Confusion when dealing with loops and surfaces with Maxwell equations

In summary, the circumference of the circle around the loop is the area in which we calculate the time derivative of the electric flux.
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greg_rack

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DISCLAIMER: in Italy, we talk about "circuitazione" of a field through a closed loop ##\gamma##, for the physical quantity $$\Gamma_\gamma(\overrightarrow{E}) = \sum_{k}\overrightarrow{\Delta l_k} \cdot \overrightarrow{E_k}$$
but after some research, I haven't managed to find the correspondent in English...

Hi guys,
I am having troubles with Maxwell equations related to "choosing" the right frame, of the right radius/dimension for calculating the varying flux of an electromagnetic field and the related loop, for the corresponding induced EM field.

For instance, let's assume a capacitor with parallel-circular plates, of radius ##R## at distance ##d##. The voltage ##V(t)## across it varies over time.
Say we want to calculate the induced mag field at a distance ##r## from the axis of the capacitor, with ##r<R##.
By the fourth Maxwell equation, we know that the "circuitazione" around the loop of radius ##r## is ##\Gamma_\gamma(\overrightarrow{B})=\mu(\epsilon \cdot x^2\pi \cdot\frac{dE}{dt})## and, by definition, ##\Gamma_\gamma(\overrightarrow{B})=2\pi r\cdot B##.

Now, my question is, which radius should I pick in place of ##x## related to the flux variation, and why?
Should it be ##r## or ##R##?
 
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  • #2
I would say "circulazione" is the closed loop line integral ##\oint \vec B \cdot d\vec l.## I have seen it called "circulation" by some people. In your expression ##x^2\pi## is the area of a circle of radius ##x##. The circulazione or line integral is taken around the boundary of that circle, so ##r## and ##x## must match. That's a consequence of Stokes' theorem, if you have studied it.
 
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  • #3
It should be ##x=r##. The way the fourth Maxwell's equation is in integral form it means that the area enclosed by the loop of radius ##r## is the area in which we calculate the time derivative of the electric flux (displacement current).
 
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