# How to compute the field lines of an induced magnetic field inside a capacitor?

Consider a capacitor with a varying voltage applied to it. As the voltage changes over time, the electrical field $$\vec{E}$$ inside the plates does too.

Assumption We assume that the direction of $$\vec{E}$$ is the same inside the capacitor, hence ignoring border effects.

According to Ampere-Maxwell (Maxwell's 4th equation):

$$\nabla \times \vec{B} = \mu_0 \vec{j} + \frac{1}{c^2}\frac{\partial \vec{E}}{\partial t}$$

A magnetic field $$\vec{B}$$ is generated. How do I find out the direction of the magnetic field? Basically, how do I find out the field lines of $$\vec{B}$$ inside the capacitor?

Ultimately, how can we find the equations of the field lines?

Assuming that the approximation that the electric field is constant in space but changes with time is reasonable, then since $$\vec j = 0$$ between the plates, a possible solution for the $$\vec \nabla \times \vec B$$ equation which also satisfies $$\vec \nabla \cdot \vec B = 0$$ is $$\vec B = \frac{1}{2c^2} \vec r \times \frac{\partial \vec E}{\partial t}$$. To see what other solutions can be, imagine another solution $$\vec B_2$$, that satisfies both of these equations. The difference $$\vec \Delta = \vec B-\vec B_2$$ satisfies $$\vec \nabla \times \vec \Delta = 0$$, so it can be written as the gradient of a scalar field. Since the divergence of $$\vec \Delta$$ is also zero, this scalar field satisfies Laplaces equation. Adding the gradient of any solution to Laplaces equation to $$\vec B$$ will give another solution. You need to add boundary conditions, i.e. deal with the shape of the capacitor plates and the charging current, to set these boundary conditions and get a unique solution.
$$\oint_{\partial \Sigma} \mathbf{B} \cdot \mathrm{d}\boldsymbol{\ell} = \mu_0 \varepsilon_0 \frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{E} \cdot \mathrm{d}\mathbf{S}$$
we can see that around the borders of the plates, the LHS is maximum, because the integral of RHS includes all the area. The B-vectors are pointing clockwise if viewed from positive to negative side when $$\mathbf E$$ is increasing. For regions closer to the center, the direction is the same, but the magnitude of $$\mathbf B$$ is smaller, due to the smaller enclosed area.