Application of Ampère-Maxwell equ.

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In summary: I would need to see an equation in order to know for certain, but from what you've said it seems like it would be a straight line.
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quasar987
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Consider the set-up illustrated in the attachement. The radius of the capacitor plates is a. The field btw the plates varies according to

[tex]\vec{E}(t) = \frac{It}{\epsilon_0 \pi a^2}\hat{z}[/tex]

(z is to the right)

So since the current enclosed is 0, Ampere-Maxwell law in its integral form after evaluation of the integral of [itex]d\Phi_E/dt[/itex] reads...

[tex]\oint \vec{B}\cdot d\vec{l} = \mu_0 I \frac{s^2}{a^2}[/tex]

s being the radius of my amperian loop.

What is the argument according to which B is solenoid and constant along the path of integration? In the case of magnetostatic, it was the right-hand thumb rule (i.e. the Biot-Savart law) that allowed us to determine the orientation of B. But now what is it that permits to conclude?
 
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Looks like I forgot to add the picture doesn't it.
 

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I'm not really sure I understand what you want to know (what do you mean exactly when saying "What is the argument according to which B is solenoid and constant along the path of integration?"), but I'll just suppose you want to know why you can assume the magnitude of the B-field to be the same around your loop. The answer is: a symmetry argument (cylindrical symmetry).
But then I don't get what you're talking about "right-hand thumb rule" and "Biot-Savart". This has nothing to do with assuming the magnitude of the B-field to be constant along your loop.

Hoping to get some clearer input next time...Cliowa
 
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cliowa said:
But then I don't get what you're talking about "right-hand thumb rule" and "Biot-Savart". This has nothing to do with assuming the magnitude of the B-field to be constant along your loop.
Hoping to get some clearer input next time...Cliowa

In the case of say a straight current wire in which a current I circulates, what allows you to conclude that the field is in the chape of concentric circles around the wire and constant in magnitude on the contour of each circles? It is the Biot-Savart law, which tells you B is in the direction of

[tex]\vec{I} \times \frac{\vec{r}-\vec{r}'}{|\vec{r}-\vec{r}'|}[/tex]

(using Griffith'Ssnotation). This is the "right-hand thumb rule": let your thumb point in the direction of I; then your fingers curl in the direction of B.

In order to say that

[tex]\oint \vec{B}\cdot d\vec{l} =B2\pi s[/tex]

one must show that

i) B is constant in magnitude on the contour of the amperian circle.

ii) B is tangent to the amperian circle (so the dot product has no [itex]\cos\theta[/itex] term)

The "right-hand thumb rule" is used to verify the property ii).

Also in the case of the straight current wire, B is constant in magnitude on the contour by a simple symetry argument: Consider a point anywhere is space. Now rotate the wire along its axis or make the point travel along a circular path centered on the wire: nothing changes AT ALL. Hence the field has no choice but to be the same everywhere along the circular path.


cliowa said:
I'm not really sure I understand what you want to know (what do you mean exactly when saying "What is the argument according to which B is solenoid and constant along the path of integration?"), but I'll just suppose you want to know why you can assume the magnitude of the B-field to be the same around your loop. The answer is: a symmetry argument (cylindrical symmetry).

I have strong reasons to believe that in the case of the magnetic field of my problem, B also satisfies the conditions i) and ii). But I don't have an equation for [itex]\vec{B}[/itex] in terms of dE/dt (i.e. the equivalent of Biot-Savart). So how do I know what the B field created by a chnaging electric field looks like?

I can see why it is constant along the amperian circle though.
 
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1. What is the Ampère-Maxwell equation used for?

The Ampère-Maxwell equation is used to describe the relationship between electric and magnetic fields in a vacuum. It is one of the four Maxwell's equations that form the basis of classical electromagnetism.

2. Can you explain the Ampère-Maxwell equation in simple terms?

The Ampère-Maxwell equation states that the circulation of the electric field around a closed loop is equal to the rate of change of magnetic flux through the surface enclosed by the loop. In other words, it describes how a changing magnetic field can create an electric field and vice versa.

3. How is the Ampère-Maxwell equation derived?

The Ampère-Maxwell equation is derived from Faraday's law of induction and the principle of charge conservation. It can also be derived from the general Maxwell's equations through the use of vector calculus.

4. What are some practical applications of the Ampère-Maxwell equation?

The Ampère-Maxwell equation has numerous practical applications, including the design and analysis of electromagnetic devices such as motors, generators, and antennas. It is also used in the study of electromagnetic waves, which are used in communication technologies like radio, television, and cell phones.

5. What are the implications of the Ampère-Maxwell equation on our understanding of electromagnetism?

The Ampère-Maxwell equation is a fundamental law of electromagnetism and has greatly enhanced our understanding of the relationship between electric and magnetic fields. It also paved the way for the development of many modern technologies, such as electric power distribution and wireless communication. Its inclusion in the Maxwell's equations helped unify electricity and magnetism into a single theory of electromagnetism.

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