The switch shown in the circuit is closed al t = 0, The current drawn from the battery by the circuit at t = 0 and `1 = oo` are in the ratio

Options

A. 2:1
B. 1:2
C. 1:1
D. 1:4

Answer

To solve the problem, we need to analyze the behavior of the circuit at two different time instances: when the switch is closed at \( t = 0 \) and as \( t \) approaches infinity.

### Step 1: Analyze the circuit at \( t = 0 \)
At \( t = 0 \):
- The capacitor behaves as a short circuit (since it has not had time to charge).
- The inductor behaves as an open circuit (since it does not allow sudden changes in current).

Given that the capacitor is short-circuited and the inductor is open-circuited, we can redraw the circuit. The only components that will affect the current are the resistors.

**Net Resistance at \( t = 0 \)**:
- If we have two resistors \( R \) and \( R' \) in series, the total resistance \( R_{\text{net}} \) is:
\[ R_{\text{net}} = R + R' = 2R \]

**Current at \( t = 0 \)**:
Using Ohm's law, the current \( I_1 \) drawn from the battery at \( t = 0 \) is:
\[ I_1 = \frac{E}{R_{\text{net}}} = \frac{E}{2R} \]

### Step 2: Analyze the circuit as \( t \to \infty \)
As \( t \) approaches infinity:
- The capacitor behaves as an open circuit (fully charged).
- The inductor behaves as a short circuit (it allows steady current to pass through).

In this case, the circuit can again be simplified to just the resistors since the capacitor does not allow current to flow through it.

**Net Resistance at \( t \to \infty \)**:
- The resistors are still in series, so the total resistance remains:
\[ R_{\text{net}} = R + R = 2R \]

**Current at \( t \to \infty \)**:
The current \( I_2 \) drawn from the battery at \( t \to \infty \) is:
\[ I_2 = \frac{E}{R_{\text{net}}} = \frac{E}{2R} \]

### Step 3: Calculate the ratio of currents
Now, we can find the ratio of the currents \( I_1 \) and \( I_2 \):
\[ \frac{I_1}{I_2} = \frac{\frac{E}{2R}}{\frac{E}{2R}} = 1 \]

### Conclusion
Thus, the ratio of the current drawn from the battery at \( t = 0 \) and \( t = \infty \) is:
\[ \text{Ratio} = 1:1 \]

### Final Answer
The ratio of the current drawn from the battery by the circuit at \( t = 0 \) and \( t = \infty \) is \( 1:1 \).

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