Physics Book - User contributions [en]

Magnetic Field of a Toroid Using Ampere's Law

2015-12-06T02:41:44Z

Kmcgorrey3: /* External links */

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid since using the Biot-Savart law would be extremely difficult due to having to integrate over all the current elements in the toroid.
===Geometry of a Toroid===
[[File:Geometry Fig.JPG |frame|right|alt=Alt text| A toroid's geometry.]]
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===
[[File:Computation Fig.JPG | left|frame|none|alt=Alt text| A toroid's view from above.]]
First we start with solving the path integral from Ampere's law:
<br /> <div style="text-align: center;"><math>{\oint\,\vec{B}•d\vec{l} = μ_{0}∑I_{inside&ensp;path}}</math></div>

<br />The magnetic field, <math>{\vec{B}}</math>, is, due to the symmetry of a toroid, constant in magnitude and always tangential to the circular path (i.e. parallel to <math>{d\vec{l}}</math>). The path of a toroid is circular, so <math>{\oint\,d\vec{l}}</math> is equal to <math>{2πr}</math>. Therefore, the path integral of the magnetic field is equal to <math>{B2πr}</math>. The amount of current piercing the soap film (i.e. <math>{∑I_{inside&ensp;path}}</math>) is <math>NI</math>, where <math>N</math> is the number of piercings (i.e. turns in the coil) and <math>I</math> is the current. Ampere's law is now this:
<br /> <div style="text-align: center;"><math>{B2πr = μ_{0}NI}</math></div>
<br /> From this, we can solve for the magnetic field for a toroid:
<br /> <div style="text-align: center;"><math>{B = \frac{μ_{0}NI}{2πr}}</math></div>

<br />
==Examples==
===Simple===
[[File:Simple Example Fig.JPG | right|frame|none|alt=Alt text| Figure for the simple example.]]
A toroid frame is made out of plastic of small square cross section and tightly wrapped uniformly with 100 turns of wire, so that the magnetic field has essentially the same magnitude throughout the plastic (radius R of the curved part is much larger than cross section width w). With a current of 2 A and radius of 5 m, what is the magnetic field inside the plastic.
[[File:Simple Example Sol.JPG | center|frame|none|alt=Alt text| The solution for the simple example.]]

===Middling - Difficult===
[[File:Mid - Diff Example Fig.JPG | right|frame|none|alt=Alt text| Figure for the middle and difficult examples.]]
The toroid shown in the diagram has an inner radius of <math>R_{i}</math> and an outer radius of <math>R_{o}</math> and is centered at the origin in the diagram. The z-axis passes through the center of the doughnut hole. This toroid is wrapped with <math>N</math> loops of current <math>I</math> flowing up the outside surface of the toroid, radially inward, down the inner surface, and then radial outward. Assume that the magnetic field produced by this toroid has the form <math>\vec{B} = B(r,z)\hat{φ}</math> '''at every point in space''' where <math>r</math> is the perpendicular distance from the z-axis and <math>\hat{φ}</math> is a unit vector which "curls" around the z-axis, i.e., it is always tangent to any circle with rotational symmetry around the z-axis.

<br />
====Middling====
(a.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>r < R_{i}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
(b.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>r > R_{o}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.
[[File:Mid - Diff Example Sol ab.JPG | center|frame|none|alt=Alt text| The solution for the middle example.]]

<br />
====Difficult====
(c.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>R_{i} < r < R_{o}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
(d.) Consider a z-axis centered Amperian loop far above the toroid <math>z >> R_{o}</math>, with a radius <math>R_{i} < r < R_{o}</math> and use it to find the magnitude of the magnetic field far above the toroid.
[[File:Mid - Diff Example Sol cd.JPG | center|frame|none|alt=Alt text| The solution for the difficult example.]]

== See also ==

[[Ampere's Law]]
<br />
[[Magnetic Field of a Long Thick Wire Using Ampere's Law]]
<br />
[[Magnetic Field of Coaxial Cable Using Ampere's Law]]

===Further reading===
Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 2). Raleigh, North Carolina: Wiley.

===External links===
*http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/toroid.html
*http://www.phys.uri.edu/gerhard/PHY204/tsl242.pdf
*https://clas-pages.uncc.edu/phys2102/online-lectures/chapter-7-magnetism/7-3-amperes-law/example-magnetic-field-of-a-toroid/

==References==
Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 2). Raleigh, North Carolina: Wiley. Pg 888-889.
<br />
Fall 2014 Test 4 from Phys 2212.
[[Category:Maxwell's Equations]]

Magnetic Field of a Toroid Using Ampere's Law

2015-12-06T02:37:38Z

Kmcgorrey3: /* References */

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid since using the Biot-Savart law would be extremely difficult due to having to integrate over all the current elements in the toroid.
===Geometry of a Toroid===
[[File:Geometry Fig.JPG |frame|right|alt=Alt text| A toroid's geometry.]]
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===
[[File:Computation Fig.JPG | left|frame|none|alt=Alt text| A toroid's view from above.]]
First we start with solving the path integral from Ampere's law:
<br /> <div style="text-align: center;"><math>{\oint\,\vec{B}•d\vec{l} = μ_{0}∑I_{inside&ensp;path}}</math></div>

<br />The magnetic field, <math>{\vec{B}}</math>, is, due to the symmetry of a toroid, constant in magnitude and always tangential to the circular path (i.e. parallel to <math>{d\vec{l}}</math>). The path of a toroid is circular, so <math>{\oint\,d\vec{l}}</math> is equal to <math>{2πr}</math>. Therefore, the path integral of the magnetic field is equal to <math>{B2πr}</math>. The amount of current piercing the soap film (i.e. <math>{∑I_{inside&ensp;path}}</math>) is <math>NI</math>, where <math>N</math> is the number of piercings (i.e. turns in the coil) and <math>I</math> is the current. Ampere's law is now this:
<br /> <div style="text-align: center;"><math>{B2πr = μ_{0}NI}</math></div>
<br /> From this, we can solve for the magnetic field for a toroid:
<br /> <div style="text-align: center;"><math>{B = \frac{μ_{0}NI}{2πr}}</math></div>

<br />
==Examples==
===Simple===
[[File:Simple Example Fig.JPG | right|frame|none|alt=Alt text| Figure for the simple example.]]
A toroid frame is made out of plastic of small square cross section and tightly wrapped uniformly with 100 turns of wire, so that the magnetic field has essentially the same magnitude throughout the plastic (radius R of the curved part is much larger than cross section width w). With a current of 2 A and radius of 5 m, what is the magnetic field inside the plastic.
[[File:Simple Example Sol.JPG | center|frame|none|alt=Alt text| The solution for the simple example.]]

===Middling - Difficult===
[[File:Mid - Diff Example Fig.JPG | right|frame|none|alt=Alt text| Figure for the middle and difficult examples.]]
The toroid shown in the diagram has an inner radius of <math>R_{i}</math> and an outer radius of <math>R_{o}</math> and is centered at the origin in the diagram. The z-axis passes through the center of the doughnut hole. This toroid is wrapped with <math>N</math> loops of current <math>I</math> flowing up the outside surface of the toroid, radially inward, down the inner surface, and then radial outward. Assume that the magnetic field produced by this toroid has the form <math>\vec{B} = B(r,z)\hat{φ}</math> '''at every point in space''' where <math>r</math> is the perpendicular distance from the z-axis and <math>\hat{φ}</math> is a unit vector which "curls" around the z-axis, i.e., it is always tangent to any circle with rotational symmetry around the z-axis.

<br />
====Middling====
(a.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>r < R_{i}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
(b.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>r > R_{o}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.
[[File:Mid - Diff Example Sol ab.JPG | center|frame|none|alt=Alt text| The solution for the middle example.]]

<br />
====Difficult====
(c.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>R_{i} < r < R_{o}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
(d.) Consider a z-axis centered Amperian loop far above the toroid <math>z >> R_{o}</math>, with a radius <math>R_{i} < r < R_{o}</math> and use it to find the magnitude of the magnetic field far above the toroid.
[[File:Mid - Diff Example Sol cd.JPG | center|frame|none|alt=Alt text| The solution for the difficult example.]]

== See also ==

[[Ampere's Law]]
<br />
[[Magnetic Field of a Long Thick Wire Using Ampere's Law]]
<br />
[[Magnetic Field of Coaxial Cable Using Ampere's Law]]

===Further reading===
Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 2). Raleigh, North Carolina: Wiley.

===External links===

Internet resources on this topic

==References==
Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 2). Raleigh, North Carolina: Wiley. Pg 888-889.
<br />
Fall 2014 Test 4 from Phys 2212.
[[Category:Maxwell's Equations]]

Magnetic Field of a Toroid Using Ampere's Law

2015-12-06T02:36:23Z

Kmcgorrey3: /* See also */

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid since using the Biot-Savart law would be extremely difficult due to having to integrate over all the current elements in the toroid.
===Geometry of a Toroid===
[[File:Geometry Fig.JPG |frame|right|alt=Alt text| A toroid's geometry.]]
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===
[[File:Computation Fig.JPG | left|frame|none|alt=Alt text| A toroid's view from above.]]
First we start with solving the path integral from Ampere's law:
<br /> <div style="text-align: center;"><math>{\oint\,\vec{B}•d\vec{l} = μ_{0}∑I_{inside&ensp;path}}</math></div>

<br />The magnetic field, <math>{\vec{B}}</math>, is, due to the symmetry of a toroid, constant in magnitude and always tangential to the circular path (i.e. parallel to <math>{d\vec{l}}</math>). The path of a toroid is circular, so <math>{\oint\,d\vec{l}}</math> is equal to <math>{2πr}</math>. Therefore, the path integral of the magnetic field is equal to <math>{B2πr}</math>. The amount of current piercing the soap film (i.e. <math>{∑I_{inside&ensp;path}}</math>) is <math>NI</math>, where <math>N</math> is the number of piercings (i.e. turns in the coil) and <math>I</math> is the current. Ampere's law is now this:
<br /> <div style="text-align: center;"><math>{B2πr = μ_{0}NI}</math></div>
<br /> From this, we can solve for the magnetic field for a toroid:
<br /> <div style="text-align: center;"><math>{B = \frac{μ_{0}NI}{2πr}}</math></div>

<br />
==Examples==
===Simple===
[[File:Simple Example Fig.JPG | right|frame|none|alt=Alt text| Figure for the simple example.]]
A toroid frame is made out of plastic of small square cross section and tightly wrapped uniformly with 100 turns of wire, so that the magnetic field has essentially the same magnitude throughout the plastic (radius R of the curved part is much larger than cross section width w). With a current of 2 A and radius of 5 m, what is the magnetic field inside the plastic.
[[File:Simple Example Sol.JPG | center|frame|none|alt=Alt text| The solution for the simple example.]]

===Middling - Difficult===
[[File:Mid - Diff Example Fig.JPG | right|frame|none|alt=Alt text| Figure for the middle and difficult examples.]]
The toroid shown in the diagram has an inner radius of <math>R_{i}</math> and an outer radius of <math>R_{o}</math> and is centered at the origin in the diagram. The z-axis passes through the center of the doughnut hole. This toroid is wrapped with <math>N</math> loops of current <math>I</math> flowing up the outside surface of the toroid, radially inward, down the inner surface, and then radial outward. Assume that the magnetic field produced by this toroid has the form <math>\vec{B} = B(r,z)\hat{φ}</math> '''at every point in space''' where <math>r</math> is the perpendicular distance from the z-axis and <math>\hat{φ}</math> is a unit vector which "curls" around the z-axis, i.e., it is always tangent to any circle with rotational symmetry around the z-axis.

<br />
====Middling====
(a.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>r < R_{i}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
(b.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>r > R_{o}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.
[[File:Mid - Diff Example Sol ab.JPG | center|frame|none|alt=Alt text| The solution for the middle example.]]

<br />
====Difficult====
(c.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>R_{i} < r < R_{o}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
(d.) Consider a z-axis centered Amperian loop far above the toroid <math>z >> R_{o}</math>, with a radius <math>R_{i} < r < R_{o}</math> and use it to find the magnitude of the magnetic field far above the toroid.
[[File:Mid - Diff Example Sol cd.JPG | center|frame|none|alt=Alt text| The solution for the difficult example.]]

== See also ==

[[Ampere's Law]]
<br />
[[Magnetic Field of a Long Thick Wire Using Ampere's Law]]
<br />
[[Magnetic Field of Coaxial Cable Using Ampere's Law]]

===Further reading===
Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 2). Raleigh, North Carolina: Wiley.

===External links===

Internet resources on this topic

==References==

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

[[Category:Maxwell's Equations]]

Magnetic Field of a Toroid Using Ampere's Law

2015-12-06T02:28:29Z

Kmcgorrey3:

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid since using the Biot-Savart law would be extremely difficult due to having to integrate over all the current elements in the toroid.
===Geometry of a Toroid===
[[File:Geometry Fig.JPG |frame|right|alt=Alt text| A toroid's geometry.]]
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===
[[File:Computation Fig.JPG | left|frame|none|alt=Alt text| A toroid's view from above.]]
First we start with solving the path integral from Ampere's law:
<br /> <div style="text-align: center;"><math>{\oint\,\vec{B}•d\vec{l} = μ_{0}∑I_{inside&ensp;path}}</math></div>

<br />The magnetic field, <math>{\vec{B}}</math>, is, due to the symmetry of a toroid, constant in magnitude and always tangential to the circular path (i.e. parallel to <math>{d\vec{l}}</math>). The path of a toroid is circular, so <math>{\oint\,d\vec{l}}</math> is equal to <math>{2πr}</math>. Therefore, the path integral of the magnetic field is equal to <math>{B2πr}</math>. The amount of current piercing the soap film (i.e. <math>{∑I_{inside&ensp;path}}</math>) is <math>NI</math>, where <math>N</math> is the number of piercings (i.e. turns in the coil) and <math>I</math> is the current. Ampere's law is now this:
<br /> <div style="text-align: center;"><math>{B2πr = μ_{0}NI}</math></div>
<br /> From this, we can solve for the magnetic field for a toroid:
<br /> <div style="text-align: center;"><math>{B = \frac{μ_{0}NI}{2πr}}</math></div>

<br />
==Examples==
===Simple===
[[File:Simple Example Fig.JPG | right|frame|none|alt=Alt text| Figure for the simple example.]]
A toroid frame is made out of plastic of small square cross section and tightly wrapped uniformly with 100 turns of wire, so that the magnetic field has essentially the same magnitude throughout the plastic (radius R of the curved part is much larger than cross section width w). With a current of 2 A and radius of 5 m, what is the magnetic field inside the plastic.
[[File:Simple Example Sol.JPG | center|frame|none|alt=Alt text| The solution for the simple example.]]

===Middling - Difficult===
[[File:Mid - Diff Example Fig.JPG | right|frame|none|alt=Alt text| Figure for the middle and difficult examples.]]
The toroid shown in the diagram has an inner radius of <math>R_{i}</math> and an outer radius of <math>R_{o}</math> and is centered at the origin in the diagram. The z-axis passes through the center of the doughnut hole. This toroid is wrapped with <math>N</math> loops of current <math>I</math> flowing up the outside surface of the toroid, radially inward, down the inner surface, and then radial outward. Assume that the magnetic field produced by this toroid has the form <math>\vec{B} = B(r,z)\hat{φ}</math> '''at every point in space''' where <math>r</math> is the perpendicular distance from the z-axis and <math>\hat{φ}</math> is a unit vector which "curls" around the z-axis, i.e., it is always tangent to any circle with rotational symmetry around the z-axis.

<br />
====Middling====
(a.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>r < R_{i}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
(b.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>r > R_{o}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.
[[File:Mid - Diff Example Sol ab.JPG | center|frame|none|alt=Alt text| The solution for the middle example.]]

<br />
====Difficult====
(c.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>R_{i} < r < R_{o}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
(d.) Consider a z-axis centered Amperian loop far above the toroid <math>z >> R_{o}</math>, with a radius <math>R_{i} < r < R_{o}</math> and use it to find the magnitude of the magnetic field far above the toroid.
[[File:Mid - Diff Example Sol cd.JPG | center|frame|none|alt=Alt text| The solution for the difficult example.]]

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

[[Category:Maxwell's Equations]]

Magnetic Field of a Toroid Using Ampere's Law

2015-12-06T02:27:56Z

Kmcgorrey3:

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid since using the Biot-Savart law would be extremely difficult due to having to integrate over all the current elements in the toroid.
===Geometry of a Toroid===
[[File:Geometry Fig.JPG |frame|right|alt=Alt text| A toroid's geometry.]]
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===
[[File:Computation Fig.JPG | left|frame|none|alt=Alt text| A toroid's view from above.]]
First we start with solving the path integral from Ampere's law:
<br /> <div style="text-align: center;"><math>{\oint\,\vec{B}•d\vec{l} = μ_{0}∑I_{inside&ensp;path}}</math></div>

<br />The magnetic field, <math>{\vec{B}}</math>, is, due to the symmetry of a toroid, constant in magnitude and always tangential to the circular path (i.e. parallel to <math>{d\vec{l}}</math>). The path of a toroid is circular, so <math>{\oint\,d\vec{l}}</math> is equal to <math>{2πr}</math>. Therefore, the path integral of the magnetic field is equal to <math>{B2πr}</math>. The amount of current piercing the soap film (i.e. <math>{∑I_{inside&ensp;path}}</math>) is <math>NI</math>, where <math>N</math> is the number of piercings (i.e. turns in the coil) and <math>I</math> is the current. Ampere's law is now this:
<br /> <div style="text-align: center;"><math>{B2πr = μ_{0}NI}</math></div>
<br /> From this, we can solve for the magnetic field for a toroid:
<br /> <div style="text-align: center;"><math>{B = \frac{μ_{0}NI}{2πr}}</math></div>

<br />
==Examples==
===Simple===
[[File:Simple Example Fig.JPG | right|frame|none|alt=Alt text| Figure for the simple example.]]
A toroid frame is made out of plastic of small square cross section and tightly wrapped uniformly with 100 turns of wire, so that the magnetic field has essentially the same magnitude throughout the plastic (radius R of the curved part is much larger than cross section width w). With a current of 2 A and radius of 5 m, what is the magnetic field inside the plastic.
[[File:Simple Example Sol.JPG | center|frame|none|alt=Alt text| The solution for the simple example.]]

===Middling - Difficult===
[[File:Mid - Diff Example Fig.JPG | right|frame|none|alt=Alt text| Figure for the middle and difficult examples.]]
The toroid shown in the diagram has an inner radius of <math>R_{i}</math> and an outer radius of <math>R_{o}</math> and is centered at the origin in the diagram. The z-axis passes through the center of the doughnut hole. This toroid is wrapped with <math>N</math> loops of current <math>I</math> flowing up the outside surface of the toroid, radially inward, down the inner surface, and then radial outward. Assume that the magnetic field produced by this toroid has the form <math>\vec{B} = B(r,z)\hat{φ}</math> '''at every point in space''' where <math>r</math> is the perpendicular distance from the z-axis and <math>\hat{φ}</math> is a unit vector which "curls" around the z-axis, i.e., it is always tangent to any circle with rotational symmetry around the z-axis.

<br />
====Middling====
(a.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>r < R_{i}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
(b.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>r > R_{o}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.
[[File:Mid - Diff Example Sol ab.JPG | center|frame|none|alt=Alt text| The solution for the middle example.]]

<br />
====Difficult====
(c.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>R_{i} < r < R_{o}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
(d.) Consider a z-axis centered Amperian loop far above the toroid <math>z >> R_{o}</math>, with a radius <math>R_{i} < r < R_{o}</math> and use it to find the magnitude of the magnetic field far above the toroid.
[[File:Mid - Diff Example Sol cd.JPG | center|frame|none|alt=Alt text| The solution for the difficult example.]]

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

[[Category:Maxwell's Equations]]

Magnetic Field of a Toroid Using Ampere's Law

2015-12-06T02:19:37Z

Kmcgorrey3:

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid since using the Biot-Savart law would be extremely difficult due to having to integrate over all the current elements in the toroid.
===Geometry of a Toroid===
[[File:Geometry Fig.JPG | right|frame|none|alt=Alt text| A toroid's geometry.]]
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===
[[File:Computation Fig.JPG | left|frame|none|alt=Alt text| A toroid's view from above.]]
First we start with solving the path integral from Ampere's law:
<br /> <div style="text-align: center;"><math>{\oint\,\vec{B}•d\vec{l} = μ_{0}∑I_{inside&ensp;path}}</math></div>

<br />The magnetic field, <math>{\vec{B}}</math>, is, due to the symmetry of a toroid, constant in magnitude and always tangential to the circular path (i.e. parallel to <math>{d\vec{l}}</math>). The path of a toroid is circular, so <math>{\oint\,d\vec{l}}</math> is equal to <math>{2πr}</math>. Therefore, the path integral of the magnetic field is equal to <math>{B2πr}</math>. The amount of current piercing the soap film (i.e. <math>{∑I_{inside&ensp;path}}</math>) is <math>NI</math>, where <math>N</math> is the number of piercings (i.e. turns in the coil) and <math>I</math> is the current. Ampere's law is now this:
<br /> <div style="text-align: center;"><math>{B2πr = μ_{0}NI}</math></div>
<br /> From this, we can solve for the magnetic field for a toroid:
<br /> <div style="text-align: center;"><math>{B = \frac{μ_{0}NI}{2πr}}</math></div>

<br />
==Examples==
===Simple===
A toroid frame is made out of plastic of small square cross section and tightly wrapped uniformly with 100 turns of wire, so that the magnetic field has essentially the same magnitude throughout the plastic (radius R of the curved part is much larger than cross section width w). With a current of 2 A and radius of 5 m, what is the magnetic field inside the plastic.

===Middling - Difficult===
The toroid shown in the diagram has an inner radius of <math>R_{i}</math> and an outer radius of <math>R_{o}</math> and is centered at the origin in the diagram. The z-axis passes through the center of the doughnut hole. This toroid is wrapped with <math>N</math> loops of current <math>I</math> flowing up the outside surface of the toroid, radially inward, down the inner surface, and then radial outward. Assume that the magnetic field produced by this toroid has the form <math>\vec{B} = B(r,z)\hat{φ}</math> '''at every point in space''' where <math>r</math> is the perpendicular distance from the z-axis and <math>\hat{φ}</math> is a unit vector which "curls" around the z-axis, i.e., it is always tangent to any circle with rotational symmetry around the z-axis.

<br />
====Middling====
(a.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>r < R_{i}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
(b.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>r > R_{o}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
====Difficult====
(c.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>R_{i} < r < R_{o}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
(d.) Consider a z-axis centered Amperian loop far above the toroid <math>z >> R_{o}</math>, with a radius <math>R_{i} < r < R_{o}</math> and use it to find the magnitude of the magnetic field far above the toroid.

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

[[Category:Maxwell's Equations]]

Magnetic Field of a Toroid Using Ampere's Law

2015-12-06T02:18:50Z

Kmcgorrey3: /* Magnetic Field of a Toroid using Ampere's Law */

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid since using the Biot-Savart law would be extremely difficult due to having to integrate over all the current elements in the toroid.
===Geometry of a Toroid===
[[File:Geometry Fig.JPG | right|frame|none|alt=Alt text| A toroid's geometry.]]
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===
[[File:Computation Fig.JPG | left|frame|none|alt=Alt text| A toroid's view from above.]]
First we start with solving the path integral from Ampere's law:
<br /> <div style="text-align: center;"><math>{\oint\,\vec{B}•d\vec{l} = μ_{0}∑I_{inside&ensp;path}}</math></div>

<br />The magnetic field, <math>{\vec{B}}</math>, is, due to the symmetry of a toroid, constant in magnitude and always tangential to the circular path (i.e. parallel to <math>{d\vec{l}}</math>). The path of a toroid is circular, so <math>{\oint\,d\vec{l}}</math> is equal to <math>{2πr}</math>. Therefore, the path integral of the magnetic field is equal to <math>{B2πr}</math>. The amount of current piercing the soap film (i.e. <math>{∑I_{inside&ensp;path}}</math>) is <math>NI</math>, where <math>N</math> is the number of piercings (i.e. turns in the coil) and <math>I</math> is the current. Ampere's law is now this:
<br /> <div style="text-align: center;"><math>{B2πr = μ_{0}NI}</math></div>
<br /> From this, we can solve for the magnetic field for a toroid:
<br /> <div style="text-align: center;"><math>{B = \frac{μ_{0}NI}{2πr}}</math></div>

==Examples==
===Simple===
A toroid frame is made out of plastic of small square cross section and tightly wrapped uniformly with 100 turns of wire, so that the magnetic field has essentially the same magnitude throughout the plastic (radius R of the curved part is much larger than cross section width w). With a current of 2 A and radius of 5 m, what is the magnetic field inside the plastic.

===Middling - Difficult===
The toroid shown in the diagram has an inner radius of <math>R_{i}</math> and an outer radius of <math>R_{o}</math> and is centered at the origin in the diagram. The z-axis passes through the center of the doughnut hole. This toroid is wrapped with <math>N</math> loops of current <math>I</math> flowing up the outside surface of the toroid, radially inward, down the inner surface, and then radial outward. Assume that the magnetic field produced by this toroid has the form <math>\vec{B} = B(r,z)\hat{φ}</math> '''at every point in space''' where <math>r</math> is the perpendicular distance from the z-axis and <math>\hat{φ}</math> is a unit vector which "curls" around the z-axis, i.e., it is always tangent to any circle with rotational symmetry around the z-axis.

<br />
====Middling====
(a.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>r < R_{i}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
(b.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>r > R_{o}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
====Difficult====
(c.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>R_{i} < r < R_{o}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
(d.) Consider a z-axis centered Amperian loop far above the toroid <math>z >> R_{o}</math>, with a radius <math>R_{i} < r < R_{o}</math> and use it to find the magnitude of the magnetic field far above the toroid.

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

[[Category:Maxwell's Equations]]

Magnetic Field of a Toroid Using Ampere's Law

2015-12-06T02:15:39Z

Kmcgorrey3: /* Geometry of a Toroid */

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid since using the Biot-Savart law would be extremely difficult due to having to integrate over all the current elements in the toroid.
===Geometry of a Toroid===
[[File:Geometry Fig.JPG | right]]
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===

First we start with solving the path integral from Ampere's law:
<br /> <div style="text-align: center;"><math>{\oint\,\vec{B}•d\vec{l} = μ_{0}∑I_{inside&ensp;path}}</math></div>

<br />The magnetic field, <math>{\vec{B}}</math>, is, due to the symmetry of a toroid, constant in magnitude and always tangential to the circular path (i.e. parallel to <math>{d\vec{l}}</math>). The path of a toroid is circular, so <math>{\oint\,d\vec{l}}</math> is equal to <math>{2πr}</math>. Therefore, the path integral of the magnetic field is equal to <math>{B2πr}</math>. The amount of current piercing the soap film (i.e. <math>{∑I_{inside&ensp;path}}</math>) is <math>NI</math>, where <math>N</math> is the number of piercings (i.e. turns in the coil) and <math>I</math> is the current. Ampere's law is now this:
<br /> <div style="text-align: center;"><math>{B2πr = μ_{0}NI}</math></div>
<br /> From this, we can solve for the magnetic field for a toroid:
<br /> <div style="text-align: center;"><math>{B = \frac{μ_{0}NI}{2πr}}</math></div>

==Examples==
===Simple===
A toroid frame is made out of plastic of small square cross section and tightly wrapped uniformly with 100 turns of wire, so that the magnetic field has essentially the same magnitude throughout the plastic (radius R of the curved part is much larger than cross section width w). With a current of 2 A and radius of 5 m, what is the magnetic field inside the plastic.

===Middling - Difficult===
The toroid shown in the diagram has an inner radius of <math>R_{i}</math> and an outer radius of <math>R_{o}</math> and is centered at the origin in the diagram. The z-axis passes through the center of the doughnut hole. This toroid is wrapped with <math>N</math> loops of current <math>I</math> flowing up the outside surface of the toroid, radially inward, down the inner surface, and then radial outward. Assume that the magnetic field produced by this toroid has the form <math>\vec{B} = B(r,z)\hat{φ}</math> '''at every point in space''' where <math>r</math> is the perpendicular distance from the z-axis and <math>\hat{φ}</math> is a unit vector which "curls" around the z-axis, i.e., it is always tangent to any circle with rotational symmetry around the z-axis.

<br />
====Middling====
(a.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>r < R_{i}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
(b.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>r > R_{o}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
====Difficult====
(c.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>R_{i} < r < R_{o}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
(d.) Consider a z-axis centered Amperian loop far above the toroid <math>z >> R_{o}</math>, with a radius <math>R_{i} < r < R_{o}</math> and use it to find the magnitude of the magnetic field far above the toroid.

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

[[Category:Maxwell's Equations]]

File:Simple Example Sol.JPG

2015-12-06T02:11:48Z

Kmcgorrey3:

File:Simple Example Fig.JPG

2015-12-06T02:11:38Z

Kmcgorrey3:

File:Mid - Diff Example Sol cd.JPG

2015-12-06T02:11:26Z

Kmcgorrey3:

File:Mid - Diff Example Sol ab.JPG

2015-12-06T02:11:16Z

Kmcgorrey3:

File:Mid - Diff Example Fig.JPG

2015-12-06T02:11:03Z

Kmcgorrey3:

File:Geometry Fig.JPG

2015-12-06T02:10:49Z

Kmcgorrey3:

File:Computation Fig.JPG

2015-12-06T02:10:34Z

Kmcgorrey3:

Magnetic Field of a Toroid Using Ampere's Law

2015-12-06T01:52:51Z

Kmcgorrey3: /* Simple */

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid since using the Biot-Savart law would be extremely difficult due to having to integrate over all the current elements in the toroid.
===Geometry of a Toroid===
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===

First we start with solving the path integral from Ampere's law:
<br /> <div style="text-align: center;"><math>{\oint\,\vec{B}•d\vec{l} = μ_{0}∑I_{inside&ensp;path}}</math></div>

<br />The magnetic field, <math>{\vec{B}}</math>, is, due to the symmetry of a toroid, constant in magnitude and always tangential to the circular path (i.e. parallel to <math>{d\vec{l}}</math>). The path of a toroid is circular, so <math>{\oint\,d\vec{l}}</math> is equal to <math>{2πr}</math>. Therefore, the path integral of the magnetic field is equal to <math>{B2πr}</math>. The amount of current piercing the soap film (i.e. <math>{∑I_{inside&ensp;path}}</math>) is <math>NI</math>, where <math>N</math> is the number of piercings (i.e. turns in the coil) and <math>I</math> is the current. Ampere's law is now this:
<br /> <div style="text-align: center;"><math>{B2πr = μ_{0}NI}</math></div>
<br /> From this, we can solve for the magnetic field for a toroid:
<br /> <div style="text-align: center;"><math>{B = \frac{μ_{0}NI}{2πr}}</math></div>

==Examples==
===Simple===
A toroid frame is made out of plastic of small square cross section and tightly wrapped uniformly with 100 turns of wire, so that the magnetic field has essentially the same magnitude throughout the plastic (radius R of the curved part is much larger than cross section width w). With a current of 2 A and radius of 5 m, what is the magnetic field inside the plastic.

===Middling - Difficult===
The toroid shown in the diagram has an inner radius of <math>R_{i}</math> and an outer radius of <math>R_{o}</math> and is centered at the origin in the diagram. The z-axis passes through the center of the doughnut hole. This toroid is wrapped with <math>N</math> loops of current <math>I</math> flowing up the outside surface of the toroid, radially inward, down the inner surface, and then radial outward. Assume that the magnetic field produced by this toroid has the form <math>\vec{B} = B(r,z)\hat{φ}</math> '''at every point in space''' where <math>r</math> is the perpendicular distance from the z-axis and <math>\hat{φ}</math> is a unit vector which "curls" around the z-axis, i.e., it is always tangent to any circle with rotational symmetry around the z-axis.

<br />
====Middling====
(a.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>r < R_{i}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
(b.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>r > R_{o}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
====Difficult====
(c.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>R_{i} < r < R_{o}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
(d.) Consider a z-axis centered Amperian loop far above the toroid <math>z >> R_{o}</math>, with a radius <math>R_{i} < r < R_{o}</math> and use it to find the magnitude of the magnetic field far above the toroid.

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

[[Category:Maxwell's Equations]]

Magnetic Field of a Toroid Using Ampere's Law

2015-12-06T01:19:03Z

Kmcgorrey3: /* Examples */

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid since using the Biot-Savart law would be extremely difficult due to having to integrate over all the current elements in the toroid.
===Geometry of a Toroid===
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===

First we start with solving the path integral from Ampere's law:
<br /> <div style="text-align: center;"><math>{\oint\,\vec{B}•d\vec{l} = μ_{0}∑I_{inside&ensp;path}}</math></div>

<br />The magnetic field, <math>{\vec{B}}</math>, is, due to the symmetry of a toroid, constant in magnitude and always tangential to the circular path (i.e. parallel to <math>{d\vec{l}}</math>). The path of a toroid is circular, so <math>{\oint\,d\vec{l}}</math> is equal to <math>{2πr}</math>. Therefore, the path integral of the magnetic field is equal to <math>{B2πr}</math>. The amount of current piercing the soap film (i.e. <math>{∑I_{inside&ensp;path}}</math>) is <math>NI</math>, where <math>N</math> is the number of piercings (i.e. turns in the coil) and <math>I</math> is the current. Ampere's law is now this:
<br /> <div style="text-align: center;"><math>{B2πr = μ_{0}NI}</math></div>
<br /> From this, we can solve for the magnetic field for a toroid:
<br /> <div style="text-align: center;"><math>{B = \frac{μ_{0}NI}{2πr}}</math></div>

==Examples==
===Simple===
Use Ampere's law to calculate the magnetic field inside a toroid with 100 turns that has an inner radius of 5 m, an outer radius of 10 m, and carries a current of 2 A?

===Middling - Difficult===
The toroid shown in the diagram has an inner radius of <math>R_{i}</math> and an outer radius of <math>R_{o}</math> and is centered at the origin in the diagram. The z-axis passes through the center of the doughnut hole. This toroid is wrapped with <math>N</math> loops of current <math>I</math> flowing up the outside surface of the toroid, radially inward, down the inner surface, and then radial outward. Assume that the magnetic field produced by this toroid has the form <math>\vec{B} = B(r,z)\hat{φ}</math> '''at every point in space''' where <math>r</math> is the perpendicular distance from the z-axis and <math>\hat{φ}</math> is a unit vector which "curls" around the z-axis, i.e., it is always tangent to any circle with rotational symmetry around the z-axis.

<br />
====Middling====
(a.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>r < R_{i}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
(b.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>r > R_{o}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
====Difficult====
(c.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>R_{i} < r < R_{o}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
(d.) Consider a z-axis centered Amperian loop far above the toroid <math>z >> R_{o}</math>, with a radius <math>R_{i} < r < R_{o}</math> and use it to find the magnitude of the magnetic field far above the toroid.

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

[[Category:Maxwell's Equations]]

Magnetic Field of a Toroid Using Ampere's Law

2015-12-06T01:09:35Z

Kmcgorrey3:

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid since using the Biot-Savart law would be extremely difficult due to having to integrate over all the current elements in the toroid.
===Geometry of a Toroid===
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===

First we start with solving the path integral from Ampere's law:
<br /> <div style="text-align: center;"><math>{\oint\,\vec{B}•d\vec{l} = μ_{0}∑I_{inside&ensp;path}}</math></div>

<br />The magnetic field, <math>{\vec{B}}</math>, is, due to the symmetry of a toroid, constant in magnitude and always tangential to the circular path (i.e. parallel to <math>{d\vec{l}}</math>). The path of a toroid is circular, so <math>{\oint\,d\vec{l}}</math> is equal to <math>{2πr}</math>. Therefore, the path integral of the magnetic field is equal to <math>{B2πr}</math>. The amount of current piercing the soap film (i.e. <math>{∑I_{inside&ensp;path}}</math>) is <math>NI</math>, where <math>N</math> is the number of piercings (i.e. turns in the coil) and <math>I</math> is the current. Ampere's law is now this:
<br /> <div style="text-align: center;"><math>{B2πr = μ_{0}NI}</math></div>
<br /> From this, we can solve for the magnetic field for a toroid:
<br /> <div style="text-align: center;"><math>{B = \frac{μ_{0}NI}{2πr}}</math></div>

==Examples==
===Simple===
Use Ampere's law to calculate the magnetic field inside a toroid with 100 turns that has an inner radius of 5 m, an outer radius of 10 m, and carries a current of 2 A?

===Middling===
===Difficult===
The toroid shown in the diagram has an inner radius of <math>R_{i}</math> and an outer radius of <math>R_{o}</math> and is centered at the origin in the diagram. The z-axis passes through the center of the doughnut hole. This toroid is wrapped with <math>N</math> loops of current <math>I</math> flowing up the outside surface of the toroid, radially inward, down the inner surface, and then radial outward. Assume that the magnetic field produced by this toroid has the form <math>\vec{B} = B(r,z)\hat{φ}</math> '''at every point in space''' where <math>r</math> is the perpendicular distance from the z-axis and <math>\hat{φ}</math> is a unit vector which "curls" around the z-axis, i.e., it is always tangent to any circle with rotational symmetry around the z-axis.

<br />
(a.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>r < R_{i}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
(b.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>r > R_{o}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
(c.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>R_{i} < r < R_{o}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

<br />
(d.) Consider a z-axis centered Amperian loop far above the toroid <math>z >> R_{o}</math>, with a radius <math>R_{i} < r < R_{o}</math> and use it to find the magnitude of the magnetic field far above the toroid.

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

[[Category:Maxwell's Equations]]

Magnetic Field of a Toroid Using Ampere's Law

2015-12-06T00:52:54Z

Kmcgorrey3: /* Examples */

Magnetic Field of a Toroid Using Ampere's Law

2015-12-06T00:52:37Z

Kmcgorrey3: /* Simple */

Magnetic Field of a Toroid Using Ampere's Law

2015-12-06T00:04:02Z

Kmcgorrey3:

Magnetic Field of a Toroid Using Ampere's Law

2015-12-02T02:51:08Z

Kmcgorrey3: /* Magnetic Field of a Toroid using Ampere's Law */

Magnetic Field of a Toroid Using Ampere's Law

2015-12-02T02:45:11Z

Kmcgorrey3: /* A Mathematical Model */

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid.

===Geometry of a Toroid===
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===

First we start with solving the path integral from Ampere's law:
<br /> <div style="text-align: center;"><math>{\oint\,\vec{B}•d\vec{l} = μ_{0}∑I_{inside&ensp;path}}</math></div>

<br />The magnetic field, <math>{\vec{B}}</math>, is, due to the symmetry of a toroid, constant in magnitude and always tangential to the circular path (i.e. parallel to <math>{d\vec{l}}</math>). The path of a toroid is circular, so <math>{\oint\,d\vec{l}}</math> is equal to <math>{2πr}</math>. Therefore, the path integral of the magnetic field is equal to <math>{B2πr}</math>. The amount of current piercing the soap film (i.e. <math>{∑I_{inside&ensp;path}}</math>) is <math>NI</math>, where <math>N</math> is the number of piercings (i.e. turns in the coil) and <math>I</math> is the current. Ampere's law is now this:
<br /> <div style="text-align: center;"><math>{B2πr = μ_{0}NI}</math></div>
<br /> From this, we can solve for the magnetic field for a toroid:
<br /> <div style="text-align: center;"><math>{B = \frac{μ_{0}NI}{2πr}}</math></div>

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

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

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

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

==History==

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

[[Category:Maxwell's Equations]]

Magnetic Field of a Toroid Using Ampere's Law

2015-12-02T02:44:34Z

Kmcgorrey3: /* A Mathematical Model */

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid.

===Geometry of a Toroid===
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===

First we start with solving the path integral from Ampere's law:
<br /> <div style="text-align: center;"><math>{\oint\,\vec{B}•d\vec{l} = μ_{0}∑I_{inside&ensp;path}}</math></div>

<br />The magnetic field, <math>{\vec{B}}</math>, is, due to the symmetry of a toroid, constant in magnitude and always tangential to the circular path (i.e. parallel to <math>{d\vec{l}}</math>). The path of a toroid is circular, so <math>{\oint\,d\vec{l}}</math> is equal to <math>{2πr}</math>. Therefore, the path integral of the magnetic field is equal to <math>{B2πr}</math>. The amount of current piercing the soap film (i.e. <math>{∑I_{inside&ensp;path}}</math>) is <math>NI</math>, where <math>N</math> is the number of piercings (i.e. turns in the coil) and <math>I</math> is the current. Ampere's law is now this:
<br /> <div style="text-align: center;"><math>{B2πr = μ_{0}NI}</math></div>
<br /> From this, we can solve for the magnetic field:
<br /> <div style="text-align: center;"><math>{B = \frac{μ_{0}NI}{2πr}}</math></div>

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

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

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

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

==History==

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

[[Category:Maxwell's Equations]]

Magnetic Field of a Toroid Using Ampere's Law

2015-12-02T02:40:31Z

Kmcgorrey3: /* References */

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid.

===Geometry of a Toroid===
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===

First we start with solving the path integral from Ampere's law:
<br /> <div style="text-align: center;"><math>{\oint\,\vec{B}•d\vec{l} = μ_{0}∑I_{inside&ensp;path}}</math></div>

<br />The magnetic field, <math>{\vec{B}}</math>, is, due to the symmetry of a toroid, constant in magnitude and always tangential to the circular path (i.e. parallel to <math>{d\vec{l}}</math>). The path of a toroid is circular, so <math>{\oint\,d\vec{l}}</math> is equal to <math>{2πr}</math>. Therefore, the path integral of the magnetic field is equal to <math>{B2πr}</math>. The amount of current piercing the soap film (i.e. <math>{∑I_{inside&ensp;path}}</math>) is <math>NI</math>, where <math>N</math> is the number of piercings (i.e. turns in the coil) and <math>I</math> is the current. Ampere's law is now this:
<br /> <div style="text-align: center;"><math>{B2πr = μ_{0}NI}</math></div>

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

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

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

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

==History==

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

[[Category:Maxwell's Equations]]

Magnetic Field of a Toroid Using Ampere's Law

2015-12-02T02:39:49Z

Kmcgorrey3: /* References */

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid.

===Geometry of a Toroid===
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===

First we start with solving the path integral from Ampere's law:
<br /> <div style="text-align: center;"><math>{\oint\,\vec{B}•d\vec{l} = μ_{0}∑I_{inside&ensp;path}}</math></div>

<br />The magnetic field, <math>{\vec{B}}</math>, is, due to the symmetry of a toroid, constant in magnitude and always tangential to the circular path (i.e. parallel to <math>{d\vec{l}}</math>). The path of a toroid is circular, so <math>{\oint\,d\vec{l}}</math> is equal to <math>{2πr}</math>. Therefore, the path integral of the magnetic field is equal to <math>{B2πr}</math>. The amount of current piercing the soap film (i.e. <math>{∑I_{inside&ensp;path}}</math>) is <math>NI</math>, where <math>N</math> is the number of piercings (i.e. turns in the coil) and <math>I</math> is the current. Ampere's law is now this:
<br /> <div style="text-align: center;"><math>{B2πr = μ_{0}NI}</math></div>

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

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

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

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

==History==

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

[[Category:Maxwells's Equations]]

Magnetic Field of a Toroid Using Ampere's Law

2015-12-02T02:35:28Z

Kmcgorrey3: /* A Mathematical Model */

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid.

===Geometry of a Toroid===
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===

First we start with solving the path integral from Ampere's law:
<br /> <div style="text-align: center;"><math>{\oint\,\vec{B}•d\vec{l} = μ_{0}∑I_{inside&ensp;path}}</math></div>

<br />The magnetic field, <math>{\vec{B}}</math>, is, due to the symmetry of a toroid, constant in magnitude and always tangential to the circular path (i.e. parallel to <math>{d\vec{l}}</math>). The path of a toroid is circular, so <math>{\oint\,d\vec{l}}</math> is equal to <math>{2πr}</math>. Therefore, the path integral of the magnetic field is equal to <math>{B2πr}</math>. The amount of current piercing the soap film (i.e. <math>{∑I_{inside&ensp;path}}</math>) is <math>NI</math>, where <math>N</math> is the number of piercings (i.e. turns in the coil) and <math>I</math> is the current. Ampere's law is now this:
<br /> <div style="text-align: center;"><math>{B2πr = μ_{0}NI}</math></div>

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

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

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

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

==History==

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

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

Magnetic Field of a Toroid Using Ampere's Law

2015-12-02T02:29:42Z

Kmcgorrey3: /* A Mathematical Model */

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid.

===Geometry of a Toroid===
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===

First we start with solving the path integral from Ampere's law: [[File:Ampere's_Law.png]]

<br />The magnetic field, <math>{\vec{B}}</math>, is, due to the symmetry of a toroid, constant in magnitude and always tangential to the circular path (i.e. parallel to <math>{d\vec{l}}</math>). The path of a toroid is circular, so <math>{\oint\,d\vec{l}}</math> is equal to <math>{2πr}</math>. Therefore, the path integral of the magnetic field is equal to <math>{B2πr}</math>. The amount of current piercing the soap film (i.e. <math>{∑I_{inside&ensp;path}}</math>) is <math>NI</math>, where <math>N</math> is the number of piercings (i.e. turns in the coil) and <math>I</math> is the current.

<br />Ampere's law is now this:
<br /> <div style="text-align: center;"><math>{B2πr = μ_{0}NI}</math></div>

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

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

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

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

==History==

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

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

Magnetic Field of a Toroid Using Ampere's Law

2015-12-02T02:24:47Z

Kmcgorrey3: /* A Mathematical Model */

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid.

===Geometry of a Toroid===
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===

First we start with solving the path integral from Ampere's Law: [[File:Ampere's_Law.png]]

<br />The magnetic field, <math>{\vec{B}}</math>, is, due to the symmetry of a toroid, constant in magnitude and always tangential to the circular path (i.e. parallel to <math>{d\vec{l}}</math>). The path of a toroid is circular, so <math>{\oint\,d\vec{l}}</math> is equal to <math>{2πr}</math>. Therefore, the path integral of the magnetic field is equal to <math>{B2πr}</math>. The amount of current piercing the soap film (i.e. <math>{∑I_{inside&ensp;path}}</math>) is <math>NI</math>, where <math>N</math> is the number of piercings (i.e. turns in the coil) and <math>I</math> is the current.

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

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

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

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

==History==

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

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

Magnetic Field of a Toroid Using Ampere's Law

2015-12-02T02:13:33Z

Kmcgorrey3: /* A Mathematical Model */

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid.

===Geometry of a Toroid===
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===

First we start with solving the path integral from Ampere's Law: [[File:Ampere's_Law.png]]

<br />The magnetic field, <math>{\vec{B}}</math>, is, due to the symmetry of a toroid, constant in magnitude and always tangential to the circular path (i.e. parallel to <math>{d\vec{l}}</math>). The path of a toroid is circular, so <math>{\oint\,d\vec{l}}</math> is equal to <math>{2πr}</math>. Therefore, the path integral of the magnetic field is equal to <math>{B2πr}</math>.

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

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

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

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

==History==

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

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

Magnetic Field of a Toroid Using Ampere's Law

2015-12-02T02:13:20Z

Kmcgorrey3: /* A Mathematical Model */

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid.

===Geometry of a Toroid===
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===

First we start with solving the path integral from Ampere's Law: [[File:Ampere's_Law.png]]

<br />The magnetic field, <math>{\vec{B}}</math>, is, due to the symmetry of a toroid, constant in magnitude and always tangential to the circular path (i.e. parallel to <math>{d\vec{l}}</math>). The path of a toroid is circular, so <math>{\oint\,d\vec{l}}</math> is equal to <math>{2πr}</math>. Therefore, the path integral of the magnetic field is equal to <math>{{B2πr}</math>.

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

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

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

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

==History==

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

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

Magnetic Field of a Toroid Using Ampere's Law

2015-12-02T02:11:01Z

Kmcgorrey3: /* A Mathematical Model */

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid.

===Geometry of a Toroid===
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===

First we start with solving the path integral from Ampere's Law: [[File:Ampere's_Law.png]]

<br />The magnetic field, <math>{\vec{B}}</math>, is, due to the symmetry of a toroid, constant in magnitude and always tangential to the circular path (i.e. parallel to <math>{d\vec{l}}</math>). The path of a toroid is circular, so <math>{\oint\,d\vec{l}}</math> is equal to <math>{2πr}</math>.

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

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

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

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

==History==

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

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

Magnetic Field of a Toroid Using Ampere's Law

2015-12-02T01:45:53Z

Kmcgorrey3: /* A Mathematical Model */

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid.

===Geometry of a Toroid===
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===

First we start with solving the path integral from Ampere's Law. For example <math>{\vec{B}}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings.

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

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

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

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

==History==

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

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

Magnetic Field of a Toroid Using Ampere's Law

2015-12-02T01:44:44Z

Kmcgorrey3: /* A Mathematical Model */

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid.

===Geometry of a Toroid===
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===

First we start with solving the path integral from Ampere's Law. For example <math>{\frac{d\vec{p}}{dt}}_{system}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings.

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

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

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

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

==History==

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

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

Magnetic Field of a Toroid Using Ampere's Law

2015-12-02T01:42:17Z

Kmcgorrey3: Undo revision 7266 by Kmcgorrey3 (talk)

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid.

===Geometry of a Toroid===
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===

First we start with solving the path integral from Ampere's Law. For example <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings.

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

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

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

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

==History==

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

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

Magnetic Field of a Toroid Using Ampere's Law

2015-12-02T01:41:55Z

Kmcgorrey3: /* A Mathematical Model */

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid.

===Geometry of a Toroid===
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===

First we start with solving the path integral from Ampere's Law. For example <math>{\vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings.

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

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

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

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

==History==

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

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

Magnetic Field of a Toroid Using Ampere's Law

2015-12-02T01:41:06Z

Kmcgorrey3: Undo revision 7260 by Kmcgorrey3 (talk)

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid.

===Geometry of a Toroid===
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===

First we start with solving the path integral from Ampere's Law. For example <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings.

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

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

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

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

==History==

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

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

Magnetic Field of a Toroid Using Ampere's Law

2015-12-02T01:39:37Z

Kmcgorrey3: /* A Mathematical Model */

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid.

===Geometry of a Toroid===
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===

First we start with solving the path integral from Ampere's Law. <math>\vec{B}<math>

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

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

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

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

==History==

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

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

Magnetic Field of a Toroid Using Ampere's Law

2015-12-02T01:04:41Z

Kmcgorrey3: /* Magnetic Field of a Toroid using Ampere's Law */

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid.

===Geometry of a Toroid===
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===

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

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

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

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

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

==History==

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

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

Magnetic Field of a Toroid Using Ampere's Law

2015-12-02T01:00:23Z

Kmcgorrey3: /* Geometry of a Toroid */

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===Geometry of a Toroid===
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===

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

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

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

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

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

==History==

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

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

Magnetic Field of a Toroid Using Ampere's Law

2015-12-02T00:42:45Z

Kmcgorrey3:

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===Geometry of a Toroid===

===A Mathematical Model===

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

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

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

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

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

==History==

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

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

Magnetic Field of a Toroid Using Ampere's Law

2015-12-02T00:37:36Z

Kmcgorrey3:

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==The Main Idea==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Mathematical Model===

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

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

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

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

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

==History==

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

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

Magnetic Field of a Toroid Using Ampere's Law

2015-12-02T00:30:04Z

Kmcgorrey3:

Claimed by Kevin McGorrey

Short Description of Topic

==The Main Idea==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Mathematical Model===

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

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

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

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

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

==History==

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

== See also ==

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

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

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

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

Main Page

2015-12-02T00:27:45Z

Kmcgorrey3:

__NOTOC__
Welcome to the Georgia Tech Wiki for Intro Physics. This resources was created so that students can contribute and curate content to help those with limited or no access to a textbook. When reading this website, please correct any errors you may come across. If you read something that isn't clear, please consider revising it!

Looking to make a contribution?
#Pick a specific topic from intro physics
#Add that topic, as a link to a new page, under the appropriate category listed below by editing this page.
#Copy and paste the default [[Template]] into your new page and start editing.

Please remember that this is not a textbook and you are not limited to expressing your ideas with only text and equations. Whenever possible embed: pictures, videos, diagrams, simulations, computational models (e.g. Glowscript), and whatever content you think makes learning physics easier for other students.

== Source Material ==
All of the content added to this resource must be in the public domain or similar free resource. If you are unsure about a source, contact the original author for permission. That said, there is a surprisingly large amount of introductory physics content scattered across the web. Here is an incomplete list of intro physics resources (please update as needed).
* A physics resource written by experts for an expert audience [https://en.wikipedia.org/wiki/Portal:Physics Physics Portal]
* A wiki book on modern physics [https://en.wikibooks.org/wiki/Modern_Physics Modern Physics Wiki]
* The MIT open courseware for intro physics [http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MITOCW Wiki]
* An online concept map of intro physics [http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics]
* Interactive physics simulations [https://phet.colorado.edu/en/simulations/category/physics PhET]
* OpenStax algebra based intro physics textbook [https://openstaxcollege.org/textbooks/college-physics College Physics]
* The Open Source Physics project is a collection of online physics resources [http://www.opensourcephysics.org/ OSP]
* A resource guide compiled by the [http://www.aapt.org/ AAPT] for educators [http://www.compadre.org/ ComPADRE]

== Organizing Categories ==
These are the broad, overarching categories, that we cover in two semester of introductory physics. You can add subcategories or make a new category as needed. A single topic should direct readers to a page in one of these catagories.

<div class="toccolours mw-collapsible mw-collapsed">
===Interactions===
<div class="mw-collapsible-content">
*[[Kinds of Matter]]
**[[Ball and Spring Model of Matter]]
*[[Detecting Interactions]]
*[[Fundamental Interactions]]
*[[System & Surroundings]]
*[[Newton's First Law of Motion]]
*[[Newton's Second Law of Motion]]
*[[Newton's Third Law of Motion]]
*[[Gravitational Force]]
*[[Electric Force]]
*[[Conservation of Charge]]
*[[Terminal Speed]]
*[[Simple Harmonic Motion]]
*[[Speed and Velocity]]
*[[Electric Polarization]]
*[[Perpetual Freefall (Orbit)]]
*[[2-Dimensional Motion]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

===Theory===
<div class="mw-collapsible-content">
*[[Einstein's Theory of Special Relativity]]
*[[Quantum Theory]]
*[[Big Bang Theory]]
*[[Maxwell's Electromagnetic Theory]]
*[[Atomic Theory]]
*[[Wave-Particle Duality]]
*[[String Theory]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

===Notable Scientists===
<div class="mw-collapsible-content">
*[[Christian Doppler]]
*[[Albert Einstein]]
*[[Ernest Rutherford]]
*[[Joseph Henry]]
*[[Michael Faraday]]
*[[J.J. Thomson]]
*[[James Maxwell]]
*[[Robert Hooke]]
*[[Carl Friedrich Gauss]]
*[[Nikola Tesla]]
*[[Andre Marie Ampere]]
*[[Sir Isaac Newton]]
*[[J. Robert Oppenheimer]]
*[[Oliver Heaviside]]
*[[Rosalind Franklin]]
*[[Erwin Schrödinger]]
*[[Enrico Fermi]]
*[[Robert J. Van de Graaff]]
*[[Charles de Coulomb]]
*[[Hans Christian Ørsted]]
*[[Philo Farnsworth]]
*[[Niels Bohr]]
*[[Georg Ohm]]
*[[Galileo Galilei]]
*[[Gustav Kirchhoff]]
*[[Max Planck]]
*[[Heinrich Hertz]]
*[[Edwin Hall]]
*[[James Watt]]
*[[Count Alessandro Volta]]
*[[Josiah Willard Gibbs]]
*[[Richard Phillips Feynman]]
*[[Sir David Brewster]]
*[[Daniel Bernoulli]]
*[[William Thomson]]
*[[Leonhard Euler]]
*[[Robert Fox Bacher]]
*[[Stephen Hawking]]
*[[Amedeo Avogadro]]
*[[Wilhelm Conrad Roentgen]]
*[[Pierre Laplace]]
*[[Thomas Edison]]
*[[Hendrik Lorentz]]
*[[Jean-Baptiste Biot]]
*[[Lise Meitner]]
*[[Lisa Randall]]
*[[Felix Savart]]
*[[Heinrich Lenz]]
*[[Max Born]]
*[[Archimedes]]
*[[Jean Baptiste Biot]]
*[[Carl Sagan]]
*[[Eugene Wigner]]
*[[Marie Curie]]
*[[Pierre Curie]]
*[[Werner Heisenberg]]
*[[Johannes Diderik van der Waals]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

===Properties of Matter===
<div class="mw-collapsible-content">
*[[Mass]]
*[[Velocity]]
*[[Relative Velocity]]
*[[Density]]
*[[Charge]]
*[[Spin]]
*[[SI Units]]
*[[Heat Capacity]]
*[[Specific Heat]]
*[[Wavelength]]
*[[Conductivity]]
*[[Malleability]]
*[[Weight]]
*[[Boiling Point]]
*[[Melting Point]]
*[[Higgs Boson]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

===Contact Interactions===
<div class="mw-collapsible-content">
* [[Young's Modulus]]
* [[Friction]]
* [[Tension]]
* [[Hooke's Law]]
*[[Centripetal Force and Curving Motion]]
*[[Compression or Normal Force]]
* [[Length and Stiffness of an Interatomic Bond]]
* [[Speed of Sound in a Solid]]
* [[Iterative Prediction of Spring-Mass System]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

===Momentum===
<div class="mw-collapsible-content">
* [[Vectors]]
* [[Kinematics]]
* [[Conservation of Momentum]]
* [[Predicting Change in multiple dimensions]]
* [[Momentum Principle]]
* [[Impulse Momentum]]
* [[Curving Motion]]
* [[Multi-particle Analysis of Momentum]]
* [[Iterative Prediction]]
* [[Newton's Laws and Linear Momentum]]
* [[Net Force]]
* [[Center of Mass]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

===Angular Momentum===
<div class="mw-collapsible-content">
* [[The Moments of Inertia]]
* [[Moment of Inertia for a ring]]
* [[Rotation]]
* [[Torque]]
* [[Systems with Zero Torque]]
* [[Systems with Nonzero Torque]]
* [[Right Hand Rule]]
* [[Angular Velocity]]
* [[Predicting the Position of a Rotating System]]
* [[Translational Angular Momentum]]
* [[The Angular Momentum Principle]]
* [[Rotational Angular Momentum]]
* [[Total Angular Momentum]]
* [[Gyroscopes]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

===Energy===
<div class="mw-collapsible-content">
*[[The Photoelectric Effect]]
*[[Photons]]
*[[The Energy Principle]]
*[[Predicting Change]]
*[[Rest Mass Energy]]
*[[Kinetic Energy]]
*[[Potential Energy]]
*[[Work]]
*[[Thermal Energy]]
*[[Conservation of Energy]]
*[[Electric Potential]]
*[[Energy Transfer due to a Temperature Difference]]
*[[Gravitational Potential Energy]]
*[[Point Particle Systems]]
*[[Real Systems]]
*[[Spring Potential Energy]]
**[[Ball and Spring Model]]
*[[Internal Energy]]
**[[Potential Energy of a Pair of Neutral Atoms]]
*[[Translational, Rotational and Vibrational Energy]]
*[[Franck-Hertz Experiment]]
*[[Power]]
*[[Energy Graphs]]
*[[Air Resistance]]
*[[Electronic Energy Levels]]
*[[Second Law of Thermodynamics and Entropy]]
*[[Specific Heat Capacity]]
*[[Electronic Energy Levels and Photons]]
*[[Energy Density]]
*[[Bohr Model]]
*[[Quantized energy levels]]
*[[Path Independence of Electric Potential]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

===Collisions===
<div class="mw-collapsible-content">
*[[Collisions]]
*[[Maximally Inelastic Collision]]
*[[Elastic Collisions]]
*[[Inelastic Collisions]]
*[[Head-on Collision of Equal Masses]]
*[[Head-on Collision of Unequal Masses]]
*[[Rutherford Experiment and Atomic Collisions]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

===Fields===
<div class="mw-collapsible-content">
* [[Electric Field]] of a
** [[Point Charge]]
** [[Electric Dipole]]
** [[Capacitor]]
** [[Charged Rod]]
** [[Charged Ring]]
** [[Charged Disk]]
** [[Charged Spherical Shell]]
** [[Charged Cylinder]]
**[[A Solid Sphere Charged Throughout Its Volume]]
*[[Electric Potential]]
**[[Potential Difference Path Independence]]
**[[Potential Difference in a Uniform Field]]
**[[Potential Difference of point charge in a non-Uniform Field]]
**[[Sign of Potential Difference]]
**[[Potential Difference in an Insulator]]
**[[Energy Density and Electric Field]]
** [[Systems of Charged Objects]]
*[[Electric Force]]
*[[Polarization]]
*[[Charge Motion in Metals]]
*[[Charge Transfer]]
*[[Magnetic Field]]
**[[Right-Hand Rule]]
**[[Direction of Magnetic Field]]
**[[Magnetic Field of a Long Straight Wire]]
**[[Magnetic Field of a Loop]]
**[[Magnetic Field of a Solenoid]]
**[[Bar Magnet]]
**[[Magnetic Dipole Moment]]
**[[Magnetic Force]]
**[[Hall Effect]]
**[[Lorentz Force]]
**[[Biot-Savart Law]]
**[[Biot-Savart Law for Currents]]
**[[Integration Techniques for Magnetic Field]]
**[[Sparks in Air]]
**[[Motional Emf]]
**[[Detecting a Magnetic Field]]
**[[Moving Point Charge]]
**[[Non-Coulomb Electric Field]]
**[[Motors and Generators]]
**[[Solenoid Applications]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

===Simple Circuits===
<div class="mw-collapsible-content">
*[[Components]]
*[[Steady State]]
*[[Non Steady State]]
*[[Charging and Discharging a Capacitor]]
*[[Thin and Thick Wires]]
*[[Node Rule]]
*[[Loop Rule]]
*[[Electrical Resistance]]
*[[Power in a circuit]]
*[[Ammeters,Voltmeters,Ohmmeters]]
*[[Current]]
**[[AC]]
*[[Ohm's Law]]
*[[Series Circuits]]
*[[Parallel Circuits]]
*[[RC]]
*[[Current in a RC circuit]]
*[[Circular Loop of Wire]]
*[[RL Circuit]]
*[[LC Circuit]]
*[[Surface Charge Distributions]]
*[[Feedback]]
*[[Transformers]]
*[[Resistors and Conductivity]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

===Maxwell's Equations===
<div class="mw-collapsible-content">
*[[Gauss's Flux Theorem]]
**[[Electric Fields]]
**[[Magnetic Fields]]
*[[Ampere's Law]]
**[[Magnetic Field of Coaxial Cable Using Ampere's Law]]
**[[Magnetic Field of a Solenoid Using Ampere's Law]]
**[[Magnetic Field of a Toroid Using Ampere's Law]]
*[[Faraday's Law]]
**[[Curly Electric Fields]]
**[[Inductance]]
***[[Transformers]]
***[[Energy Density]]
**[[Lenz's Law]]
***[[Lenz Effect and the Jumping Ring]]
**[[Motional Emf using Faraday's Law]]
*[[Ampere-Maxwell Law]]
*[[Superconductors]]
**[[Meissner effect]]
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</div>

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===Radiation===
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*[[Producing a Radiative Electric Field]]
*[[Sinusoidal Electromagnetic Radiaton]]
*[[Lenses]]
*[[Energy and Momentum Analysis in Radiation]]
*[[Electromagnetic Propagation]]
**[[Wavelength and Frequency]]
*[[Snell's Law]]
*[[Light Propagation Through a Medium]]
*[[Light Scaterring: Why is the Sky Blue]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

===Sound===
<div class="mw-collapsible-content">
*[[Doppler Effect]]
*[[Nature, Behavior, and Properties of Sound]]
*[[Resonance]]
*[[Sound Barrier]]
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</div>
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===Waves===
<div class="mw-collapsible-content">
*[[Multisource Interference: Diffraction]]
*[[Gravitational waves]]
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*[[blahb]]
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<div class="toccolours mw-collapsible mw-collapsed">

===Real Life Applications of Electromagnetic Principles===
<div class="mw-collapsible-content">
*[[Electromagnetic Junkyard Cranes]]
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</div>

== Resources ==
* Commonly used wiki commands [https://en.wikipedia.org/wiki/Help:Cheatsheet Wiki Cheatsheet]
* A guide to representing equations in math mode [https://en.wikipedia.org/wiki/Help:Displaying_a_formula Wiki Math Mode]
* A page to keep track of all the physics [[Constants]]
* An overview of [[VPython]]

Magnetic Field of a Toroid Using Ampere's Law

2015-12-02T00:26:30Z

Kmcgorrey3: Created page with "Claimed by Kevin McGorrey"

Claimed by Kevin McGorrey