How does the current in an RC circuit vary when the charge on the capacitor build up ?

Options

A. It decreases linearly
B. It increases linearly
C. it decreases exponentially
D. It increases exponentially

Answer

To understand how the current in an RC circuit varies when the charge on the capacitor builds up, we can analyze the charging process of the capacitor step by step.

### Step-by-Step Solution:

1. **Understanding the RC Circuit**:
- An RC circuit consists of a resistor (R) and a capacitor (C) connected in series with a voltage source (V). Initially, when the switch is closed, the capacitor is uncharged.

2. **Initial Condition (T = 0)**:
- At time \( T = 0 \), the capacitor has no charge. When the switch is closed, the capacitor behaves like a short circuit (it offers zero resistance), allowing maximum current to flow through the circuit.
- The maximum current \( I_0 \) can be calculated using Ohm's Law:
\[ I_0 = \frac{V}{R} \]

3. **Charging Process**:
- As time progresses, the capacitor begins to charge. The charge \( Q \) on the capacitor increases over time.
- The relationship between the charge \( Q \) on the capacitor and the voltage across it is given by:
\[ Q = C \cdot V_C \]
- Here, \( V_C \) is the voltage across the capacitor.

4. **Current Decay**:
- The current \( I \) in the circuit is related to the rate of change of charge on the capacitor:
\[ I = -\frac{dQ}{dt} \]
- As the capacitor charges, the rate of change of charge decreases, leading to a decrease in current.

5. **Exponential Decay of Current**:
- The current in the RC circuit during charging follows an exponential decay function:
\[ I(t) = I_0 e^{-\frac{t}{RC}} \]
- This equation shows that the current decreases exponentially over time as the capacitor charges.

6. **Final Condition (T → ∞)**:
- As time approaches infinity, the capacitor becomes fully charged, and the current in the circuit approaches zero:
\[ I(t \to \infty) = 0 \]
- At this point, the capacitor behaves like an open circuit, preventing any current from flowing.

### Conclusion:
- The current in an RC circuit decreases exponentially as the charge on the capacitor builds up. Therefore, the correct answer to the question is that the current decreases exponentially.