Question

demonstrate that the magnetic field induced by the displacement current present between the plates of a parallel plate capacitor of radius R, separated at a distance d, which is being charged by a current I, is maximum just at the edge of the plates. Draw a picture that shows the Amperian spiral you are considering, as well as the surface related to it. Suggestion beware of different surfaces, characterize each with a different radius.

Answer 1

The calculation of the magnetic field of a current distribution can, in principle, be carried out using Ampere's law which relates the path integral of the magnetic field around a closed path to the current intercepted by an arbitrary surface that spans this path:

Ampere's law is independent of the shape of the surface chosen as long as the current flows along a continuous, unbroken circuit. However, consider the case in which the current wire is broken and connected to a parallel-plate capacitor. A current will flow through the wire during the charging process of the capacitor. This current will generate a magnetic field and if we are far away from the capacitor, this field should be very similar to the magnetic field produced by an infinitely long, continuous, wire. However, the current intercepted by an arbitrary surface now depends on the surface chosen. For example, the surface does not intercept any current. Clearly, Ampere's law can not be applied in this case to find the magnetic field generated by the current.

Ampere's law in a capacitor circuit.

Although the surface does not intercept any current, it intercepts electric flux. Suppose the capacitor is an ideal capacitor, with a homogeneous electric field E between the plates and no electric field outside the plates. At a certain time t the charge on the capacitor plates is Q. If the plates have a surface area A then the electric field between the plates is equal to

The electric field outside the capacitor is equal to zero. The electric flux, [Phi]_E, intercepted by the surface is equal to

If a current I is flowing through the wire, then the charge on the capacitor plates will be time dependent. The electric flux will therefore also be time dependent, and the rate of change of electric flux is equal to

The magnetic field around the wire can now be found by modifying Ampere's law

where [Phi]_E is the electric flux through the surface. In the most general case, the surface spanned by the integration path of the magnetic field can intercept current and electric flux. In such a case, the effects of the electric flux and the electric current must be combined, and Ampere's law becomes

The current I is the current intercepted by whatever surface is used in the calculation, and is not necessarily the same as the current in the wires.

where I_d is called the displacement current and is defined as