Understanding Displacement Current in Capacitors

Displacement current ($I_d$) is a concept introduced by Maxwell, crucial in understanding electromagnetic waves and circuits containing capacitors. In a capacitor, displacement current flows specifically when the voltage across its plates is changing.

Capacitor Displacement Current Formula

For a parallel plate capacitor with capacitance $C$, the relationship between charge $Q$ and voltage $V$ is $Q = CV$. The displacement current is defined as $I_d = \epsilon_0 \frac{d\Phi_E}{dt}$, where $\Phi_E$ is the electric flux. For a capacitor, this simplifies to the rate of change of charge:

$I_d = \frac{dQ}{dt}$

Substituting $Q = CV$, we get:

$I_d = \frac{d(CV)}{dt} = C \frac{dV}{dt}$

This formula shows that displacement current is directly proportional to the rate at which the voltage ($V$) across the capacitor plates changes over time ($t$).

Analyzing the Options

• Option 1: becomes zero - Displacement current is zero only if $\frac{dV}{dt} = 0$, meaning the voltage is constant (or momentarily zero but not changing).
• Option 2: is increasing with time - If voltage is increasing, $\frac{dV}{dt} > 0$. Therefore, $I_d = C \frac{dV}{dt} > 0$, and displacement current flows.
• Option 3: is decreasing with time - If voltage is decreasing, $\frac{dV}{dt} < 0$. Therefore, $I_d = C \frac{dV}{dt} < 0$, and displacement current flows (in the opposite direction).
• Option 4: attains a constant value - If voltage attains a constant value, $\frac{dV}{dt} = 0$. Therefore, $I_d = 0$, and displacement current does not flow.

Conclusion on Displacement Current Flow

Based on the formula $I_d = C \frac{dV}{dt}$, displacement current flows whenever the voltage across the capacitor plates is changing. This occurs when the voltage is either increasing or decreasing with time.

Therefore, the conditions under which displacement current flows are when the voltage:

• is increasing with time (Option B)
• is decreasing with time (Option C)