Capacitors And Capacitance

The capacitance of the system can be calculated using the formula for capacitance: where:
,
.
Therefore, the capacitance is:

(ii) When the dielectric constant is inserted, the capacitance increases by a factor of . The new capacitance becomes: Since , and the charge remains the same, the new potential difference is:

(iii) The capacitance of the system depends only on the geometry of the conductors and the dielectric constant of the medium between them, not the charges. Therefore, if the charges are doubled, the capacitance remains unchanged. The capacitance will still be , because capacitance is independent of the charge.

Charge Storage in a Capacitor

The amount of charge that a capacitor can store is limited by the breakdown voltage of the dielectric material between the plates.

When the applied voltage across the capacitor reaches a certain value (known as the breakdown voltage), the dielectric material between the plates begins to break down, causing an electrical discharge. This breakdown limits the amount of charge the capacitor can hold.

Relationship Between Charge, Capacitance, and Voltage:

The relationship between the charge , capacitance , and potential difference is given by the formula:

Thus, for a given capacitor, the maximum charge that can be stored is determined by its capacitance and the maximum voltage it can withstand before dielectric breakdown occurs.

In an AC circuit with a parallel plate capacitor, the capacitor is charged by the alternating current (AC) source. In this scenario, we need to show that the sum of the conduction current ( ) and the displacement current ( ) is the same at all points of the circuit. Let's break down the process and derive the relation step by step.

1. Conduction Current ( ) in the Circuit:

When the capacitor is connected to the AC source, the current from the source charges the capacitor. The conduction current ( ) is the current supplied by the AC source that flows through the wires leading to the capacitor. This current fluctuates sinusoidally because the voltage supplied by the AC source is time-varying.

The conduction current in the circuit is related to the voltage supplied by the AC source. For a sinusoidal AC source, the current is given by Ohm's law or the capacitive reactance, where the relationship is:

• is the conduction current at time ,
• is the capacitance of the capacitor, and
• is the time rate of change of the voltage across the capacitor.

2. Displacement Current ( ) in the Capacitor:

The displacement current is a concept introduced to account for the time-varying electric field inside the capacitor. In a parallel plate capacitor, the electric field between the plates changes as the charge on the plates changes with time, due to the time-varying voltage from the AC source.

The displacement current is defined by the relation:

• is the displacement current,
• is the permittivity of free space,
• is the area of each plate of the capacitor, and
• is the time rate of change of the electric field between the plates.

Since the electric field between the plates is related to the voltage by the equation , where is the separation between the plates, the displacement current can also be written as:

Since is constant (the distance between the plates does not change), we can simplify this to:

3. Relating Conduction Current and Displacement Current:

We can now compare the expressions for the conduction current and the displacement current . From the previous equations:

Since the capacitance of the capacitor is related to the area of the plates and the distance between them by the equation:

Substituting this into the equation for , we get:

Now, comparing this expression for with the expression for , we see that:

4. Conclusion:

Therefore, the conduction current and the displacement current have the same value at all points in the circuit. This is because the conduction current in the circuit is responsible for charging the capacitor, and the displacement current arises due to the changing electric field inside the capacitor. Since the electric field and the voltage are related, the time rate of change of the voltage determines both currents, and thus they must be equal at every point in the circuit.