Ampere-Maxwell Law

2015-11-30T23:20:46Z

Mrivero7: /* Conceptual Question */

Claimed by Maria Rivero

== Ampere-Maxwell Law==

James Maxwell discovered that a time-varying electric field could be accompanied by a magnetic field. He thought of this after Faraday discovered that a time-varying magnetic field was accompanied by an electric field.

The time rate that is used for this equation is given by the derivative of the electric flux with respect to time. Because The derivative of the flux gives current over epsilon nod, the derivative of the electric flux times epsilon nod will also have units of amperes.

[[File: Ampere-Maxwell.png]]

Where
B is the magnetic field
dl is the change in path
The sum of I is the sum of the charges inside the path.

==Example==

Pick a closed, rectangular path in the xz plane with a height h and a width w. Calculate the speed v of the slab.

[[File: AMexample.png]]

At a time change in t, we can calculate the area as the height (h) times the speed over the change in time. ⩟ A = v (⩟ t) h
Because the electric field is constant in this region we can also calculate the change in electric flux over time as Evh(⩟ t) /(⩟ t) which is the same as Evh
Calculating the path integral for the magnetic field we get that ∮B . dl = Bh cos 0 = Bh
An important thing to notice is that there is no current I, so, using the Ampere-Maxwell Law we can see that

Bh = μ. [I+ ε. (vEh)]

but since there is no current,

B = μ. ε. (vE)

From Faraday's law we get that the emf equals the rate of change of the magnetic flux: Eh = Bvh

Substituting E = Bv into our previous equation we get that

B = μ. ε. (v(vB))

Solving for v we get that:

[[File: AMvelocity.png]]

From the picture we see that the speed of light relates the a time varying electric and magnetic field.

==What this implies==

In the example above we saw that the speed of light relates the a time varying electric and magnetic field. This translates to the fact that an electromagnetic wave propagates at the speed of light.

[[File: AMwave.png]]

==Conceptual Question==
[[File: AMconceptual.png|500px|Image: 500 pixels]]

The plot above shows a side and a top view of a capacitor with charge Q with electric and magnetic fields E and B at time t. The charge Q is:

1) Increasing in time

2) Decreasing in time

3) Constant in time

4) I don't know

Answer:

The unit vector N points out of the plane. Given that the magnetic field curls clockwise, the electric flux would be positive and decreasing. Hence E is decreasing. Thus Q must be decreasing, since E is proportional to Q.

Consider a circular capacitor, with an Amperian circular loop (radius r) in the plane midway between the plates. When the capacitor is charging, the line integral of the magnetic field around the circle (in direction shown) is

==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==

http://bulldog2.redlands.edu/fac/eric_hill/Phys232/Lectures/Ch%2023%20lect%201.pdf
http://ocw.mit.edu/high-school/physics/exam-prep/electromagnetism/maxwells-equations/
http://ocw.mit.edu/courses/physics/8-02-physics-ii-electricity-and-magnetism-spring-2007/readings/summary_w13d1.pdf
http://www.schoolphysics.co.uk/age16-19/Wave%20properties/Wave%20properties/text/Electromagnetic_radiation/index.html

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

Ampere-Maxwell Law

2015-11-30T23:16:28Z

Mrivero7: /* Conceptual Question */

Claimed by Maria Rivero

== Ampere-Maxwell Law==

James Maxwell discovered that a time-varying electric field could be accompanied by a magnetic field. He thought of this after Faraday discovered that a time-varying magnetic field was accompanied by an electric field.

The time rate that is used for this equation is given by the derivative of the electric flux with respect to time. Because The derivative of the flux gives current over epsilon nod, the derivative of the electric flux times epsilon nod will also have units of amperes.

[[File: Ampere-Maxwell.png]]

Where
B is the magnetic field
dl is the change in path
The sum of I is the sum of the charges inside the path.

==Example==

Pick a closed, rectangular path in the xz plane with a height h and a width w. Calculate the speed v of the slab.

[[File: AMexample.png]]

At a time change in t, we can calculate the area as the height (h) times the speed over the change in time. ⩟ A = v (⩟ t) h
Because the electric field is constant in this region we can also calculate the change in electric flux over time as Evh(⩟ t) /(⩟ t) which is the same as Evh
Calculating the path integral for the magnetic field we get that ∮B . dl = Bh cos 0 = Bh
An important thing to notice is that there is no current I, so, using the Ampere-Maxwell Law we can see that

Bh = μ. [I+ ε. (vEh)]

but since there is no current,

B = μ. ε. (vE)

From Faraday's law we get that the emf equals the rate of change of the magnetic flux: Eh = Bvh

Substituting E = Bv into our previous equation we get that

B = μ. ε. (v(vB))

Solving for v we get that:

[[File: AMvelocity.png]]

From the picture we see that the speed of light relates the a time varying electric and magnetic field.

==What this implies==

In the example above we saw that the speed of light relates the a time varying electric and magnetic field. This translates to the fact that an electromagnetic wave propagates at the speed of light.

[[File: AMwave.png]]

==Conceptual Question==
[[File: AMconceptual.png|500px|Image: 500 pixels]]

The plot above shows a side and a top view of a capacitor with charge Q with electric and magnetic fields E and B at time t. The charge Q is:

1) Increasing in time

2) Decreasing in time

3) Constant in time

4) I don't know

Answer:

The unit vector N points out of the plane. Given that the magnetic field curls clockwise, the electric flux would be positive and decreasing. Hence E is decreasing. Thus Q must be decreasing, since E is proportional to Q.

==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==

http://bulldog2.redlands.edu/fac/eric_hill/Phys232/Lectures/Ch%2023%20lect%201.pdf
http://ocw.mit.edu/high-school/physics/exam-prep/electromagnetism/maxwells-equations/
http://ocw.mit.edu/courses/physics/8-02-physics-ii-electricity-and-magnetism-spring-2007/readings/summary_w13d1.pdf
http://www.schoolphysics.co.uk/age16-19/Wave%20properties/Wave%20properties/text/Electromagnetic_radiation/index.html

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

Ampere-Maxwell Law

2015-11-30T23:14:01Z

Mrivero7: /* Conceptual Questions */

Claimed by Maria Rivero

== Ampere-Maxwell Law==

James Maxwell discovered that a time-varying electric field could be accompanied by a magnetic field. He thought of this after Faraday discovered that a time-varying magnetic field was accompanied by an electric field.

The time rate that is used for this equation is given by the derivative of the electric flux with respect to time. Because The derivative of the flux gives current over epsilon nod, the derivative of the electric flux times epsilon nod will also have units of amperes.

[[File: Ampere-Maxwell.png]]

Where
B is the magnetic field
dl is the change in path
The sum of I is the sum of the charges inside the path.

==Example==

Pick a closed, rectangular path in the xz plane with a height h and a width w. Calculate the speed v of the slab.

[[File: AMexample.png]]

At a time change in t, we can calculate the area as the height (h) times the speed over the change in time. ⩟ A = v (⩟ t) h
Because the electric field is constant in this region we can also calculate the change in electric flux over time as Evh(⩟ t) /(⩟ t) which is the same as Evh
Calculating the path integral for the magnetic field we get that ∮B . dl = Bh cos 0 = Bh
An important thing to notice is that there is no current I, so, using the Ampere-Maxwell Law we can see that

Bh = μ. [I+ ε. (vEh)]

but since there is no current,

B = μ. ε. (vE)

From Faraday's law we get that the emf equals the rate of change of the magnetic flux: Eh = Bvh

Substituting E = Bv into our previous equation we get that

B = μ. ε. (v(vB))

Solving for v we get that:

[[File: AMvelocity.png]]

From the picture we see that the speed of light relates the a time varying electric and magnetic field.

==What this implies==

In the example above we saw that the speed of light relates the a time varying electric and magnetic field. This translates to the fact that an electromagnetic wave propagates at the speed of light.

[[File: AMwave.png]]

==Conceptual Question==
[[File: AMconceptual.png|300px|Image: 300 pixels]]

The plot above shows a side and a top view of a capacitor with charge Q with electric and magnetic fields E and B at time t. The charge Q is:

1) Increasing in time

2) Decreasing in time

3) Constant in time

4) I don't know

Answer:

The unit vector N points out of the plane. Given that the magnetic field curls clockwise, the electric flux would be positive and decreasing. Hence E is decreasing. Thus Q must be decreasing, since E is proportional to Q.

==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==

http://bulldog2.redlands.edu/fac/eric_hill/Phys232/Lectures/Ch%2023%20lect%201.pdf
http://ocw.mit.edu/high-school/physics/exam-prep/electromagnetism/maxwells-equations/
http://ocw.mit.edu/courses/physics/8-02-physics-ii-electricity-and-magnetism-spring-2007/readings/summary_w13d1.pdf
http://www.schoolphysics.co.uk/age16-19/Wave%20properties/Wave%20properties/text/Electromagnetic_radiation/index.html

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

Ampere-Maxwell Law

2015-11-30T23:08:48Z

Mrivero7: /* Conceptual Questions */

Ampere-Maxwell Law

2015-11-30T23:04:10Z

Mrivero7: /* Conceptual Questions */

Ampere-Maxwell Law

2015-11-30T23:03:53Z

2015-11-30T22:31:57Z

Mrivero7:

Ampere-Maxwell Law

2015-11-30T22:31:17Z

Mrivero7:

Claimed by Maria Rivero

== Ampere-Maxwell Law==

James Maxwell discovered that a time-varying electric field could be accompanied by a magnetic field. He thought of this after Faraday discovered that a time-varying magnetic field was accompanied by an electric field.

The time rate that is used for this equation is given by the derivative of the electric flux with respect to time. Because The derivative of the flux gives current over epsilon nod, the derivative of the electric flux times epsilon nod will also have units of amperes.

[[File: Ampere-Maxwell.png]]

Where
B is the magnetic field
dl is the change in path
The sum of I is the sum of the charges inside the path.

==Example==

Pick a closed, rectangular path in the xz plane with a height h and a width w. Calculate the speed v of the slab.

[[File: AMexample.png]]

At a time change in t, we can calculate the area as the height (h) times the speed over the change in time. ⩟ A = v (⩟ t) h
Because the electric field is constant in this region we can also calculate the change in electric flux over time as Evh(⩟ t) /(⩟ t) which is the same as Evh
Calculating the path integral for the magnetic field we get that ∮B . dl = Bh cos 0 = Bh
An important thing to notice is that there is no current I, so, using the Ampere-Maxwell Law we can see that

Bh = μ. [I+ ε. (vEh)]

but since there is no current,

B = μ. ε. (vE)

From Faraday's law we get that the emf equals the rate of change of the magnetic flux: Eh = Bvh

Substituting E = Bv into our previous equation we get that

B = μ. ε. (v(vB))

Solving for v we get that:

==A Mathematical Model==

In order to find the potential difference between two locations, we use this formula <math> dV = -\left(E_x*dx + E_y*dy + E_z*dz\right) </math>, where '''E''' is the electric field with components in the x, y, and z directions. Delta x, y, and z are the components of final location minus to the components of the initial location.

==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]

=Simple Example=
[[File:pathindependence.png]]

In this example, the electric field is equal to <math> E = \left(E_x, 0, 0\right)</math>. The initial location is A and the final location is C. In order to find the potential difference between A and C, we use <math>dV = V_C - V_A </math>.

Since there are no y and z components of the electric field, the potential difference is <math> dV = -\left(E_x*\left(x_1 - 0\right) + 0*\left(-y_1 - 0\right) + 0*0\right) = -E_x*x_1</math>

[[File:BC.png]]

Let's say there is a location B at <math> \left(x_1, 0, 0\right) </math>. Now in order to find the potential difference between A and C, we need to find the potential difference between A and B and then between B and C.

The potential difference between A and B is <math>dV = V_B - V_A = -\left(E_x*\left(x_1 - 0\right) + 0*0 + 0*0\right) = -E_x*x_1</math>.

The potential difference between B and C is <math>dV = V_C - V_B = -\left(E_x*0 + 0*\left(-y_1 - 0\right) + 0*0\right) = 0</math>.

Therefore, the potential difference A and C is <math>V_C - V_A = \left(V_C - V_B\right) + \left(V_B - V_A\right) = E_x*x_1 </math>, which is the same answer that we got when we did not use location B.

==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?]]

Ampere-Maxwell Law

2015-11-30T22:19:45Z

Mrivero7:

Claimed by Maria Rivero

== Ampere-Maxwell Law==

James Maxwell discovered that a time-varying electric field could be accompanied by a magnetic field. He thought of this after Faraday discovered that a time-varying magnetic field was accompanied by an electric field.

The time rate that is used for this equation is given by the derivative of the electric flux with respect to time. Because The derivative of the flux gives current over epsilon nod, the derivative of the electric flux times epsilon nod will also have units of amperes.

[[File: Ampere-Maxwell.png]]

Where
B is the magnetic field
dl is the change in path
The sum of I is the sum of the charges inside the path.

==Example==

Pick a closed, rectangular path in the xz plane with a height h and a width w. Calculate the speed v of the slab.

[[File: AMexample.png]]

At a time change in t, we can calculate the area as the height (h) times the speed over the change in time. ⩟ A = v (⩟ t) h
Because the electric field is constant in this region we can also calculate the change in electric flux over time as Evh(⩟ t) /(⩟ t) which is the same as Evh
Calculating the path integral for the magnetic field we get that ∮B . dl = Bh cos 0 = Bh
An important thing to notice is that there is no current I, so, using the Ampere-Maxwell Law we can see that

Bh = μ. [I+ ε. (vEh)]

but since there is no current,

B = μ. ε. (vE)

==A Mathematical Model==

In order to find the potential difference between two locations, we use this formula <math> dV = -\left(E_x*dx + E_y*dy + E_z*dz\right) </math>, where '''E''' is the electric field with components in the x, y, and z directions. Delta x, y, and z are the components of final location minus to the components of the initial location.

==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]

=Simple Example=
[[File:pathindependence.png]]

In this example, the electric field is equal to <math> E = \left(E_x, 0, 0\right)</math>. The initial location is A and the final location is C. In order to find the potential difference between A and C, we use <math>dV = V_C - V_A </math>.

Since there are no y and z components of the electric field, the potential difference is <math> dV = -\left(E_x*\left(x_1 - 0\right) + 0*\left(-y_1 - 0\right) + 0*0\right) = -E_x*x_1</math>

[[File:BC.png]]

Let's say there is a location B at <math> \left(x_1, 0, 0\right) </math>. Now in order to find the potential difference between A and C, we need to find the potential difference between A and B and then between B and C.

The potential difference between A and B is <math>dV = V_B - V_A = -\left(E_x*\left(x_1 - 0\right) + 0*0 + 0*0\right) = -E_x*x_1</math>.

The potential difference between B and C is <math>dV = V_C - V_B = -\left(E_x*0 + 0*\left(-y_1 - 0\right) + 0*0\right) = 0</math>.

Therefore, the potential difference A and C is <math>V_C - V_A = \left(V_C - V_B\right) + \left(V_B - V_A\right) = E_x*x_1 </math>, which is the same answer that we got when we did not use location B.

==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?]]

Ampere-Maxwell Law

2015-11-30T22:06:28Z

File:Ampere-Maxwell.png

2015-11-30T21:39:02Z

Mrivero7:

Ampere-Maxwell Law

2015-11-30T20:55:21Z

Mrivero7:

Claimed by Maria Rivero

==Path Independence==

The potential difference between two locations does not depend on the path taken between the locations chosen.

===A Mathematical Model===

In order to find the potential difference between two locations, we use this formula <math> dV = -\left(E_x*dx + E_y*dy + E_z*dz\right) </math>, where '''E''' is the electric field with components in the x, y, and z directions. Delta x, y, and z are the components of final location minus to the components of the initial location.

===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]

=Simple Example=
[[File:pathindependence.png]]

In this example, the electric field is equal to <math> E = \left(E_x, 0, 0\right)</math>. The initial location is A and the final location is C. In order to find the potential difference between A and C, we use <math>dV = V_C - V_A </math>.

Since there are no y and z components of the electric field, the potential difference is <math> dV = -\left(E_x*\left(x_1 - 0\right) + 0*\left(-y_1 - 0\right) + 0*0\right) = -E_x*x_1</math>

[[File:BC.png]]

Let's say there is a location B at <math> \left(x_1, 0, 0\right) </math>. Now in order to find the potential difference between A and C, we need to find the potential difference between A and B and then between B and C.

The potential difference between A and B is <math>dV = V_B - V_A = -\left(E_x*\left(x_1 - 0\right) + 0*0 + 0*0\right) = -E_x*x_1</math>.

The potential difference between B and C is <math>dV = V_C - V_B = -\left(E_x*0 + 0*\left(-y_1 - 0\right) + 0*0\right) = 0</math>.

Therefore, the potential difference A and C is <math>V_C - V_A = \left(V_C - V_B\right) + \left(V_B - V_A\right) = E_x*x_1 </math>, which is the same answer that we got when we did not use location B.

==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?]]

Ampere-Maxwell Law

2015-11-02T18:56:30Z

Mrivero7: Created page with "Claimed by Maria Rivero"

Claimed by Maria Rivero

Main Page

2015-11-02T18:56:10Z

Mrivero7: /* Maxwell's Equations */

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== Organizing Catagories ==
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">
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<div class="mw-collapsible-content">
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<div class="mw-collapsible-content">
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<div class="toccolours mw-collapsible mw-collapsed">

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<div class="mw-collapsible-content">
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<div class="mw-collapsible-content">
*[[Gauss's Flux Theorem]]
**[[Electric Fields]]
**[[Magnetic Fields]]
*[[Faraday's Law]]
*[[Ampere-Maxwell Law]]
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===Radiation===
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== 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]]