When a capacitor discharges through a resistance R, the time constant is `tau` and the maximum current in the circuit is `i_0`. Then,

Options

A. the initial charge on the capacitor was `i_0 tau`
B. the initial charge on the capacitor was `1/2i_0tau`
C. the initial energy stored in the capacitor was `t_0^2 Rtau`
D. the initial energy stored in the capacitor was `1/2t_0^2 Rtau`

Answer

To solve the problem step by step, we will analyze the discharge of a capacitor through a resistor and derive the initial charge and energy stored in the capacitor.

### Step-by-Step Solution:

1. **Understanding the Circuit**:
- We have a capacitor discharging through a resistor \( R \).
- The time constant \( \tau \) is defined as \( \tau = R \cdot C \), where \( C \) is the capacitance.

2. **Identifying Given Quantities**:
- The maximum current in the circuit is \( I_0 \).
- The time constant is \( \tau \).
- The resistance is \( R \).

3. **Finding Initial Voltage**:
- The initial current \( I_0 \) can be expressed in terms of the initial voltage \( V_0 \) across the capacitor:
\[ I_0 = \frac{V_0}{R} \]
- Rearranging gives:
\[ V_0 = I_0 \cdot R \]

4. **Expressing Capacitance in Terms of Time Constant**:
- From the time constant formula \( \tau = R \cdot C \), we can express capacitance \( C \):
\[ C = \frac{\tau}{R} \]

5. **Calculating Initial Charge**:
- The initial charge \( Q_0 \) on the capacitor can be calculated using:
\[ Q_0 = C \cdot V_0 \]
- Substituting the expressions for \( C \) and \( V_0 \):
\[ Q_0 = \left(\frac{\tau}{R}\right) \cdot (I_0 \cdot R) \]
- Simplifying this gives:
\[ Q_0 = \tau \cdot I_0 \]

6. **Calculating Initial Energy**:
- The initial energy \( U \) stored in the capacitor is given by:
\[ U = \frac{1}{2} C V_0^2 \]
- Substituting \( C \) and \( V_0 \):
\[ U = \frac{1}{2} \left(\frac{\tau}{R}\right) \cdot (I_0 \cdot R)^2 \]
- This simplifies to:
\[ U = \frac{1}{2} \cdot \frac{\tau}{R} \cdot I_0^2 R^2 \]
- Further simplification yields:
\[ U = \frac{1}{2} \tau I_0^2 R \]

7. **Final Answers**:
- The initial charge \( Q_0 \) is \( \tau \cdot I_0 \).
- The initial energy \( U \) is \( \frac{1}{2} \tau I_0^2 R \).