A capacitor is charged to a certain potential and then allowed to discharge through a resistance R. The ratio of charge on the capacitor to current in the circuit

Options

A. changes with time
B. does not change with time and it is equal to time constant of circuit
C. does not change with time, but not equal to time constant of circuit
D. may or may not change depending upon the charge given to the capacitor

Explanation

To solve the problem regarding the ratio of charge on a capacitor to the current in the circuit during the discharge process, we can follow these steps:

### Step 1: Understand the Discharge Equation of a Capacitor
When a capacitor discharges through a resistor, the charge \( Q \) on the capacitor at any time \( t \) is given by the equation:
\[ Q(t) = Q_0 e^{-\frac{t}{RC}} \]
where:
- \( Q_0 \) is the initial charge on the capacitor,
- \( R \) is the resistance,
- \( C \) is the capacitance,
- \( t \) is the time,
- \( e \) is the base of the natural logarithm.

### Step 2: Find the Current in the Circuit
The current \( I \) in the circuit can be found by differentiating the charge with respect to time:
\[ I(t) = -\frac{dQ}{dt} = -\frac{d}{dt}(Q_0 e^{-\frac{t}{RC}}) \]
Using the chain rule, we get:
\[ I(t) = \frac{Q_0}{RC} e^{-\frac{t}{RC}} \]

### Step 3: Calculate the Ratio of Charge to Current
Now, we need to find the ratio of charge \( Q \) to current \( I \):
\[ \frac{Q}{I} = \frac{Q_0 e^{-\frac{t}{RC}}}{\frac{Q_0}{RC} e^{-\frac{t}{RC}}} \]
Here, the \( Q_0 e^{-\frac{t}{RC}} \) terms cancel out:
\[ \frac{Q}{I} = RC \]

### Step 4: Analyze the Result
The ratio \( \frac{Q}{I} = RC \) shows that:
- The ratio of charge on the capacitor to the current in the circuit is equal to the time constant \( \tau = RC \).
- This ratio does not change with time since \( R \) and \( C \) are constants.

### Conclusion
Thus, the ratio of charge on the capacitor to the current in the circuit is equal to the time constant \( RC \) and does not change with time.

### Final Answer
The ratio of charge on the capacitor to the current in the circuit is \( RC \), which is constant and equal to the time constant.

---