# Confusion when dealing with loops and surfaces with Maxwell equations

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

#### greg_rack

Gold Member
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##?

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.

vanhees71 and greg_rack
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).

greg_rack