A 3 mega ohm resistor and an uncharged `1 mu F` capacitor are connected in a single loop circuit with a constant source of 4 volt. At one second after the connection is made what are the rates at which,
(i) The charge on the capacitor is increasing.

Options

A. `4(1-e^(-1//3))mu C//s`
B. `4e^(-1//3) mu C//s`
C. `(4)/(3)e^(-1//3) mu C//s`
D. `(4)/(3)(1-e^(-1//3)) mu C//s`

Answer

To solve the problem step by step, we need to find the rate at which the charge on the capacitor is increasing after one second.

### Step 1: Identify the components of the circuit
We have:
- A resistor \( R = 3 \, \text{M}\Omega = 3 \times 10^6 \, \Omega \)
- A capacitor \( C = 1 \, \mu F = 1 \times 10^{-6} \, F \)
- A voltage source \( V = 4 \, V \)

### Step 2: Calculate the time constant \( \tau \)
The time constant \( \tau \) for an RC circuit is given by:
\[ \tau = R \cdot C \]
Substituting the values:
\[ \tau = (3 \times 10^6 \, \Omega) \cdot (1 \times 10^{-6} \, F) = 3 \, s \]

### Step 3: Determine the maximum current \( I_0 \)
The maximum current \( I_0 \) at the moment the circuit is closed (when the capacitor is uncharged) is given by:
\[ I_0 = \frac{V}{R} \]
Substituting the values:
\[ I_0 = \frac{4 \, V}{3 \times 10^6 \, \Omega} = \frac{4}{3 \times 10^6} \, A \]

### Step 4: Write the expression for the current as a function of time
The current \( I(t) \) in the circuit as the capacitor charges is given by:
\[ I(t) = I_0 e^{-t/\tau} \]
Substituting the expression for \( I_0 \):
\[ I(t) = \left( \frac{4}{3 \times 10^6} \right) e^{-t/3} \]

### Step 5: Calculate the current at \( t = 1 \, s \)
Now, we need to find the current at \( t = 1 \, s \):
\[ I(1) = \left( \frac{4}{3 \times 10^6} \right) e^{-1/3} \]
Calculating \( e^{-1/3} \) (approximately \( 0.7165 \)):
\[ I(1) \approx \left( \frac{4}{3 \times 10^6} \right) \times 0.7165 \]
\[ I(1) \approx \frac{4 \times 0.7165}{3 \times 10^6} \approx \frac{2.866}{3 \times 10^6} \approx 9.55 \times 10^{-7} \, A \]

### Step 6: Convert the current to microamperes
Since \( 1 \, A = 10^6 \, \mu A \):
\[ I(1) \approx 9.55 \, \mu A \]

### Final Answer
The rate at which the charge on the capacitor is increasing after one second is approximately \( 9.55 \, \mu A \).

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