A charged capacitor is discharged through a resistance. The time constant of the circuit is `eta`. Then the value of time constant for the power dissipated through the resistance will be

Options

A. `eta`
B. `2eta`
C. `eta/2`
D. zero

Explanation

To solve the problem of finding the time constant for the power dissipated through a resistance when a charged capacitor is discharged, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Circuit**:
A capacitor discharges through a resistor. The time constant (τ) of the RC circuit is defined as τ = R * C, where R is the resistance and C is the capacitance.

2. **Current Equation**:
The current (I) flowing through the circuit at any time t during the discharge can be expressed as:
\[ I(t) = I_0 e^{-t/\eta} \]
where \( I_0 \) is the initial current and \( \eta \) is the time constant.

3. **Power Dissipation**:
The power (P) dissipated in the resistor can be calculated using the formula:
\[ P = I^2 R \]
Substituting the expression for current, we get:
\[ P(t) = (I_0 e^{-t/\eta})^2 R \]
This simplifies to:
\[ P(t) = I_0^2 R e^{-2t/\eta} \]

4. **Identifying the New Time Constant**:
The expression for power can be rewritten as:
\[ P(t) = (I_0^2 R) e^{-2t/\eta} \]
From this equation, we can see that the power dissipated also follows an exponential decay, but with a new time constant.

5. **Finding the New Time Constant**:
The new time constant for the power dissipation can be identified from the exponent of the exponential term. In this case, the time constant for power is:
\[ \text{New Time Constant} = \frac{\eta}{2} \]

### Final Answer:
The time constant for the power dissipated through the resistance is \( \frac{\eta}{2} \).

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