (velocity drawn). Find the direction of -dB/dt and the direction of the induced current. Remember to use the right hand rule.

[[File:LenzLaw.jpg]]
Graphic created by Nicole Romer

'''Solution'''

A)-y, clockwise B)+y, counterclockwise C)+y, counterclockwise D)-y, clockwise

The magnetic field in this graphic is decreasing at a rate of 5.0mT/s. What is the direction of the current in the circle of wire?

[[File:hm235.jpg]]

'''Solution'''
The current in the circle of wire will produce a magnetic field that needs to supplement the existing diminishing field. This is in accordance with Lenz's Law. Therefore, the magnetic field produced needs to be into the page. Using the right hand rule, to produce a field going into the page the current in the circle of the wire must be in the clockwise direction.

===Example 3: Difficult===
The magnetic field is decreasing at a rate of 5.0mT/s. The radius of the loop of wire is 5.0m, and the resistance is 5 ohms. What is the magnitude and direction of the current?

[[File:hm235.jpg]]

'''Solution'''

To find the magnitude of the current we must first use the formula <math>\epsilon = -{\frac{d\phi}{dt}} </math> to find the <math>\epsilon</math> representing the emf of the system. We know, more specifically, that <math>\epsilon = -NAcos\theta{\frac{d\phi}{dt}} </math>. Therefore, <math>\epsilon = -1*5.0^2*\pi*1*-5*10^{-3} </math> which resolves to <math>\epsilon = .392699</math>. From there, to find the current we know that <math>I = {\frac{\epsilon}{R}}</math>. Plugging in the values we know we find, <math>I = {\frac{.392699}{5.0}} = .07854A</math>. That is the magnitude of the current. To find the direction we must use Lenz's Law. The current in the circle of wire will produce a magnetic field that needs to supplement the existing diminishing field. Therefore, the magnetic field produced needs to be into the page. Using the right hand rule, to produce a field going into the page the current in the circle of the wire must be in the clockwise direction.

==Connectedness==

An application of Lenz's Law is to cause rotation to create energy. In industry, Lenz's Law can be applied to electric generators or motors. When a current is induced in a generator, the direction of the induced current will flow in opposition of the magnetic field that created it. This causes rotation in the generator.

[[File:Electricmotor1.jpeg]]

Another application of Lenz's Law is in electromagnetic braking for vehicles. This process begins with electromagnets inducing eddy currents into the spinning rotor. Magnetic fields that oppose the initial change in magnetic flux are created from these eddy currents. This slows the rotor.

[[File:Electromagbraking.png]]

One last application of Lenz's Law is induction stove tops. These cooktops heat up as a result of changing magnetic fields and eddy currents operating according to Lenz's Law.

[[File:inductionstove.jpeg]]

==History==

Henrich Friefrich Emil Lenz (1804-1865), a Russian physicist of German origin was born in Dorpat, nowadays Tartu, Estonia. Henrich studied chemistry and physics at the University of Dorpat in 1820 after his secondary education. From 1823 to 1826, he traveled with the navigator, Otto von Kotzebue on his third expedition around the world. During this journey he studied climate conditions, and properties of seawater. After his travels, he worked at the University of St. Petersburg, Russia, where he later became the Dean of Mathematics and Physics from 1840 to 1863. In the year of 1831, he started studying electromagnetism, and soon after in 1835, what is known today as Lenz's Law was created. Lenz was also know for carefully checking his work, testing any variable that might effect his results. Lenz died on February 10, 1865, just two days before his 61st birthday, after suffering a stroke, while in Rome.

===Further reading===

Faraday's Law
[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html]
Conservation Laws
[http://hyperphysics.phy-astr.gsu.edu/hbase/conser.html]

===More Videos For Study===

https://www.youtube.com/watch?v=xxZenoBs2Pg This links to a video by Khan Academy that explains how Lenz's Law works through an example.

https://www.youtube.com/watch?v=Vs3afgStVy4 This links to a video by Grand Illusions that displays Lenz's Law using a magnetic falling down a tube.

==References==

https://nationalmaglab.org/education/magnet-academy/history-of-electricity-magnetism/pioneers/heinrich-friedrich-emil-lenz
http://hyperphysics.phy-astr.gsu.edu/hbase/conser.html
http://regentsprep.org/regents/physics/phys08/clenslaw/
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html
http://farside.ph.utexas.edu/teaching/302l/lectures/node85.html
http://www.electrical4u.com/lenz-law-of-electromagnetic-induction/

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

File:Inductionstove.jpeg

2019-04-14T18:03:27Z

Acasado3:

File:Electromagbraking.png

2019-04-14T18:01:41Z

Acasado3:

File:Electricmotor1.jpeg

2019-04-14T17:58:24Z

Acasado3:

Lenz's Law

2019-04-13T22:34:10Z

Acasado3:

"Edited by Alex Casado (Spring 2019)"

==The Main Idea==

Lenz's Law is applies the law of conservation of energy to Faraday's Law, which states that any change in the magnetic field will cause an induced current. Lenz's law accounts for the direction of the new current - with the change coming from either the strength of the magnetic field, the direction of the magnetic field, the position of a circuit, the shape of a circuit, or the orientation of a circuit. To keep the magnetic flux of a loop of wires constant, there is an induced magnetic field. Change in magnetic flux results in an equal and opposite change in the loop which is why we use negative <math>{\frac{dB}{dt}}</math>. Below is a basic visualization of Lenz's Law where there is an existing magnetic field.

[[File:lenzlaw1.png]]
(image obtained from https://brilliant.org/wiki/lenzs-law/)

This Law is applied to determine the direction of current and the field it creates. When an induced field is generated by a change in magnetic flux (Faraday's Law), the induced current will flow creating its own magnetic field that opposes the magnetic field that created it. Direction is important when dealing with the law of conservation of energy. It can be seen applied when looking at electromagnetism such as the direction of voltage induced in an inductor.

In summary, whenever there is a change in magnetic flux in a conducting loop, the charge is balanced by the induction of a magnetic field. The integral of B * n remains constant.

==Theory==

===Mathematical Model===

Lenz's Law is mathematically modeled as part of Faraday's Law. The negative sign in the equation represents the opposing induced field.
<math>\epsilon = -{\frac{d\phi}{dt}} </math> where '''<math>\epsilon</math>''' is the emf of the system and '''<math>d\phi</math>''' is the change in the magnetic field. This means the induced electromotive force and rate of change in magnetic flux both have opposite signs. While there is a formula, it is important to note that this is mainly a qualitative law and does not incorporate magnitude, only direction.

Example:
The currents seen within strong magnets can create currents rotating in the opposite direction in metals like cooper or aluminum. If you were to drop a strong magnet through a cooper or aluminum pipe, you would observe that the velocity is decently slower than if you were to drop the magnet outside of the pipe.

=== Right Hand Rule ===

The Right Hand Rule is a heuristic used to determine the direction of angular momentum as a vector. The direction of angular momentum is determined by calculating the cross product of the vectors. To use the Right Hand Rule, place your thumb in the direction of <math> -dB </math> and curl your fingers. The direction of the non-coulombic electric field is the same direction as your fingers curl. An example of how the cross product is performed is below.

[[File:crossproduct.png]]

==Examples==
'''Right Hand Rule'''

In the examples above, the direction of the electric field is the pink arrows. This can be determined by first placing your thumb in the direction of the <math> -dB </math> and then curling your fingers. The direction your fingers curl is the direction of the electric field.

===Simple===

A solenoid has current <math> I_1 </math> flowing around it increasing from 0 to 40A. A plain loop of wire is placed around the solenoid, perpendicular to the axis of the solenoid. An emf is produced, therefore producing an <math> I_2 </math>. Are <math> I_1 </math> and <math> I_2 </math> flowing in the same direction or opposite?

'''Solution'''
The original current <math> I_1 </math> produced a magnetic field. In order to maintain the conservation of energy and Newton's Third Law, the magnetic field produced by <math> I_2 </math> must oppose this field. This is in accordance with Lenz's Law. Therefore, <math> I_2 </math> must oppose <math> I_1 </math> in direction.

===Middling===

An magnet is moving through a copper tube (velocity drawn). Find the direction of -dB/dt and the direction of the induced current. Remember to use the right hand rule.

[[File:LenzLaw.jpg]]
Graphic created by Nicole Romer

'''Solution'''
A)-y, clockwise B)+y, counterclockwise C)+y, counterclockwise D)-y, clockwise

The magnetic field in this graphic is decreasing at a rate of 5.0mT/s. What is the direction of the current in the circle of wire?

[[File:hm235.jpg]]

'''Solution'''
The current in the circle of wire will produce a magnetic field that needs to supplement the existing diminishing field. This is in accordance with Lenz's Law. Therefore, the magnetic field produced needs to be into the page. Using the right hand rule, to produce a field going into the page the current in the circle of the wire must be in the clockwise direction.

===Difficult===
The magnetic field is decreasing at a rate of 5.0mT/s. The radius of the loop of wire is 5.0m, and the resistance is 5 ohms. What is the magnitude and direction of the current?

[[File:hm235.jpg]]

'''Solution'''

To find the magnitude of the current we must first use the formula <math>\epsilon = -{\frac{d\phi}{dt}} </math> to find the <math>\epsilon</math> representing the emf of the system. We know, more specifically, that <math>\epsilon = -NAcos\theta{\frac{d\phi}{dt}} </math>. Therefore, <math>\epsilon = -1*5.0^2*\pi*1*-5*10^{-3} </math> which resolves to <math>\epsilon = .392699</math>. From there, to find the current we know that <math>I = {\frac{\epsilon}{R}}</math>. Plugging in the values we know we find, <math>I = {\frac{.392699}{5.0}} = .07854A</math>. That is the magnitude of the current. To find the direction we must use Lenz's Law. The current in the circle of wire will produce a magnetic field that needs to supplement the existing diminishing field. Therefore, the magnetic field produced needs to be into the page. Using the right hand rule, to produce a field going into the page the current in the circle of the wire must be in the clockwise direction.

==Connectedness==
This topic has many applications in the real world that are very interesting. For example, an application for Lenz's Law is to cause rotation to create energy. In an industry setting, Lenz's Law can be applied to electric generators or electric motors. When a current is induced in a generator, the direction of the induced current will flow in opposition of the magnetic field that created it, causing rotation of the generator.
[[File:use235.jpg]]

Another real world application of Lenz's Law is in electromagnetic braking in vehicles. This process begins with electromagnets inducing eddy currents into the spinning rotor. Magnetic fields that oppose the initial change in magnetic flux are created from these eddy currents (Lenz's Law). This ultimately slows the rotor.

[[File:engine.jpg]]

Another example of Lenz's Law in the real world is in Induction stovetops. These cooktops heat up as a result of changing magnetic fields and eddy currents operating according to Lenz's Law.

[[File:stovetop.jpg]]

==History==

Henrich Friefrich Emil Lenz (1804-1865), a Russian physicist of German origin was born in Dorpat, nowadays Tartu, Estonia. Henrich studied chemistry and physics at the University of Dorpat in 1820 after his secondary education. From 1823 to 1826, he traveled with the navigator, Otto von Kotzebue on his third expedition around the world. During this journey he studied climate conditions, and properties of seawater. After his travels, he worked at the University of St. Petersburg, Russia, where he later became the Dean of Mathematics and Physics from 1840 to 1863. In the year of 1831, he started studying electromagnetism, and soon after in 1835, what is known today as Lenz's Law was created. Lenz was also know for carefully checking his work, testing any variable that might effect his results. Lenz died on February 10, 1865, just two days before his 61st birthday, after suffering a stroke, while in Rome.

== See also ==

Since Lenz's Law and Farady's Law go hand in hand, Faraday's Law would be great supplemental information to read about. Newton's Third Law would also be a topic to read on for further understanding why Lenz's Law exists.

===Further reading===

Faraday's Law
[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html]
Conservation Laws
[http://hyperphysics.phy-astr.gsu.edu/hbase/conser.html]

===More Videos For Study===

https://www.youtube.com/watch?v=xxZenoBs2Pg This links to a video by Khan Academy that explains how Lenz's Law works through an example.

https://www.youtube.com/watch?v=Vs3afgStVy4 This links to a video by Grand Illusions that displays Lenz's Law using a magnetic falling down a tube.

==References==

https://nationalmaglab.org/education/magnet-academy/history-of-electricity-magnetism/pioneers/heinrich-friedrich-emil-lenz
http://hyperphysics.phy-astr.gsu.edu/hbase/conser.html
http://regentsprep.org/regents/physics/phys08/clenslaw/
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html
http://farside.ph.utexas.edu/teaching/302l/lectures/node85.html
http://www.electrical4u.com/lenz-law-of-electromagnetic-induction/

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

Lenz's Law

2019-04-13T22:28:59Z

Acasado3:

"Edited by Alex Casado (Spring 2019)"

==The Main Idea==

Lenz's Law is applies the law of conservation of energy to Faraday's Law, which states that any change in the magnetic field will cause an induced current. Lenz's law accounts for the direction of the new current - with the change coming from either the strength of the magnetic field, the direction of the magnetic field, the position of a circuit, the shape of a circuit, or the orientation of a circuit. To keep the magnetic flux of a loop of wires constant, there is an induced magnetic field. Change in magnetic flux results in an equal and opposite change in the loop which is why we use negative <math>{\frac{dB}{dt}}</math>. Below is a basic visualization of Lenz's Law where there is an existing magnetic field.

[[File:lenzlaw1.png]]
(image obtained from https://brilliant.org/wiki/lenzs-law/)

This Law is applied to determine the direction of current and the field it creates. When an induced field is generated by a change in magnetic flux (Faraday's Law), the induced current will flow creating its own magnetic field that opposes the magnetic field that created it. Direction is important when dealing with the law of conservation of energy. It can be seen applied when looking at electromagnetism such as the direction of voltage induced in an inductor.

In summary, whenever there is a change in magnetic flux in a conducting loop, the charge is balanced by the induction of a magnetic field. The integral of B * n remains constant.

==Theory==

===Mathematical Model===

Lenz's Law is mathematically modeled as part of Faraday's Law. The negative sign in the equation represents the opposing induced field.
<math>\epsilon = -{\frac{d\phi}{dt}} </math> where '''<math>\epsilon</math>''' is the emf of the system and '''<math>d\phi</math>''' is the change in the magnetic field. This means the induced electromotive force and rate of change in magnetic flux both have opposite signs. While there is a formula, it is important to note that this is mainly a qualitative law and does not incorporate magnitude, only direction.

Example:
The currents seen within strong magnets can create currents rotating in the opposite direction in metals like cooper or aluminum. If you were to drop a strong magnet through a cooper or aluminum pipe, you would observe that the velocity is decently slower than if you were to drop the magnet outside of the pipe.

=== Right Hand Rule ===

The Right Hand Rule is a heuristic used to determine the direction of angular momentum as a vector. The direction of angular momentum is determined by calculating the cross product of the vectors. An example of how the cross product is performed is below.

[[File:crossproduct.png]]

==Examples==
'''Example Using Right Hand Rule'''

Use the right hand rule to find the non-coulombic electric field in the given situations.

[[File:Non-Coulobmic_Fields.gif]]

Place your thumb in the direction of <math> -dB </math> and curl your fingers. The direction of the non-coulombic electric field is the same direction as your fingers curl. In the examples above, the direction of the electric field is the pink arrows.

===Simple===

A solenoid has current <math> I_1 </math> flowing around it increasing from 0 to 40A. A plain loop of wire is placed around the solenoid, perpendicular to the axis of the solenoid. An emf is produced, therefore producing an <math> I_2 </math>. Are <math> I_1 </math> and <math> I_2 </math> flowing in the same direction or opposite?

'''Solution'''
The original current <math> I_1 </math> produced a magnetic field. In order to maintain the conservation of energy and Newton's Third Law, the magnetic field produced by <math> I_2 </math> must oppose this field. This is in accordance with Lenz's Law. Therefore, <math> I_2 </math> must oppose <math> I_1 </math> in direction.

===Middling===

An magnet is moving through a copper tube (velocity drawn). Find the direction of -dB/dt and the direction of the induced current. Remember to use the right hand rule.

[[File:LenzLaw.jpg]]
Graphic created by Nicole Romer

'''Solution'''
A)-y, clockwise B)+y, counterclockwise C)+y, counterclockwise D)-y, clockwise

The magnetic field in this graphic is decreasing at a rate of 5.0mT/s. What is the direction of the current in the circle of wire?

[[File:hm235.jpg]]

'''Solution'''
The current in the circle of wire will produce a magnetic field that needs to supplement the existing diminishing field. This is in accordance with Lenz's Law. Therefore, the magnetic field produced needs to be into the page. Using the right hand rule, to produce a field going into the page the current in the circle of the wire must be in the clockwise direction.

===Difficult===
The magnetic field is decreasing at a rate of 5.0mT/s. The radius of the loop of wire is 5.0m, and the resistance is 5 ohms. What is the magnitude and direction of the current?

[[File:hm235.jpg]]

'''Solution'''

To find the magnitude of the current we must first use the formula <math>\epsilon = -{\frac{d\phi}{dt}} </math> to find the <math>\epsilon</math> representing the emf of the system. We know, more specifically, that <math>\epsilon = -NAcos\theta{\frac{d\phi}{dt}} </math>. Therefore, <math>\epsilon = -1*5.0^2*\pi*1*-5*10^{-3} </math> which resolves to <math>\epsilon = .392699</math>. From there, to find the current we know that <math>I = {\frac{\epsilon}{R}}</math>. Plugging in the values we know we find, <math>I = {\frac{.392699}{5.0}} = .07854A</math>. That is the magnitude of the current. To find the direction we must use Lenz's Law. The current in the circle of wire will produce a magnetic field that needs to supplement the existing diminishing field. Therefore, the magnetic field produced needs to be into the page. Using the right hand rule, to produce a field going into the page the current in the circle of the wire must be in the clockwise direction.

==Connectedness==
This topic has many applications in the real world that are very interesting. For example, an application for Lenz's Law is to cause rotation to create energy. In an industry setting, Lenz's Law can be applied to electric generators or electric motors. When a current is induced in a generator, the direction of the induced current will flow in opposition of the magnetic field that created it, causing rotation of the generator.
[[File:use235.jpg]]

Another real world application of Lenz's Law is in electromagnetic braking in vehicles. This process begins with electromagnets inducing eddy currents into the spinning rotor. Magnetic fields that oppose the initial change in magnetic flux are created from these eddy currents (Lenz's Law). This ultimately slows the rotor.

[[File:engine.jpg]]

Another example of Lenz's Law in the real world is in Induction stovetops. These cooktops heat up as a result of changing magnetic fields and eddy currents operating according to Lenz's Law.

[[File:stovetop.jpg]]

==History==

Henrich Friefrich Emil Lenz (1804-1865), a Russian physicist of German origin was born in Dorpat, nowadays Tartu, Estonia. Henrich studied chemistry and physics at the University of Dorpat in 1820 after his secondary education. From 1823 to 1826, he traveled with the navigator, Otto von Kotzebue on his third expedition around the world. During this journey he studied climate conditions, and properties of seawater. After his travels, he worked at the University of St. Petersburg, Russia, where he later became the Dean of Mathematics and Physics from 1840 to 1863. In the year of 1831, he started studying electromagnetism, and soon after in 1835, what is known today as Lenz's Law was created. Lenz was also know for carefully checking his work, testing any variable that might effect his results. Lenz died on February 10, 1865, just two days before his 61st birthday, after suffering a stroke, while in Rome.

== See also ==

Since Lenz's Law and Farady's Law go hand in hand, Faraday's Law would be great supplemental information to read about. Newton's Third Law would also be a topic to read on for further understanding why Lenz's Law exists.

===Further reading===

Faraday's Law
[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html]
Conservation Laws
[http://hyperphysics.phy-astr.gsu.edu/hbase/conser.html]

===More Videos For Study===

https://www.youtube.com/watch?v=xxZenoBs2Pg This links to a video by Khan Academy that explains how Lenz's Law works through an example.

https://www.youtube.com/watch?v=Vs3afgStVy4 This links to a video by Grand Illusions that displays Lenz's Law using a magnetic falling down a tube.

==References==

https://nationalmaglab.org/education/magnet-academy/history-of-electricity-magnetism/pioneers/heinrich-friedrich-emil-lenz
http://hyperphysics.phy-astr.gsu.edu/hbase/conser.html
http://regentsprep.org/regents/physics/phys08/clenslaw/
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html
http://farside.ph.utexas.edu/teaching/302l/lectures/node85.html
http://www.electrical4u.com/lenz-law-of-electromagnetic-induction/

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

Lenz's Law

2019-04-13T22:26:59Z

Acasado3:

"Edited by Alex Casado (Spring 2019)"

==The Main Idea==

Lenz's Law is applies the law of conservation of energy to Faraday's Law, which states that any change in the magnetic field will cause an induced current. Lenz's law accounts for the direction of the new current - with the change coming from either the strength of the magnetic field, the direction of the magnetic field, the position of a circuit, the shape of a circuit, or the orientation of a circuit. To keep the magnetic flux of a loop of wires constant, there is an induced magnetic field. Change in magnetic flux results in an equal and opposite change in the loop which is why we use negative <math>{\frac{dB}{dt}}</math>. Below is a basic visualization of Lenz's Law where there is an existing magnetic field.

[[File:lenzlaw1.png]]
(image obtained from https://brilliant.org/wiki/lenzs-law/)

This Law is applied to determine the direction of current and the field it creates. When an induced field is generated by a change in magnetic flux (Faraday's Law), the induced current will flow creating its own magnetic field that opposes the magnetic field that created it. Direction is important when dealing with the law of conservation of energy. It can be seen applied when looking at electromagnetism such as the direction of voltage induced in an inductor.

In summary, whenever there is a change in magnetic flux in a conducting loop, the charge is balanced by the induction of a magnetic field. The integral of B * n remains constant.

===Theory===

Lenz's Law is mathematically modeled as part of Faraday's Law. The negative sign in the equation represents the opposing induced field.
<math>\epsilon = -{\frac{d\phi}{dt}} </math> where '''<math>\epsilon</math>''' is the emf of the system and '''<math>d\phi</math>''' is the change in the magnetic field. This means the induced electromotive force and rate of change in magnetic flux both have opposite signs. While there is a formula, it is important to note that this is mainly a qualitative law and does not incorporate magnitude, only direction.

Example:
The currents seen within strong magnets can create currents rotating in the opposite direction in metals like cooper or aluminum. If you were to drop a strong magnet through a cooper or aluminum pipe, you would observe that the velocity is decently slower than if you were to drop the magnet outside of the pipe.

=== Right Hand Rule ===

The Right Hand Rule is a heuristic used to determine the direction of angular momentum as a vector. The direction of angular momentum is determined by calculating the cross product of the vectors. An example of how the cross product is performed is below.

[[File:crossproduct.png]]

==Examples==
'''Example Using Right Hand Rule'''

Use the right hand rule to find the non-coulombic electric field in the given situations.

[[File:Non-Coulobmic_Fields.gif]]

Place your thumb in the direction of <math> -dB </math> and curl your fingers. The direction of the non-coulombic electric field is the same direction as your fingers curl. In the examples above, the direction of the electric field is the pink arrows.

===Simple===

A solenoid has current <math> I_1 </math> flowing around it increasing from 0 to 40A. A plain loop of wire is placed around the solenoid, perpendicular to the axis of the solenoid. An emf is produced, therefore producing an <math> I_2 </math>. Are <math> I_1 </math> and <math> I_2 </math> flowing in the same direction or opposite?

'''Solution'''
The original current <math> I_1 </math> produced a magnetic field. In order to maintain the conservation of energy and Newton's Third Law, the magnetic field produced by <math> I_2 </math> must oppose this field. This is in accordance with Lenz's Law. Therefore, <math> I_2 </math> must oppose <math> I_1 </math> in direction.

===Middling===

An magnet is moving through a copper tube (velocity drawn). Find the direction of -dB/dt and the direction of the induced current. Remember to use the right hand rule.

[[File:LenzLaw.jpg]]
Graphic created by Nicole Romer

'''Solution'''
A)-y, clockwise B)+y, counterclockwise C)+y, counterclockwise D)-y, clockwise

The magnetic field in this graphic is decreasing at a rate of 5.0mT/s. What is the direction of the current in the circle of wire?

[[File:hm235.jpg]]

'''Solution'''
The current in the circle of wire will produce a magnetic field that needs to supplement the existing diminishing field. This is in accordance with Lenz's Law. Therefore, the magnetic field produced needs to be into the page. Using the right hand rule, to produce a field going into the page the current in the circle of the wire must be in the clockwise direction.

===Difficult===
The magnetic field is decreasing at a rate of 5.0mT/s. The radius of the loop of wire is 5.0m, and the resistance is 5 ohms. What is the magnitude and direction of the current?

[[File:hm235.jpg]]

'''Solution'''

To find the magnitude of the current we must first use the formula <math>\epsilon = -{\frac{d\phi}{dt}} </math> to find the <math>\epsilon</math> representing the emf of the system. We know, more specifically, that <math>\epsilon = -NAcos\theta{\frac{d\phi}{dt}} </math>. Therefore, <math>\epsilon = -1*5.0^2*\pi*1*-5*10^{-3} </math> which resolves to <math>\epsilon = .392699</math>. From there, to find the current we know that <math>I = {\frac{\epsilon}{R}}</math>. Plugging in the values we know we find, <math>I = {\frac{.392699}{5.0}} = .07854A</math>. That is the magnitude of the current. To find the direction we must use Lenz's Law. The current in the circle of wire will produce a magnetic field that needs to supplement the existing diminishing field. Therefore, the magnetic field produced needs to be into the page. Using the right hand rule, to produce a field going into the page the current in the circle of the wire must be in the clockwise direction.

==Connectedness==
This topic has many applications in the real world that are very interesting. For example, an application for Lenz's Law is to cause rotation to create energy. In an industry setting, Lenz's Law can be applied to electric generators or electric motors. When a current is induced in a generator, the direction of the induced current will flow in opposition of the magnetic field that created it, causing rotation of the generator.
[[File:use235.jpg]]

Another real world application of Lenz's Law is in electromagnetic braking in vehicles. This process begins with electromagnets inducing eddy currents into the spinning rotor. Magnetic fields that oppose the initial change in magnetic flux are created from these eddy currents (Lenz's Law). This ultimately slows the rotor.

[[File:engine.jpg]]

Another example of Lenz's Law in the real world is in Induction stovetops. These cooktops heat up as a result of changing magnetic fields and eddy currents operating according to Lenz's Law.

[[File:stovetop.jpg]]

==History==

Henrich Friefrich Emil Lenz (1804-1865), a Russian physicist of German origin was born in Dorpat, nowadays Tartu, Estonia. Henrich studied chemistry and physics at the University of Dorpat in 1820 after his secondary education. From 1823 to 1826, he traveled with the navigator, Otto von Kotzebue on his third expedition around the world. During this journey he studied climate conditions, and properties of seawater. After his travels, he worked at the University of St. Petersburg, Russia, where he later became the Dean of Mathematics and Physics from 1840 to 1863. In the year of 1831, he started studying electromagnetism, and soon after in 1835, what is known today as Lenz's Law was created. Lenz was also know for carefully checking his work, testing any variable that might effect his results. Lenz died on February 10, 1865, just two days before his 61st birthday, after suffering a stroke, while in Rome.

== See also ==

Since Lenz's Law and Farady's Law go hand in hand, Faraday's Law would be great supplemental information to read about. Newton's Third Law would also be a topic to read on for further understanding why Lenz's Law exists.

===Further reading===

Faraday's Law
[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html]
Conservation Laws
[http://hyperphysics.phy-astr.gsu.edu/hbase/conser.html]

===More Videos For Study===

https://www.youtube.com/watch?v=xxZenoBs2Pg This links to a video by Khan Academy that explains how Lenz's Law works through an example.

https://www.youtube.com/watch?v=Vs3afgStVy4 This links to a video by Grand Illusions that displays Lenz's Law using a magnetic falling down a tube.

==References==

https://nationalmaglab.org/education/magnet-academy/history-of-electricity-magnetism/pioneers/heinrich-friedrich-emil-lenz
http://hyperphysics.phy-astr.gsu.edu/hbase/conser.html
http://regentsprep.org/regents/physics/phys08/clenslaw/
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html
http://farside.ph.utexas.edu/teaching/302l/lectures/node85.html
http://www.electrical4u.com/lenz-law-of-electromagnetic-induction/

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

Lenz's Law

2019-04-13T17:45:25Z

Acasado3:

"Edited by Alex Casado (Spring 2019)"

==The Main Idea==

Lenz's Law is applies the law of conservation of energy to Faraday's Law, which states that any change in the magnetic field will cause an induced current. Lenz's law accounts for the direction of the new current - with the change coming from either the strength of the magnetic field, the direction of the magnetic field, the position of a circuit, the shape of a circuit, or the orientation of a circuit. To keep the magnetic flux of a loop of wires constant, there is an induced magnetic field. Change in magnetic flux results in an equal and opposite change in the loop which is why we use negative <math>{\frac{dB}{dt}}</math>. Below is a basic visualization of Lenz's Law where there is an existing magnetic field.

[[File:lenzlaw1.png]]
(image obtained from https://brilliant.org/wiki/lenzs-law/)

This Law is applied to determine the direction of current and the field it creates. When an induced field is generated by a change in magnetic flux (Faraday's Law), the induced current will flow creating its own magnetic field that opposes the magnetic field that created it. Direction is important when dealing with the law of conservation of energy. It can be seen applied when looking at electromagnetism such as the direction of voltage induced in an inductor.

In summary, whenever there is a change in magnetic flux in a conducting loop, the charge is balanced by the induction of a magnetic field. The integral of B * n remains constant.

===A Mathematical Model===

Lenz's Law is mathematically modeled as part of Faraday's Law. The negative sign in the equation represents the opposing induced field.
<math>\epsilon = -{\frac{d\phi}{dt}} </math> where '''<math>\epsilon</math>''' is the emf of the system and '''<math>d\phi</math>''' is the change in the magnetic field. This means the induced electromotive force and rate of change in magnetic flux both have opposite signs. While there is a formula, it is important to note that this is mainly a qualitative law and does not incorporate magnitude, only direction.

Example:
The currents seen within strong magnets can create currents rotating in the opposite direction in metals like cooper or aluminum.

===A Computational Model===

https://www.youtube.com/watch?v=xxZenoBs2Pg This links to a video by Khan Academy that explains how Lenz's Law works through an example.

https://www.youtube.com/watch?v=Vs3afgStVy4 This links to a video by Grand Illusions that displays Lenz's Law using a magnetic falling down a tube.

==Examples==
'''Example Using Right Hand Rule'''

Use the right hand rule to find the non-coulombic electric field in the given situations.
[[File:Non-Coulobmic_Fields.gif]]

'''Solution'''

To use the right hand rule: place your thumb in the direction of the <math> -dB </math>, then curl your fingers. The direction in which your fingers curl is the direction of the non-coulombic electric field. The non-coulombic field is represented by the pink arrows.

===Simple===

A solenoid has current <math> I_1 </math> flowing around it increasing from 0 to 40A. A plain loop of wire is placed around the solenoid, perpendicular to the axis of the solenoid. An emf is produced, therefore producing an <math> I_2 </math>. Are <math> I_1 </math> and <math> I_2 </math> flowing in the same direction or opposite?

'''Solution'''
The original current <math> I_1 </math> produced a magnetic field. In order to maintain the conservation of energy and Newton's Third Law, the magnetic field produced by <math> I_2 </math> must oppose this field. This is in accordance with Lenz's Law. Therefore, <math> I_2 </math> must oppose <math> I_1 </math> in direction.

===Middling===

An magnet is moving through a copper tube (velocity drawn). Find the direction of -dB/dt and the direction of the induced current. Remember to use the right hand rule.

[[File:LenzLaw.jpg]]
Graphic created by Nicole Romer

'''Solution'''
A)-y, clockwise B)+y, counterclockwise C)+y, counterclockwise D)-y, clockwise

The magnetic field in this graphic is decreasing at a rate of 5.0mT/s. What is the direction of the current in the circle of wire?

[[File:hm235.jpg]]

'''Solution'''
The current in the circle of wire will produce a magnetic field that needs to supplement the existing diminishing field. This is in accordance with Lenz's Law. Therefore, the magnetic field produced needs to be into the page. Using the right hand rule, to produce a field going into the page the current in the circle of the wire must be in the clockwise direction.

===Difficult===
The magnetic field is decreasing at a rate of 5.0mT/s. The radius of the loop of wire is 5.0m, and the resistance is 5 ohms. What is the magnitude and direction of the current?

[[File:hm235.jpg]]

'''Solution'''

To find the magnitude of the current we must first use the formula <math>\epsilon = -{\frac{d\phi}{dt}} </math> to find the <math>\epsilon</math> representing the emf of the system. We know, more specifically, that <math>\epsilon = -NAcos\theta{\frac{d\phi}{dt}} </math>. Therefore, <math>\epsilon = -1*5.0^2*\pi*1*-5*10^{-3} </math> which resolves to <math>\epsilon = .392699</math>. From there, to find the current we know that <math>I = {\frac{\epsilon}{R}}</math>. Plugging in the values we know we find, <math>I = {\frac{.392699}{5.0}} = .07854A</math>. That is the magnitude of the current. To find the direction we must use Lenz's Law. The current in the circle of wire will produce a magnetic field that needs to supplement the existing diminishing field. Therefore, the magnetic field produced needs to be into the page. Using the right hand rule, to produce a field going into the page the current in the circle of the wire must be in the clockwise direction.

==Connectedness==
This topic has many applications in the real world that are very interesting. For example, an application for Lenz's Law is to cause rotation to create energy. In an industry setting, Lenz's Law can be applied to electric generators or electric motors. When a current is induced in a generator, the direction of the induced current will flow in opposition of the magnetic field that created it, causing rotation of the generator.
[[File:use235.jpg]]

Another real world application of Lenz's Law is in electromagnetic braking in vehicles. This process begins with electromagnets inducing eddy currents into the spinning rotor. Magnetic fields that oppose the initial change in magnetic flux are created from these eddy currents (Lenz's Law). This ultimately slows the rotor.

[[File:engine.jpg]]

Another example of Lenz's Law in the real world is in Induction stovetops. These cooktops heat up as a result of changing magnetic fields and eddy currents operating according to Lenz's Law.

[[File:stovetop.jpg]]

==History==

Henrich Friefrich Emil Lenz (1804-1865), a Russian physicist of German origin was born in Dorpat, nowadays Tartu, Estonia. Henrich studied chemistry and physics at the University of Dorpat in 1820 after his secondary education. From 1823 to 1826, he traveled with the navigator, Otto von Kotzebue on his third expedition around the world. During this journey he studied climate conditions, and properties of seawater. After his travels, he worked at the University of St. Petersburg, Russia, where he later became the Dean of Mathematics and Physics from 1840 to 1863. In the year of 1831, he started studying electromagnetism, and soon after in 1835, what is known today as Lenz's Law was created. Lenz was also know for carefully checking his work, testing any variable that might effect his results. Lenz died on February 10, 1865, just two days before his 61st birthday, after suffering a stroke, while in Rome.

== See also ==

Since Lenz's Law and Farady's Law go hand in hand, Faraday's Law would be great supplemental information to read about. Newton's Third Law would also be a topic to read on for further understanding why Lenz's Law exists.

===Further reading===

Faraday's Law
[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html]
Conservation Laws
[http://hyperphysics.phy-astr.gsu.edu/hbase/conser.html]

==References==

https://nationalmaglab.org/education/magnet-academy/history-of-electricity-magnetism/pioneers/heinrich-friedrich-emil-lenz
http://hyperphysics.phy-astr.gsu.edu/hbase/conser.html
http://regentsprep.org/regents/physics/phys08/clenslaw/
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html
http://farside.ph.utexas.edu/teaching/302l/lectures/node85.html
http://www.electrical4u.com/lenz-law-of-electromagnetic-induction/

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

Lenz's Law

2019-04-13T17:32:15Z

Acasado3:

"Edited by Alex Casado (Spring 2019)"

==The Main Idea==

Lenz's Law is applies the law of conservation of energy to Faraday's Law, which states that any change in the magnetic field will cause an induced current. Lenz's law accounts for the direction of the new current - with the change coming from either the strength of the magnetic field, the direction of the magnetic field, the position of a circuit, the shape of a circuit, or the orientation of a circuit. To keep the magnetic flux of a loop of wires constant, there is an induced magnetic field. Change in magnetic flux results in an equal and opposite change in the loop which is why we use negative <math>{\frac{dB}{dt}}</math>.

[[File:lenzlaw1.png]]
[[File:sc235.jpg]]

===A Mathematical Model===

Lenz's Law is mathematically modeled as part of Faraday's Law. The negative sign in the equation represents the opposing induced field.
<math>\epsilon = -{\frac{d\phi}{dt}} </math> where '''<math>\epsilon</math>''' is the emf of the system and '''<math>d\phi</math>''' is the change in the magnetic field.

===A Computational Model===

https://www.youtube.com/watch?v=xxZenoBs2Pg This links to a video by Khan Academy that explains how Lenz's Law works through an example.

https://www.youtube.com/watch?v=Vs3afgStVy4 This links to a video by Grand Illusions that displays Lenz's Law using a magnetic falling down a tube.

==Examples==
'''Example Using Right Hand Rule'''

Use the right hand rule to find the non-coulombic electric field in the given situations.
[[File:Non-Coulobmic_Fields.gif]]

'''Solution'''

To use the right hand rule: place your thumb in the direction of the <math> -dB </math>, then curl your fingers. The direction in which your fingers curl is the direction of the non-coulombic electric field. The non-coulombic field is represented by the pink arrows.

===Simple===

A solenoid has current <math> I_1 </math> flowing around it increasing from 0 to 40A. A plain loop of wire is placed around the solenoid, perpendicular to the axis of the solenoid. An emf is produced, therefore producing an <math> I_2 </math>. Are <math> I_1 </math> and <math> I_2 </math> flowing in the same direction or opposite?

'''Solution'''
The original current <math> I_1 </math> produced a magnetic field. In order to maintain the conservation of energy and Newton's Third Law, the magnetic field produced by <math> I_2 </math> must oppose this field. This is in accordance with Lenz's Law. Therefore, <math> I_2 </math> must oppose <math> I_1 </math> in direction.

===Middling===

An magnet is moving through a copper tube (velocity drawn). Find the direction of -dB/dt and the direction of the induced current. Remember to use the right hand rule.

[[File:LenzLaw.jpg]]
Graphic created by Nicole Romer

'''Solution'''
A)-y, clockwise B)+y, counterclockwise C)+y, counterclockwise D)-y, clockwise

The magnetic field in this graphic is decreasing at a rate of 5.0mT/s. What is the direction of the current in the circle of wire?

[[File:hm235.jpg]]

'''Solution'''
The current in the circle of wire will produce a magnetic field that needs to supplement the existing diminishing field. This is in accordance with Lenz's Law. Therefore, the magnetic field produced needs to be into the page. Using the right hand rule, to produce a field going into the page the current in the circle of the wire must be in the clockwise direction.

===Difficult===
The magnetic field is decreasing at a rate of 5.0mT/s. The radius of the loop of wire is 5.0m, and the resistance is 5 ohms. What is the magnitude and direction of the current?

[[File:hm235.jpg]]

'''Solution'''

To find the magnitude of the current we must first use the formula <math>\epsilon = -{\frac{d\phi}{dt}} </math> to find the <math>\epsilon</math> representing the emf of the system. We know, more specifically, that <math>\epsilon = -NAcos\theta{\frac{d\phi}{dt}} </math>. Therefore, <math>\epsilon = -1*5.0^2*\pi*1*-5*10^{-3} </math> which resolves to <math>\epsilon = .392699</math>. From there, to find the current we know that <math>I = {\frac{\epsilon}{R}}</math>. Plugging in the values we know we find, <math>I = {\frac{.392699}{5.0}} = .07854A</math>. That is the magnitude of the current. To find the direction we must use Lenz's Law. The current in the circle of wire will produce a magnetic field that needs to supplement the existing diminishing field. Therefore, the magnetic field produced needs to be into the page. Using the right hand rule, to produce a field going into the page the current in the circle of the wire must be in the clockwise direction.

==Connectedness==
This topic has many applications in the real world that are very interesting. For example, an application for Lenz's Law is to cause rotation to create energy. In an industry setting, Lenz's Law can be applied to electric generators or electric motors. When a current is induced in a generator, the direction of the induced current will flow in opposition of the magnetic field that created it, causing rotation of the generator.
[[File:use235.jpg]]

Another real world application of Lenz's Law is in electromagnetic braking in vehicles. This process begins with electromagnets inducing eddy currents into the spinning rotor. Magnetic fields that oppose the initial change in magnetic flux are created from these eddy currents (Lenz's Law). This ultimately slows the rotor.

[[File:engine.jpg]]

Another example of Lenz's Law in the real world is in Induction stovetops. These cooktops heat up as a result of changing magnetic fields and eddy currents operating according to Lenz's Law.

[[File:stovetop.jpg]]

==History==

Henrich Friefrich Emil Lenz (1804-1865), a Russian physicist of German origin was born in Dorpat, nowadays Tartu, Estonia. Henrich studied chemistry and physics at the University of Dorpat in 1820 after his secondary education. From 1823 to 1826, he traveled with the navigator, Otto von Kotzebue on his third expedition around the world. During this journey he studied climate conditions, and properties of seawater. After his travels, he worked at the University of St. Petersburg, Russia, where he later became the Dean of Mathematics and Physics from 1840 to 1863. In the year of 1831, he started studying electromagnetism, and soon after in 1835, what is known today as Lenz's Law was created. Lenz was also know for carefully checking his work, testing any variable that might effect his results. Lenz died on February 10, 1865, just two days before his 61st birthday, after suffering a stroke, while in Rome.

== See also ==

Since Lenz's Law and Farady's Law go hand in hand, Faraday's Law would be great supplemental information to read about. Newton's Third Law would also be a topic to read on for further understanding why Lenz's Law exists.

===Further reading===

Faraday's Law
[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html]
Conservation Laws
[http://hyperphysics.phy-astr.gsu.edu/hbase/conser.html]

==References==

https://nationalmaglab.org/education/magnet-academy/history-of-electricity-magnetism/pioneers/heinrich-friedrich-emil-lenz
http://hyperphysics.phy-astr.gsu.edu/hbase/conser.html
http://regentsprep.org/regents/physics/phys08/clenslaw/
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html
http://farside.ph.utexas.edu/teaching/302l/lectures/node85.html
http://www.electrical4u.com/lenz-law-of-electromagnetic-induction/

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

File:Lenzlaw1.png

2019-04-13T17:30:32Z

Acasado3:

Lenz's Law

2019-04-13T17:29:59Z

Acasado3:

"Edited by Alex Casado (Spring 2019)"

==The Main Idea==

Lenz's Law is applies the law of conservation of energy to Faraday's Law, which states that any change in the magnetic field will cause an induced current. Lenz's law accounts for the direction of the new current - with the change coming from either the strength of the magnetic field, the direction of the magnetic field, the position of a circuit, the shape of a circuit, or the orientation of a circuit. To keep the magnetic flux of a loop of wires constant, there is an induced magnetic field. Change in magnetic flux results in an equal and opposite change in the loop which is why we use negative <math>{\frac{dB}{dt}}</math>.

[[File:sc235.jpg]]

===A Mathematical Model===

Lenz's Law is mathematically modeled as part of Faraday's Law. The negative sign in the equation represents the opposing induced field.
<math>\epsilon = -{\frac{d\phi}{dt}} </math> where '''<math>\epsilon</math>''' is the emf of the system and '''<math>d\phi</math>''' is the change in the magnetic field.

===A Computational Model===

https://www.youtube.com/watch?v=xxZenoBs2Pg This links to a video by Khan Academy that explains how Lenz's Law works through an example.

https://www.youtube.com/watch?v=Vs3afgStVy4 This links to a video by Grand Illusions that displays Lenz's Law using a magnetic falling down a tube.

==Examples==
'''Example Using Right Hand Rule'''

Use the right hand rule to find the non-coulombic electric field in the given situations.
[[File:Non-Coulobmic_Fields.gif]]

'''Solution'''

To use the right hand rule: place your thumb in the direction of the <math> -dB </math>, then curl your fingers. The direction in which your fingers curl is the direction of the non-coulombic electric field. The non-coulombic field is represented by the pink arrows.

===Simple===

A solenoid has current <math> I_1 </math> flowing around it increasing from 0 to 40A. A plain loop of wire is placed around the solenoid, perpendicular to the axis of the solenoid. An emf is produced, therefore producing an <math> I_2 </math>. Are <math> I_1 </math> and <math> I_2 </math> flowing in the same direction or opposite?

'''Solution'''
The original current <math> I_1 </math> produced a magnetic field. In order to maintain the conservation of energy and Newton's Third Law, the magnetic field produced by <math> I_2 </math> must oppose this field. This is in accordance with Lenz's Law. Therefore, <math> I_2 </math> must oppose <math> I_1 </math> in direction.

===Middling===

An magnet is moving through a copper tube (velocity drawn). Find the direction of -dB/dt and the direction of the induced current. Remember to use the right hand rule.

[[File:LenzLaw.jpg]]
Graphic created by Nicole Romer

'''Solution'''
A)-y, clockwise B)+y, counterclockwise C)+y, counterclockwise D)-y, clockwise

The magnetic field in this graphic is decreasing at a rate of 5.0mT/s. What is the direction of the current in the circle of wire?

[[File:hm235.jpg]]

'''Solution'''
The current in the circle of wire will produce a magnetic field that needs to supplement the existing diminishing field. This is in accordance with Lenz's Law. Therefore, the magnetic field produced needs to be into the page. Using the right hand rule, to produce a field going into the page the current in the circle of the wire must be in the clockwise direction.

===Difficult===
The magnetic field is decreasing at a rate of 5.0mT/s. The radius of the loop of wire is 5.0m, and the resistance is 5 ohms. What is the magnitude and direction of the current?

[[File:hm235.jpg]]

'''Solution'''

To find the magnitude of the current we must first use the formula <math>\epsilon = -{\frac{d\phi}{dt}} </math> to find the <math>\epsilon</math> representing the emf of the system. We know, more specifically, that <math>\epsilon = -NAcos\theta{\frac{d\phi}{dt}} </math>. Therefore, <math>\epsilon = -1*5.0^2*\pi*1*-5*10^{-3} </math> which resolves to <math>\epsilon = .392699</math>. From there, to find the current we know that <math>I = {\frac{\epsilon}{R}}</math>. Plugging in the values we know we find, <math>I = {\frac{.392699}{5.0}} = .07854A</math>. That is the magnitude of the current. To find the direction we must use Lenz's Law. The current in the circle of wire will produce a magnetic field that needs to supplement the existing diminishing field. Therefore, the magnetic field produced needs to be into the page. Using the right hand rule, to produce a field going into the page the current in the circle of the wire must be in the clockwise direction.

==Connectedness==
This topic has many applications in the real world that are very interesting. For example, an application for Lenz's Law is to cause rotation to create energy. In an industry setting, Lenz's Law can be applied to electric generators or electric motors. When a current is induced in a generator, the direction of the induced current will flow in opposition of the magnetic field that created it, causing rotation of the generator.
[[File:use235.jpg]]

Another real world application of Lenz's Law is in electromagnetic braking in vehicles. This process begins with electromagnets inducing eddy currents into the spinning rotor. Magnetic fields that oppose the initial change in magnetic flux are created from these eddy currents (Lenz's Law). This ultimately slows the rotor.

[[File:engine.jpg]]

Another example of Lenz's Law in the real world is in Induction stovetops. These cooktops heat up as a result of changing magnetic fields and eddy currents operating according to Lenz's Law.

[[File:stovetop.jpg]]

==History==

Henrich Friefrich Emil Lenz (1804-1865), a Russian physicist of German origin was born in Dorpat, nowadays Tartu, Estonia. Henrich studied chemistry and physics at the University of Dorpat in 1820 after his secondary education. From 1823 to 1826, he traveled with the navigator, Otto von Kotzebue on his third expedition around the world. During this journey he studied climate conditions, and properties of seawater. After his travels, he worked at the University of St. Petersburg, Russia, where he later became the Dean of Mathematics and Physics from 1840 to 1863. In the year of 1831, he started studying electromagnetism, and soon after in 1835, what is known today as Lenz's Law was created. Lenz was also know for carefully checking his work, testing any variable that might effect his results. Lenz died on February 10, 1865, just two days before his 61st birthday, after suffering a stroke, while in Rome.

== See also ==

Since Lenz's Law and Farady's Law go hand in hand, Faraday's Law would be great supplemental information to read about. Newton's Third Law would also be a topic to read on for further understanding why Lenz's Law exists.

===Further reading===

Faraday's Law
[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html]
Conservation Laws
[http://hyperphysics.phy-astr.gsu.edu/hbase/conser.html]

==References==

https://nationalmaglab.org/education/magnet-academy/history-of-electricity-magnetism/pioneers/heinrich-friedrich-emil-lenz
http://hyperphysics.phy-astr.gsu.edu/hbase/conser.html
http://regentsprep.org/regents/physics/phys08/clenslaw/
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html
http://farside.ph.utexas.edu/teaching/302l/lectures/node85.html
http://www.electrical4u.com/lenz-law-of-electromagnetic-induction/

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

Lenz's Law

2019-04-13T17:22:25Z

Acasado3:

"Edited by Alex Casado (Spring 2019)"

==The Main Idea==

Lenz's Law is applies the law of conservation of energy to Faraday's Law, which states that any change in the magnetic field will cause an induced current. Lenz's law accounts for the direction of the new current - with the change coming from either the strength of the magnetic field, the direction of the magnetic field, the position of a circuit, the shape of a circuit, or the orientation of a circuit. To keep the magnetic flux of a loop of wires constant, there is an induced magnetic field. Change in magnetic flux results in an equal and opposite change in the loop which is why we use negative <math>{\frac{dB}{dt}}</math>
[[File:sc235.jpg]]

===A Mathematical Model===

Lenz's Law is mathematically modeled as part of Faraday's Law. The negative sign in the equation represents the opposing induced field.
<math>\epsilon = -{\frac{d\phi}{dt}} </math> where '''<math>\epsilon</math>''' is the emf of the system and '''<math>d\phi</math>''' is the change in the magnetic field.

===A Computational Model===

https://www.youtube.com/watch?v=xxZenoBs2Pg This links to a video by Khan Academy that explains how Lenz's Law works through an example.

https://www.youtube.com/watch?v=Vs3afgStVy4 This links to a video by Grand Illusions that displays Lenz's Law using a magnetic falling down a tube.

==Examples==
'''Example Using Right Hand Rule'''

Use the right hand rule to find the non-coulombic electric field in the given situations.
[[File:Non-Coulobmic_Fields.gif]]

'''Solution'''

To use the right hand rule: place your thumb in the direction of the <math> -dB </math>, then curl your fingers. The direction in which your fingers curl is the direction of the non-coulombic electric field. The non-coulombic field is represented by the pink arrows.

===Simple===

A solenoid has current <math> I_1 </math> flowing around it increasing from 0 to 40A. A plain loop of wire is placed around the solenoid, perpendicular to the axis of the solenoid. An emf is produced, therefore producing an <math> I_2 </math>. Are <math> I_1 </math> and <math> I_2 </math> flowing in the same direction or opposite?

'''Solution'''
The original current <math> I_1 </math> produced a magnetic field. In order to maintain the conservation of energy and Newton's Third Law, the magnetic field produced by <math> I_2 </math> must oppose this field. This is in accordance with Lenz's Law. Therefore, <math> I_2 </math> must oppose <math> I_1 </math> in direction.

===Middling===

An magnet is moving through a copper tube (velocity drawn). Find the direction of -dB/dt and the direction of the induced current. Remember to use the right hand rule.

[[File:LenzLaw.jpg]]
Graphic created by Nicole Romer

'''Solution'''
A)-y, clockwise B)+y, counterclockwise C)+y, counterclockwise D)-y, clockwise

The magnetic field in this graphic is decreasing at a rate of 5.0mT/s. What is the direction of the current in the circle of wire?

[[File:hm235.jpg]]

'''Solution'''
The current in the circle of wire will produce a magnetic field that needs to supplement the existing diminishing field. This is in accordance with Lenz's Law. Therefore, the magnetic field produced needs to be into the page. Using the right hand rule, to produce a field going into the page the current in the circle of the wire must be in the clockwise direction.

===Difficult===
The magnetic field is decreasing at a rate of 5.0mT/s. The radius of the loop of wire is 5.0m, and the resistance is 5 ohms. What is the magnitude and direction of the current?

[[File:hm235.jpg]]

'''Solution'''

To find the magnitude of the current we must first use the formula <math>\epsilon = -{\frac{d\phi}{dt}} </math> to find the <math>\epsilon</math> representing the emf of the system. We know, more specifically, that <math>\epsilon = -NAcos\theta{\frac{d\phi}{dt}} </math>. Therefore, <math>\epsilon = -1*5.0^2*\pi*1*-5*10^{-3} </math> which resolves to <math>\epsilon = .392699</math>. From there, to find the current we know that <math>I = {\frac{\epsilon}{R}}</math>. Plugging in the values we know we find, <math>I = {\frac{.392699}{5.0}} = .07854A</math>. That is the magnitude of the current. To find the direction we must use Lenz's Law. The current in the circle of wire will produce a magnetic field that needs to supplement the existing diminishing field. Therefore, the magnetic field produced needs to be into the page. Using the right hand rule, to produce a field going into the page the current in the circle of the wire must be in the clockwise direction.

==Connectedness==
This topic has many applications in the real world that are very interesting. For example, an application for Lenz's Law is to cause rotation to create energy. In an industry setting, Lenz's Law can be applied to electric generators or electric motors. When a current is induced in a generator, the direction of the induced current will flow in opposition of the magnetic field that created it, causing rotation of the generator.
[[File:use235.jpg]]

Another real world application of Lenz's Law is in electromagnetic braking in vehicles. This process begins with electromagnets inducing eddy currents into the spinning rotor. Magnetic fields that oppose the initial change in magnetic flux are created from these eddy currents (Lenz's Law). This ultimately slows the rotor.

[[File:engine.jpg]]

Another example of Lenz's Law in the real world is in Induction stovetops. These cooktops heat up as a result of changing magnetic fields and eddy currents operating according to Lenz's Law.

[[File:stovetop.jpg]]

==History==

Henrich Friefrich Emil Lenz (1804-1865), a Russian physicist of German origin was born in Dorpat, nowadays Tartu, Estonia. Henrich studied chemistry and physics at the University of Dorpat in 1820 after his secondary education. From 1823 to 1826, he traveled with the navigator, Otto von Kotzebue on his third expedition around the world. During this journey he studied climate conditions, and properties of seawater. After his travels, he worked at the University of St. Petersburg, Russia, where he later became the Dean of Mathematics and Physics from 1840 to 1863. In the year of 1831, he started studying electromagnetism, and soon after in 1835, what is known today as Lenz's Law was created. Lenz was also know for carefully checking his work, testing any variable that might effect his results. Lenz died on February 10, 1865, just two days before his 61st birthday, after suffering a stroke, while in Rome.

== See also ==

Since Lenz's Law and Farady's Law go hand in hand, Faraday's Law would be great supplemental information to read about. Newton's Third Law would also be a topic to read on for further understanding why Lenz's Law exists.

===Further reading===

Faraday's Law
[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html]
Conservation Laws
[http://hyperphysics.phy-astr.gsu.edu/hbase/conser.html]

==References==

https://nationalmaglab.org/education/magnet-academy/history-of-electricity-magnetism/pioneers/heinrich-friedrich-emil-lenz
http://hyperphysics.phy-astr.gsu.edu/hbase/conser.html
http://regentsprep.org/regents/physics/phys08/clenslaw/
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html
http://farside.ph.utexas.edu/teaching/302l/lectures/node85.html
http://www.electrical4u.com/lenz-law-of-electromagnetic-induction/

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

Gauss's Flux Theorem

2019-04-13T17:06:09Z

Acasado3:

Claimed by Courtney Smith (Fall 2018)
Edited by Alex Casado (Spring 2019)

==The Main Idea==

Gauss's Flux Theorem describes the relationship between the charges inside of an object and the electric field acting on the surface(s) of that object. This theorem makes it possible to determine the sum of total charges inside an object; by calculating the direction and magnitude of all electric fields acting on an object, you can use a ratio to determine the charges inside that object. There is a proportional relationship between electric fields acting on an object and the charges inside that object, and that ratio is shown below in the Mathematical Model section. This relationship applies to all shapes with any number of sides, as long as the object is enclosed.

Note that Gauss's Flux Theorem only tells you the relationship between electric fields outside an object and the charges inside an object. It does not tell you anything about the charges outside of the object. Therefore, any outside charges do not affect Gauss's Flux Theorem calculations. This is because Gauss's Law uses the superposition principle. So no charges outside the object will produce any electric field (and in turn, zero electric flux) acting on the object.

Essentially, electric flux can be thought of as the flow rate of the electric field through a given area.

===A Mathematical Model===

The electric flux of a closed object is equal to the total charge enclosed over the permittivity of free space constant (epsilon naught). The electric flux of a closed surface is also equal to the surface integral of an electric field evaluated over the object's surfaces.

<math>\Phi_E = \frac{\Sigma q}{\varepsilon_0} = \oint_C E\bullet \hat{n} dA</math>

Where E is the electric field, dA is the infinitesimal area in the direction of the electric field, <math> \hat{n} </math> is the unit normal vector, sigma q is the sum of the charges inside the closed surface, and the dot denotes a dot product.

For the special case of a constant electric field, the electric flux is equal to the electric field multiplied by the area and the cosine of the angle between the two vectors.

<math>\Phi_E = EAcos(\theta)</math>

Where E is the electric field, A is area of the surface, and <math> \theta</math> is the angle between the E and A.
Note that this is the electric flux for one side of an object. If you want to know the electric flux of the total object, you must use this formula for calculating flux on each side of the object.

====Gaussian Surface====

To find the electric field and charge enclosed of different shapes and surfaces, often times a Gaussian surface must be drawn that encompasses a part of the main surface. When trying to find the electric field at a point outside of the surface at hand, a surface including the point should be drawn. The Gaussian surfaces are usually spheres or cylinders, although it usually doesn't matter in the end as the area of the Gaussian surface will cancel out. Spheres are used when one is trying to find the electric field of a point charge, a spherical shell of uniform charge, or any other object with a symmetric charge distribution. A cylinder should be used as the Gaussian surface when one is trying to find the electric field of an infinitely long line of charge, and infinite plane or sheet of charge, or a cylinder of charge. When analyzing the Gaussian surface, only the surfaces where the normal vector of the area is not perpendicular to the electric field vector should be considered for calculations. When trying to find the electric field of a shape inside of the total object, finding the ratio of volumes and multiplying that by Q enclosed should be done to find the charge of the smaller object.

===A Computational Model===
Gauss's Flux Theorem is able to be visualized by observing the direction and angle of an electric field traveling through a given surface. Typically in Gauss's Flux calculations, you are concerned with two vectors: n-hat and the electric field vector. N-hat always points perpendicular from the surface of interest, while the electric field can point in any direction and will vary from problem to problem.

It is always important to note the angle between the n-hat vector and the electric field. If that angle is perfectly perpendicular, the contributed electric flux will be zero.
[[File:electric flux.jpg]]

It is also important to note the sign (positive or negative) of the electric flux, with respect to the direction of the electric field. If the electric field is pointing into the surface, the flux will be negative. If the electric field is pointing out of the surface, the electric flux will be positive.
[[File:sign of flux.jpg]]

==Examples==

===Easy===
===One Surface and Uniform Electric Field===

A disk of radius 5 centimeters is in an area of uniform electric field with magnitude 400 Volts/Meter. The angle between the electric field and the disk is 35 degrees.

[[File:GL-Ex-1.S.JPG]]

Using the simplified version of Gauss's Law because the electric field is uniform: <math>\Phi_E = EAcos(\theta)</math>, fill out the known values, which in the case is all values needed.

<math>\Phi_E = (400)(\pi0.05^2)cos(35)</math>

<math>\Phi_E = 2.573</math> Volt Meters

===Middling===
===Multiple Surfaces and Uniform Electric field===

An equilateral prism lies on top of a rectangular prism with dimensions l = 5cm, w = 7cm, h = 19cm, and is in an area of uniform electric field with magnitude 560 Volts/Meter perpendicular to the long sides of the rectangular prism.

[[File:Ex-2-2.JPG]]

Using the simplified version of Gauss's Law because the electric field is uniform: <math>\Phi_E = EAcos(\theta)</math>

First of all it is important to note that there are four surfaces that contribute to the net electric flux. two of the two long sides of the equilateral prism and two of the long sides of the rectangular prism. We know the angles between E and A because the top prism is an equilateral triangle, so all the side angles are 60 degrees. With this equation we can write:

<math>\Phi_E = E[(w*h)cos(120) + (w*h)cos(60) + (w*h)cos(180) + (w*h)cos(0)]</math>

<math>\Phi_E = 0</math>

===Difficult===
===Non-Uniform Electric Field===

A solid sphere of radius R has a charge +Q uniformly distributed throughout. Find the Electric field at locations r1<R.

[[File:GL-Ex-3.JPG]]

For this problem we can utilize the equality of <math>\frac{Q}{\varepsilon_0} = \oint_C E\bullet dA</math>

For r1<R, we know the line integral of dA is just the circumference of the gaussian surface we took, a sphere of radius r1.

<math>\frac{Q}{\varepsilon_0} = E\bullet 4\pi r_1^2</math>

Since we're taking a portion of the sphere we need to find the portion of the charge that is inside of the Gaussian surface. In order to find the charge of the smaller sphere, we divide the total charge by the total volume to get the charge per unit volume. We then multiply that value q by the volume of the smaller sphere (radius r1) to get the charge contained in that volume.

The work is shown below.

<math>q = \frac{3Q}{4\pi r_1^3}*\frac{4\pi R^3}{3}</math>

<math>q = Q\frac{r_1^3}{R^3}</math>

<math>E = Q\frac{r_1}{4\pi R^3}</math>

===Electric Field of Infinite Line Charge===

What is the electric field on a point from an infinitely long line charge?

[[File:LineCharge.png]]

For this problem, the Gaussian surface we use is a cylinder encompassing a portion of the wire.

For this problem, we can use the Gauss's law to relate the enclosed charge and electric field of the wire: <math>\frac{Q}{\varepsilon_0} = \oint_C E\bullet dA</math>

For the left side of the equation, the linear charge density can be applied to ultimately replace Q:

<math>{\lambda} = Q/L</math>

<math>Q = {\lambda}L</math>

So, the left side of the equation simplifies to: <math>\frac{{\lambda}L}{\varepsilon_0}</math>

On the right side of the equation, E can be taken out of the integral and dA is replaced with the surface area of the curved portion of the cylinder, which is the gaussian surface:

<math>\oint_C E\bullet dA = E(2{\pi}rL)</math>

From here, the two simplified versions of each side of the equation can be combined: <math>\frac{{\lambda}L}{\varepsilon_0} = E(2{\pi}rL)</math>

Finally, through algebraic manipulation, E is found to be: <math>E = \frac{{\lambda}}{2{\pi}r{\varepsilon_0}}</math>

===Electric Field of Sheet of Charge===

What is the electric field on a point above a sheet of charge?

[[File:SheetCharge.jpg]]

For this problem, the Gaussian surface is a cylinder going through both sides of the sheet.

First Gauss's law can be applied: <math>\frac{Q}{\varepsilon_0} = \oint_C E\bullet dA</math>

On the left side of the equation, the 2D charge density can be used to replace Q:

<math>{\sigma} = \frac{q}{A}</math>

<math>q = {\sigma}A</math>

On the right side of the equation, the statement <math>\oint_C E\bullet dA</math> can be replaced by <math>2EA</math>. The 2 is included because the electric field going out the bottom must be accounted for.

From here, after combining both sides of the equation, we get: <math>\frac{{\sigma}A}{\varepsilon_0} = 2EA</math>

Next, the A on each side cancels out, and we can isolate E to find the final solution: <math> E = \frac{{\sigma}}{2{\varepsilon_0}}</math>

This answer can also be left in terms of q and Area. In this case, the dimensions of the sheet are LxL, so area can be replaced by <math>L^2</math>: <math>E = \frac{q}{2{\varepsilon_0}L^2}</math>

===Electric Field inside Cylinder of Charge===

What is the electric field at a point inside of a cylinder of charge?

[[File:InnerCylinder.jpg]]

For this problem, we can create a Gaussian surface of a smaller cylinder inside the larger one. From here, we can first use Gauss's Law: <math>\oint_C E\bullet dA = \frac{Q}{\varepsilon_0}</math>

For the left part of the equation, E can be brought out of the integral and dA is replaced with the curved surface area of the inner cylinder:

<math>\oint_C E\bullet dA = E(2{\pi}rL)</math>

For the right side of the equation, Q for the smaller Gaussian surface can be found by multiplying the charge of the larger cylinder by the ratio of volumes:

<math>Q' = Q(\frac{Vr}{VR}) = Q(\frac{{\pi}r^2L}{{\pi}R^2L}) = Q(\frac{r^2}{R^2})</math>

Next, both sides of the equation can be combined to form a new equation: <math>E(2{\pi}rL) = \frac{Qr^2}{{\varepsilon_0}R^2}</math>

From here, the r on the left side of the equation and one r from the right side cancel out. Finally, we can isolate E to find the electric field:

<math>E = \frac{Qr}{2{\pi}{\varepsilon_0}LR^2}</math>

This answer can also be in a different form if the linear charge density is used: <math>{\lambda} = \frac{Q}{L} </math>

So, <math>Q = {\lambda}L</math>

Finally, the <math>\frac{Q}{L}</math> can be replaced by <math>{\lambda}</math> in the electric field equation:

<math>E = \frac{{\lambda}r}{2{\pi}{\varepsilon_0}R^2}</math>

===Complex Surface===

For this problem, we are asked to use Gauss's Law to determine the flux of a net.

[[File:NetExample.jpg]]

==Differential Form==

Coulomb's law has a critical flaw when it comes to calculating the electric field at some location near a quick-moving charge. This problem called Relativistic Retardation occurs at pretty much all occasions, unless the charge is not moving at all or moving extremely slow as compared to the speed of light. Relativistic Retardation is due to the fact that the electric field calculation depends on the origin of a charge (i.e. the location where it started moving), not the current location of the charge. So if the charge has moved from its starting location, then the electric field calculation is wrong in Coulomb's law, which also means that it is inconsistent with the theory of special relativity.

Gauss's Law does not have this critical flaw and is consistent with relativity due to a property called Divergence. Conceptually, Divergence is a way to measure electric fields at some point in time as compared to a source charge at that same point in time; so the property of Divergence is the reason that Gauss's Law does not experience Relativistic Retardation. Divergence is defined as the amount of electric flux in an object as the limit of that object's volume goes to zero.

The formula which expresses divergence is called The Differential Form of Gauss's Law:

:<math>\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}</math>

where the del is the differential operator, E is the electric field vector, rho is the electric charge density, the constant epsilon naught is the electric permittivity of free space, and the dot denotes a dot product.

==Connectedness==
Most of the applications of Gauss's Law include finding the electrical fields of different shapes or the charges inside those shapes. Gauss's Law can be applied to a variety of objects such as a sphere, sheet of charge, and cylinder. This use of Gauss's Law is especially helpful in the field of Physics. Furthermore, the use of electric flux in Gauss's Flux Theorem assists in studying and understanding Faraday's Law and its applications to objects such as electrical generators, transformers, and inductors.

Being an industrial engineer, Gauss's Law is particularly interesting to me. It proportionally relates electric field to charge within an object. Industrial engineering deals with a lot of probability and numerical methods, so proportionality relationships really interest me and help when understanding the "big-picture" of a concept. It is relatively intuitive to think that electric fields and interior charges would some-how be related, but being able to actually quantify that relationship through Gauss's Law is very important and interesting.

==History==

Friedrich Gauss was born in 1777 in Brunswick, Germany. He was an extremely intelligent child, many considered him a prodigy. In fact, he discovered the exact date of his birthday by solving a puzzle about his birthdate in the context of the date of Easter and deriving methods to compute past and future dates. He started making mathematical computations and discoveries in his teen years. And in 1835, he formulated Gauss's Law as a different way to derive Coulomb's Law. He made many other discoveries in the fields of mathematics, statistics, probability, and number theory.

Today, Gauss's Law remains as one of Maxwell's equations, which are the four equations that make up classical electrodynamics. Maxwell's four equations are Gauss's Law for electricity, Gauss's Law for magnetism, incomplete version of Faraday's Law, and Ampere's Law. In addition to discovering Gauss's Law for Electricity and Gauss's Law for magnetism, Gauss also discovered other physics laws and equations including a method for measuring the horizontal intensity of the Earth's magnetic field and a mathematical theory for separating the inner and outer sources of the Earth's magnetic field.

== See also ==

===Further reading and Examples===
http://www.physics.umd.edu/courses/Phys263/wth/fall04/downloads/Gauss/divergence.pdf

http://phys.libretexts.org/TextMaps/Map%3A_Conceptual_Physics_(Crowell)/10%3A_Fields/10.7_Gauss'_Law_In_Differential_Form

https://www.youtube.com/watch?v=zn5ObHtGWeg

===External links===

http://web.hep.uiuc.edu/home/serrede/P435/Lecture_Notes/A_Brief_History_of_Electromagnetism.pdf

http://ethw.org/Gauss'_Law

http://www.storyofmathematics.com/19th_gauss.html

==References==

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/gaulaw.html

http://www.physnet.org/modules/pdf_modules/m132.pdf

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elesht.html

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elecyl.html

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

Vectors

2017-04-09T18:25:17Z

Acasado3:

Written by Elizabeth Robelo

Improved by Aparajita Satapathy

'''Improved by Lichao Tang'''

Improved by Jimin Yoon

Improved by Sabrina Seibel

Claimed: Alex Casado Spring 2017

In physics, a vector is a quantity with a magnitude and a direction.

==The Main Idea==

A vector is a quantity with a magnitude and a direction. The magnitude of a vector is a scalar which represents the length of the vector. A vector is represented by an arrow. The orientation of the vector represents its direction. When a vector is drawn, the starting point is the tail and the ending point is called the head of the vector. In physics, a vector always starts at the source and directs to the observation location. Refer to the image below for a visual representation:

[[File:Mathinsight.png|300px|thumb|center|Visual Representation]]

We can also add and subtract vectors. To add two vectors you align them head to tail and make sure that the tails of both vectors coincide with each other. The new added vector is the connecting arrow starting from the tail of one to the head of the other vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the vector you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A vector can also be multiplied by a scalar. To multiply a vector by a scalar we can strech, compress or reverse the direction of a vector. If the scalar is between 0 and 1, the vector will be compresseed. If the scalar is greater than 1, the vector will get streched. If the scalar is a negative number, then the vector reverses its direction.

[[File:Ibguides.png|400px|thumb|center|Scalar Multiplication]]

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or (xi + yj - zk).
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Dot Product: Ia·bI = IaI*IbI*cos(<math>theta</math>)

<math>\mathbf{A}\cdot\mathbf{B}=\sum_{i=1}^n A_iB_i=A_1B_1+A_2B_2+\cdots+A_nB_n</math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

Cross Product: Ia x bI =IaI*IbI*sin(<math>theta</math>)

a x b= <a1, a2, a3> x <b1, b2, b3> = <a2*b3 - a3*b2> i - <a1*b3 - a3*b1> j + <a1*b2 - a2*b1> k

Multiplication between vectors and scalars: c<a1, a2, a3> = <c*a1, c*a2, c*a3>

===A Computational Model===

VIDLE is an interactive editor for VPython. It is commonly used in Physics to visualize 3D motion and preform repetitive calculations from fundamental principles. Codes on VIDLE contain a sequence of instructions for a computer to follow.
[[File:vectorcode.jpeg]]
In VIDLE code, arrow objects usually represent vector components. Arrows contain 3 parts: pos, axis, and color. Each part can be manipulated to achieve different results. The pos and axis of arrows are vectors, so you can multiply them by scalar quantities to scale them. Arrows are often used to represent relative position vectors, starting at position A and ending at position B. This is found the commonly use phrase "final minus initial" (B-A). In the above code, the relative position vector is of the tennis ball with respect to the baseball, so the arrow points from the baseball to the tennis ball. 3 arrows (vectors) can be used as components on a coordinate system for a vector. They can represent the position vector in each coordinate direction. You can reference the z component vector using the formula vectorname.z. same goes for y and z.

In Physics 2, you need to know that a vector always starts at a source and points to an observation location you want to get physical quantities(such as electric field, magnetic field, etc.) at. Also, you should be able to calculate the magnitude and direction of the vector in 3D space. Here is an example of a VPython model that computationally calculates such values. (The green arrow represents the position vector that starts from the source, which is a proton(red ball), to the arbitrary observation location.) (Click Run on the upper left corner in order to display the model.)
[https://trinket.io/embed/glowscript/e17d933a59?outputOnly=true 3D Vector]

==Examples==

===Simple===
Which of the following statements is correct? (Can be more than one)

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{a} + \overrightarrow{b}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{c} + \overrightarrow{b}</math>

4. <math>\overrightarrow{b} = \overrightarrow{c} + \overrightarrow{a}</math>

The answer is option number 2 and option number 4.

===Middling===
1. What is the magnitude of the vector C = A - B if A = <10, 5, 8> and B = <9, 4, 3>?

First we need to find the vector C:
A - B = <math> <(10-9), (5-4), (8-3)> = <1, 1, 5> = C</math>

<math>\sqrt{(1)^2 + 1^2 + 5^2} = 5.196</math>

2. What is the cross product of A = <1,2,3> and B = <9,4,5>?

Use the equation for cross product: a x b= <a1, a2, a3> x <b1, b2, b3> = <a2*b3 - a3*b2> i - <a1*b3 - a3*b1> j + <a1*b2 - a2*b1> k

A x B<math>=<2*5 - 3*4, 1*5 - 3*9, 1*4 - 2*9> = <-2, -22, -14></math>

===Difficult===
What is the unit vector in the direction of the vector <10, 5, 8>?
First you have to find the magnitude of the vector given:
<math>\sqrt{10^2 + (5)^2 + 8^2} = 13.74</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<10, 5, 8>}{13.74}</math>

= <.727, .364, .582>
Notice that the magnitude of the unit vector is equal to 1

==Connectedness==
1.Vectors will be used in many applications in most calculation based fields when movement and position are involved. Vectors can be two dimensional or three dimensional. Vectors are used to represent forces, fields, and momentum.

2.Vectors has been used in many application problems in engineering majors. In engineering applications, vectors are used to a lot of quantities which have both magnitude and direction. For example, in Biomedcial Engineering applications, vectors are used to represent the velocity of a flow to further calculate the flow rate and some other related quantities.

3. Vectors play a huge part in industry. For example, in process flow, vectors play a huge part in most calculations.

==History==
The discovery and use of vectors can date back to the ancient philosophers, Aristotle and Heron. The theory can also be found in the first article of Newtons Principia Mathematica. In the early 19th century Caspar Wessel, Jean Robert Argand, Carl Friedrich Gauss, and a few more depicted and worked with complex numbers as points on a 2D plane. in 1827, August Ferdinand published a book introducing line segments labelled with letters. he wrote about vectors without the name "vector".

In 1835, Giusto Bellavitis abstracted the basic idea of a vector while establishing the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent (meaning equal length). He found a relationship and created the first set of vectors.
Also in 1835 Hamilton founded "quaternions", which were 4D planes and equations with vectors.

William Rowan Hamilton devised the name "vector" as part of his system of quaternions consisting of three dimensional vectors.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today.

VPython was released by David Scherer in the year 2000. He came up with the idea after taking a physics class at Carnegie Mellon University. Previous programs only allowed for 2D modeling, so he took it upon himself to make something better. VPython, also known as Visual Python, allows for 3D modeling.
== See also ==

===External links===
Here is a link on more mathematical computations on vectors:
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

Here is a link on more computational work with vectors:
[http://vpython.org/contents/docs/vector.html]

===Further Reading===

Vector Analysis by Josiah Willard Gibbs

Introduction to Matrices and Vectors by Jacob T. Schwartz

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[http://mathinsight.org/vector_introduction]

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

Vectors

2017-04-09T18:22:54Z

Acasado3:

Written by Elizabeth Robelo

Improved by Aparajita Satapathy

'''Improved by Lichao Tang'''

Improved by Jimin Yoon

Improved by Sabrina Seibel

Claimed: Alex Casado Spring 2017

In physics, a vector is a quantity with a magnitude and a direction.

==The Main Idea==

A vector is a quantity with a magnitude and a direction. The magnitude of a vector is a scalar which represents the length of the vector. A vector is represented by an arrow. The orientation of the vector represents its direction. When a vector is drawn, the starting point is the tail and the ending point is called the head of the vector. In physics, a vector always starts at the source and directs to the observation location. Refer to the image below for a visual representation:

[[File:Mathinsight.png|300px|thumb|center|Visual Representation]]

We can also add and subtract vectors. To add two vectors you align them head to tail and make sure that the tails of both vectors coincide with each other. The new added vector is the connecting arrow starting from the tail of one to the head of the other vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the vector you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A vector can also be multiplied by a scalar. To multiply a vector by a scalar we can strech, compress or reverse the direction of a vector. If the scalar is between 0 and 1, the vector will be compresseed. If the scalar is greater than 1, the vector will get streched. If the scalar is a negative number, then the vector reverses its direction.

[[File:Ibguides.png|400px|thumb|center|Scalar Multiplication]]

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or (xi + yj - zk).
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Dot Product: Ia·bI = IaI*IbI*cos(<math>theta</math>)

<math>\mathbf{A}\cdot\mathbf{B}=\sum_{i=1}^n A_iB_i=A_1B_1+A_2B_2+\cdots+A_nB_n</math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

Cross Product: Ia x bI =IaI*IbI*sin(<math>theta</math>)

a x b= <a1, a2, a3> x <b1, b2, b3> = <a2*b3 - a3*b2> i - <a1*b3 - a3*b1> j + <a1*b2 - a2*b1> k

Multiplication between vectors and scalars: c<a1, a2, a3> = <c*a1, c*a2, c*a3>

===A Computational Model===

VIDLE is an interactive editor for VPython. It is commonly used in Physics to visualize 3D motion and preform repetitive calculations from fundamental principles. Codes on VIDLE contain a sequence of instructions for a computer to follow.
[[File:vectorcode.jpeg]]
In VIDLE code, arrow objects usually represent vector components. Arrows contain 3 parts: pos, axis, and color. Each part can be manipulated to achieve different results. The pos and axis of arrows are vectors, so you can multiply them by scalar quantities to scale them. Arrows are often used to represent relative position vectors, starting at position A and ending at position B. This is found the commonly use phrase "final minus initial" (B-A). In the above code, the relative position vector is of the tennis ball with respect to the baseball, so the arrow points from the baseball to the tennis ball. 3 arrows (vectors) can be used as components on a coordinate system for a vector. They can represent the position vector in each coordinate direction. You can reference the z component vector using the formula vectorname.z. same goes for y and z.

In Physics 2, you need to know that a vector always starts at a source and points to an observation location you want to get physical quantities(such as electric field, magnetic field, etc.) at. Also, you should be able to calculate the magnitude and direction of the vector in 3D space. Here is an example of a VPython model that computationally calculates such values. (The green arrow represents the position vector that starts from the source, which is a proton(red ball), to the arbitrary observation location.) (Click Run on the upper left corner in order to display the model.)
[https://trinket.io/embed/glowscript/e17d933a59?outputOnly=true 3D Vector]

==Examples==

===Simple===
Which of the following statements is correct? (Can be more than one)

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{a} + \overrightarrow{b}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{c} + \overrightarrow{b}</math>

4. <math>\overrightarrow{b} = \overrightarrow{c} + \overrightarrow{a}</math>

The answer is option number 2 and option number 4.

===Middling===
1. What is the magnitude of the vector C = A - B if A = <10, 5, 8> and B = <9, 4, 3>?

First we need to find the vector C:
A - B = <math> <(10-9), (5-4), (8-3)> = <1, 1, 5> = C</math>

<math>\sqrt{(1)^2 + 1^2 + 5^2} = 5.196</math>

2. What is the cross product of A = <1,2,3> and B = <9,4,5>?

Use the equation for cross product: a x b= <a1, a2, a3> x <b1, b2, b3> = <a2*b3 - a3*b2> i - <a1*b3 - a3*b1> j + <a1*b2 - a2*b1> k

A x B<math>=<2*5 - 3*4, 1*5 - 3*9, 1*4 - 2*9> = <-2, -22, -14></math>

===Difficult===
What is the unit vector in the direction of the vector <10, 5, 8>?
First you have to find the magnitude of the vector given:
<math>\sqrt{10^2 + (5)^2 + 8^2} = 13.74</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<10, 5, 8>}{13.74}</math>

= <.727, .364, .582>
Notice that the magnitude of the unit vector is equal to 1

==Connectedness==
1.Vectors will be used in many applications in most calculation based fields when movement and position are involved. Vectors can be two dimensional or three dimensional. Vectors are used to represent forces, fields, and momentum.

2.Vectors has been used in many application problems in engineering majors. In engineering applications, vectors are used to a lot of quantities which have both magnitude and direction. For example, in Biomedcial Engineering applications, vectors are used to represent the velocity of a flow to further calculate the flow rate and some other related quantities.

3. Vectors play a huge part in industry. For example, in process flow, vectors play a huge part in most calculations.

==History==
The discovery and use of vectors can date back to the ancient philosophers, Aristotle and Heron. The theory can also be found in the first article of Newtons Principia Mathematica. In the early 19th century Caspar Wessel, Jean Robert Argand, Carl Friedrich Gauss, and a few more depicted and worked with complex numbers as points on a 2D plane. in 1827, August Ferdinand published a book introducing line segments labelled with letters. he wrote about vectors without the name "vector".

In 1835, Giusto Bellavitis abstracted the basic idea of a vector while establishing the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent (meaning equal length). He found a relationship and created the first set of vectors.
Also in 1835 Hamilton founded "quaternions", which were 4D planes and equations with vectors.

William Rowan Hamilton devised the name "vector" as part of his system of quaternions consisting of three dimensional vectors.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today.

VPython was released by David Scherer in the year 2000. He came up with the idea after taking a physics class at Carnegie Mellon University. Previous programs only allowed for 2D modeling, so he took it upon himself to make something better. VPython, also known as Visual Python, allows for 3D modeling.
== See also ==

===External links===
Here is a link on more mathematical computations on vectors:
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

Here is a link on more computational work with vectos:
[http://vpython.org/contents/docs/vector.html]

===Further Reading===

Vector Analysis by Josiah Willard Gibbs

Introduction to Matrices and Vectors by Jacob T. Schwartz

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[http://mathinsight.org/vector_introduction]

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

Vectors

2017-04-09T18:19:35Z

Acasado3:

Written by Elizabeth Robelo

Improved by Aparajita Satapathy

'''Improved by Lichao Tang'''

Improved by Jimin Yoon

Improved by Sabrina Seibel

Claimed: Alex Casado Spring 2017

In physics, a vector is a quantity with a magnitude and a direction.

==The Main Idea==

A vector is a quantity with a magnitude and a direction. The magnitude of a vector is a scalar which represents the length of the vector. A vector is represented by an arrow. The orientation of the vector represents its direction. When a vector is drawn, the starting point is the tail and the ending point is called the head of the vector. In physics, a vector always starts at the source and directs to the observation location. Refer to the image below for a visual representation:

[[File:Mathinsight.png|300px|thumb|center|Visual Representation]]

We can also add and subtract vectors. To add two vectors you align them head to tail and make sure that the tails of both vectors coincide with each other. The new added vector is the connecting arrow starting from the tail of one to the head of the other vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the vector you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A vector can also be multiplied by a scalar. To multiply a vector by a scalar we can strech, compress or reverse the direction of a vector. If the scalar is between 0 and 1, the vector will be compresseed. If the scalar is greater than 1, the vector will get streched. If the scalar is a negative number, then the vector reverses its direction.

[[File:Ibguides.png|400px|thumb|center|Scalar Multiplication]]

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or (xi + yj - zk).
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Dot Product: Ia·bI = IaI*IbI*cos(<math>theta</math>)

<math>\mathbf{A}\cdot\mathbf{B}=\sum_{i=1}^n A_iB_i=A_1B_1+A_2B_2+\cdots+A_nB_n</math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

Cross Product: Ia x bI =IaI*IbI*sin(<math>theta</math>)

a x b= <a1, a2, a3> x <b1, b2, b3> = <a2*b3 - a3*b2> i - <a1*b3 - a3*b1> j + <a1*b2 - a2*b1> k

Multiplication between vectors and scalars: c<a1, a2, a3> = <c*a1, c*a2, c*a3>

===A Computational Model===

VIDLE is an interactive editor for VPython. It is commonly used in Physics to visualize 3D motion and preform repetitive calculations from fundamental principles. Codes on VIDLE contain a sequence of instructions for a computer to follow.
[[File:vectorcode.jpeg]]
In VIDLE code, arrow objects usually represent vector components. Arrows contain 3 parts: pos, axis, and color. Each part can be manipulated to achieve different results. The pos and axis of arrows are vectors, so you can multiply them by scalar quantities to scale them. Arrows are often used to represent relative position vectors, starting at position A and ending at position B. This is found the commonly use phrase "final minus initial" (B-A). In the above code, the relative position vector is of the tennis ball with respect to the baseball, so the arrow points from the baseball to the tennis ball. 3 arrows (vectors) can be used as components on a coordinate system for a vector. They can represent the position vector in each coordinate direction. You can reference the z component vector using the formula vectorname.z. same goes for y and z.

In Physics 2, you need to know that a vector always starts at a source and points to an observation location you want to get physical quantities(such as electric field, magnetic field, etc.) at. Also, you should be able to calculate the magnitude and direction of the vector in 3D space. Here is an example of a VPython model that computationally calculates such values. (The green arrow represents the position vector that starts from the source, which is a proton(red ball), to the arbitrary observation location.) (Click Run on the upper left corner in order to display the model.)
[https://trinket.io/embed/glowscript/e17d933a59?outputOnly=true 3D Vector]

==Examples==

===Simple===
Which of the following statements is correct? (Can be more than one)

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{a} + \overrightarrow{b}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{c} + \overrightarrow{b}</math>

4. <math>\overrightarrow{b} = \overrightarrow{c} + \overrightarrow{a}</math>

The answer is option number 2 and option number 4.

===Middling===
1. What is the magnitude of the vector C = A - B if A = <10, 5, 8> and B = <9, 4, 3>?

First we need to find the vector C:
A - B = <math> <(10-9), (5-4), (8-3)> = <1, 1, 5> = C</math>

<math>\sqrt{(1)^2 + 1^2 + 5^2} = 5.196</math>

2. What is the cross product of A = <1,2,3> and B = <9,4,5>?

Use the equation for cross product: a x b= <a1, a2, a3> x <b1, b2, b3> = <a2*b3 - a3*b2> i - <a1*b3 - a3*b1> j + <a1*b2 - a2*b1> k

A x B<math>=<2*5 - 3*4, 1*5 - 3*9, 1*4 - 2*9> = <-2, -22, -14></math>

===Difficult===
What is the unit vector in the direction of the vector <10, 5, 8>?
First you have to find the magnitude of the vector given:
<math>\sqrt{10^2 + (5)^2 + 8^2} = 13.74</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<10, 5, 8>}{13.74}</math>

= <.727, .364, .582>
Notice that the magnitude of the unit vector is equal to 1

==Connectedness==
1.Vectors will be used in many applications in physics. These can be two dimenstional or three dimensional vectors. Vectors are used to represent forces, fields and momentum.

2.Vectors has been used in many application problems in engineering majors. In engineering applications, vectors are used to a lot of quantities which have both magnitude and direction. For example in Biomedcial Engineering applications, vectors are used to represent the velocity of a flow to further calculate the flow rate and some other related quantities.

==History==
The discovery and use of vectors can date back to the ancient philosophers, Aristotle and Heron. The theory can also be found in the first article of Newtons Principia Mathematica. In the early 19th century Caspar Wessel, Jean Robert Argand, Carl Friedrich Gauss, and a few more depicted and worked with complex numbers as points on a 2D plane. in 1827, August Ferdinand published a book introducing line segments labelled with letters. he wrote about vectors without the name "vector".

In 1835, Giusto Bellavitis abstracted the basic idea of a vector while establishing the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent (meaning equal length). He found a relationship and created the first set of vectors.
Also in 1835 Hamilton founded "quaternions", which were 4D planes and equations with vectors.

William Rowan Hamilton devised the name "vector" as part of his system of quaternions consisting of three dimensional vectors.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today.

VPython was released by David Scherer in the year 2000. He came up with the idea after taking a physics class at Carnegie Mellon University. Previous programs only allowed for 2D modeling, so he took it upon himself to make something better. VPython, also known as Visual Python, allows for 3D modeling.
== See also ==

===External links===
Here is a link on more mathematical computations on vectors:
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

Here is a link on more computational work with vectos:
[http://vpython.org/contents/docs/vector.html]

===Further Reading===

Vector Analysis by Josiah Willard Gibbs

Introduction to Matrices and Vectors by Jacob T. Schwartz

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[http://mathinsight.org/vector_introduction]

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