A C. R series circuit is connected to a battery of e.m.f E. The time required by the capacitor to acquire maximum charge, depends upon -

Options

A. R only
B. C only
C. RC
D. applied potential difference

Answer

To solve the problem of how the time required for a capacitor to acquire maximum charge in a C-R series circuit depends on certain factors, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Circuit**:
- We have a capacitor (C) and a resistor (R) in series connected to a battery with an electromotive force (e.m.f) E. Initially, the capacitor is uncharged.

2. **Applying Kirchhoff's Voltage Law (KVL)**:
- According to KVL, the sum of the potential differences in a closed loop is equal to zero. For our circuit, we can write:
\[ E - \frac{Q}{C} - I \cdot R = 0 \]
- Here, \( Q \) is the charge on the capacitor at time \( t \), and \( I \) is the current flowing through the circuit.

3. **Relating Current and Charge**:
- The current \( I \) can be expressed in terms of the charge \( Q \):
\[ I = \frac{dQ}{dt} \]
- Substituting this into the KVL equation gives:
\[ E - \frac{Q}{C} - R \frac{dQ}{dt} = 0 \]

4. **Rearranging the Equation**:
- Rearranging the equation, we get:
\[ \frac{dQ}{dt} = \frac{E - \frac{Q}{C}}{R} \]

5. **Separating Variables**:
- We can separate variables to integrate:
\[ \frac{dQ}{E - \frac{Q}{C}} = \frac{dt}{R} \]

6. **Integrating Both Sides**:
- Integrate the left side from \( 0 \) to \( Q \) and the right side from \( 0 \) to \( t \):
\[ \int_0^Q \frac{dQ}{E - \frac{Q}{C}} = \int_0^t \frac{dt}{R} \]

7. **Calculating the Integral**:
- The left side integral results in:
\[ -C \ln\left(E - \frac{Q}{C}\right) \bigg|_0^Q = \frac{t}{R} \]
- This leads to:
\[ -C \left(\ln(E - \frac{Q}{C}) - \ln(E)\right) = \frac{t}{R} \]

8. **Finding Maximum Charge**:
- The maximum charge \( Q_0 \) on the capacitor is given by:
\[ Q_0 = CE \]
- As \( t \) approaches infinity, the capacitor will charge towards \( Q_0 \).

9. **Determining Time for Practical Charging**:
- For practical purposes, we consider the capacitor to be fully charged when it reaches about 98% of \( Q_0 \):
\[ Q \approx 0.98 Q_0 \]
- From the earlier equations, we can derive the time \( t \) required to reach this charge.

10. **Final Conclusion**:
- The time required for the capacitor to charge significantly depends on both the resistance \( R \) and the capacitance \( C \). Thus, we conclude that the time taken by the capacitor to acquire maximum charge depends on the product \( RC \).