A compass needle is placed in the gap of a parallel plate
capacitor. The capacitor is connected to a battery
through a resistance. The compass needle

Options

A. does not deflect
B. deflects for a very short time and then comes back to the original position
C. deflects and remains deflected as long as the battery is connected
D. deflects and gradually comes to the original position in a time which is large compared to the time constant.

Answer

To solve the problem regarding the behavior of a compass needle placed in the gap of a parallel plate capacitor connected to a battery through a resistance, we can analyze the situation step by step.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- A parallel plate capacitor is connected to a battery through a resistor.
- A compass needle is placed in the gap between the plates of the capacitor.

2. **Behavior of the Capacitor**:
- When the capacitor is connected to the battery, it starts charging. The charge \( Q \) on the plates of the capacitor at any time \( t \) is given by the formula:
\[ Q(t) = C \cdot V \cdot \left(1 - e^{-\frac{t}{RC}}\right) \]
- Here, \( C \) is the capacitance, \( V \) is the voltage of the battery, \( R \) is the resistance, and \( t \) is the time.

3. **Electric Field and Magnetic Field**:
- A changing electric field between the plates of the capacitor induces a magnetic field according to Maxwell's equations. This magnetic field is what affects the compass needle.

4. **Time Constant**:
- The time constant \( \tau \) of the circuit is given by:
\[ \tau = R \cdot C \]
- The capacitor charges gradually over time, and the significant changes occur within a time frame of \( \tau \).

5. **Effect on the Compass Needle**:
- Initially, when the capacitor starts charging, the electric field changes, and thus a magnetic field is generated. This causes the compass needle to deflect.
- As the capacitor approaches its full charge, the rate of change of the electric field decreases, leading to a gradual reduction in the magnetic field.

6. **Final Behavior of the Compass Needle**:
- After a time much larger than the time constant \( \tau \), the capacitor will be fully charged, and the electric field will stabilize. Consequently, the magnetic field will also stabilize, and the compass needle will gradually return to its original position.

7. **Conclusion**:
- Therefore, the compass needle deflects initially due to the changing electric field and then gradually returns to its original position over a time period that is large compared to the time constant.

### Answer:
The correct option is that the compass needle deflects and gradually comes to the original position in time which is large compared to the time constant.

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