Question

A parallel plate capacitor is being charged by a time varying current. Explain briefly how Ampere’s circuital law is generalized to incorporate the effect due to the displacement current ?

Answer 1

When a capacitor is connected to an alternating current, it offers a resistance `X_e (1/(omegac))`and allows the current to pass through.

As the current is moving from plate (1) to plate (2) there should be a magnetic field associated with this current as explained by Ampere circuital law, but as there can’t be movement of actual electrons from plate (2) to plate (1), there will be no physical current, and still there will be a magnetic field.

To solve this paradox Maxwell altered the form of Ampere’s law as followed.

Original form

`int_c vecE*vecdl =mu_0i +mu_0 in_0 (dphi)/(dt)`

                         `=mu_0 (i +i_D)`

                      `i_D = in_0 (dphi_(E))/(dt)`(Displacement current)

                      `phi_E` = Electric flux

By introducing displacement current, Maxwell argued that due to change in electric field associated with alternating voltage, there will be a time dependent electric flux. This flux will cause a displacement current and hence there will be a magnetic field.