Confusion when dealing with loops and surfaces with Maxwell equations

[greg_rack]

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 $R$?

 

[kuruman]

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.

 

[Delta2]

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).