Displacement current goes through the gap between the plates of a capacitor when the charge of the capacitor

Options

A. increase
B. decrease
C. does not change
D. is zero.

Answer

To solve the question regarding displacement current in a capacitor, we will follow these steps:

### Step 1: Understand the Concept of Displacement Current
Displacement current is a concept introduced by James Clerk Maxwell to account for the changing electric field in regions where there is no conduction current, such as the gap between the plates of a capacitor. It is defined as:
\[ I_d = \epsilon_0 \frac{d\Phi_E}{dt} \]
where \( I_d \) is the displacement current, \( \epsilon_0 \) is the permittivity of free space, and \( \Phi_E \) is the electric flux.

### Step 2: Relate Electric Flux to Charge
According to Gauss's law, the electric flux \( \Phi_E \) through a surface is related to the charge \( Q \) enclosed by that surface:
\[ \Phi_E = \frac{Q}{\epsilon_0} \]
Substituting this into the displacement current equation gives:
\[ I_d = \epsilon_0 \frac{d}{dt}\left(\frac{Q}{\epsilon_0}\right) = \frac{dQ}{dt} \]
This shows that the displacement current is directly proportional to the rate of change of charge \( Q \) on the capacitor plates.

### Step 3: Analyze the Options
1. **If the charge increases**: \( \frac{dQ}{dt} > 0 \) → Displacement current exists.
2. **If the charge decreases**: \( \frac{dQ}{dt} < 0 \) → Displacement current exists.
3. **If the charge does not change**: \( \frac{dQ}{dt} = 0 \) → Displacement current is zero.
4. **If the charge is zero**: This is a special case, but if the charge is not changing, the displacement current is still zero.

### Step 4: Conclusion
From the analysis, we conclude that the displacement current goes through the gap between the plates of a capacitor when the charge of the capacitor is either increasing or decreasing. Therefore, the correct answers are:
- Option A: Increased charge → Displacement current exists.
- Option B: Decreased charge → Displacement current exists.
- Option C: Charge does not change → Displacement current is zero.
- Option D: Charge is zero → Displacement current is zero (if not changing).

Thus, the correct options are A and B.