# Magnetic field between the plates of a charging capacitor

Considering a parallel plate capacitor initially charged, with circular plates of radius $$a$$ and with a distance $$d$$ from each other. A battery with internal resistance $$r$$ and EMF $$ε$$ is then connected to the capacitor and it starts charging.

In a teacher's lecture notes, he describes that the induced magnetic field as a function of time and distance to the central axis of the capacitor is given by $$B(r) = \frac{\mu_0Ir}{2\pi a^2}$$ However, only the final result is present, I would like to understand how to develop until this result. Any thoughts?

The underlying principle is that a time-varying electric field induces a magnetic field. This is stated in Maxwell's equations as

$$\operatorname{curl}\textbf{B} = \frac{1}{c^2}\frac{\partial \textbf{E}}{\partial t}.$$

Applying Stokes's theorem to a disk of radius $$r$$ between the plates (concentric with and parallel to the plates), we get that the line integral of the magnetic field around the edge of this disk, $$2\pi r B$$, relates to the rate of change of the electric flux through this disk. If you neglect fringing fields and take the electric field to be uniform between the plates, then the rate of change of this electric flux is proportional to the area $$\pi r^2$$ and also to the rate of change of the charge, $$dq/dt$$. Recognizing $$dq/dt$$ as the current $$I$$, and sorting out the powers of $$r$$, the magnetic field is proportional to $$Ir$$, as claimed.

As an example, suppose that the capacitor is charging up with some RC time constant. Then then $$I$$ will approach zero as $$t\rightarrow\infty$$, and the magnetic field will go to zero, as it should when we reach electrostatic equilibrium.

I don't think the formula is right. First of all where is the time dependence? When the capacitor starts charging, then it has a maximum magnetic field due to a maximum current in the cable connecting it and maximum electric field derivative inside the capacitor. When it is fully charged the magnetic field will be null (Of cours theoretically this happens for $$t \rightarrow \infty$$).

I would suggest you to read this post: Magnetic field in a capacitor.