Question

When an ideal capacitor is charged by a dc battery, no current flows. However, when an ac source is used, the current flows continuously. How does one explain this, based on the concept of displacement current?

Answer 1

Hint: 1. A capacitor offers infinite resistance to a DC.
2. A DC flows in one direction only.
3. There is no electric field inside electric conductors
4. Capacitors are charge storing devices.

Complete solution:
As we know displacement current is defined as
\[{i_d} = {\varepsilon _0}\dfrac{{d{\phi _E}}}{{dt}}\]
Where \[{\phi _E}\] is the electric flux
Now as we can see from the above equation if there is any change in electric flux $\left( {{\phi _E}} \right)$ the value \[\dfrac{{d{\phi _E}}}{{dt}}\]will be finite
Displacement current exists when there is electric flux i.e, displacement current cannot be in an electric conductor.
So we can say that \[{\phi _E}\]inside a conductor is zero since E inside a conductor is zero
Here it is given that an ideal capacitor is fully charged no current flows
Consider a capacitor of capacitance C connected to a dc source v now charge q is flowing inside the circuit and the capacitor is getting charged after some time the capacitor gets fully charged it will achieve a steady state.

Now since the capacitor is fully charged that means their es enough current on either side of the capacitors nowhere an electric field will be set up from positive to negative
Since the capacitor has reached a steady-state there will be no change in charge of the capacitor so there will be no change in the electric field so there will not be any change in electric flux either
i.e. \[\dfrac{{d{\phi _E}}}{{dt}}\]will be zero
$\because {i_d} = {\varepsilon _0}\dfrac{{d{\phi _E}}}{{dt}}$
$\therefore {i_d} = 0$
So this is the reason why there is no flow of current for a capacitor connected to dc source
Now consider the same capacitor connected to the ac source
We know that the AC is of changing polarity i.e. its polarity changes continuously
Now in on polarity, the positive charge is stored on the right plate of the capacitor and the second plate negative change is stored now as the polarity changes now negative current is getting stored on the right plate and positive current on the other plate
In this situation, we can say that the charge on either plate of the conductor is regularly changing. now this change in charge on plates s changes the electric field between two plates and thus there is a change in electric flux across plates

Now as we know displacement current is only present when there is a change in electric flux here we see that due to alternating polarity there is a regular change in electric flux
So,\[\dfrac{{d{\phi _E}}}{{dt}} \ne 0\]
So there will be a flow of current.

The currents do not flow when a capacitor is connected to a dc source because the is no change in electric flux i.e. \[\dfrac{{d{\phi _E}}}{{dt}} = 0\].

Note: Ideal capacitors have infinite resistance when connected to dc. When it is used with an ac source, the current flows continuously, ideally, there is no current, and the circuit would be incomplete because we know that the capacitor has a dielectric between its plates. So, a real capacitor is charged due to the contribution of the varying electric field. The current between the capacitor plates is given by displacement current.