When a battery is connected across a series combination of self inductance L and resistance R , the variation in the current i with time t is best represented by

Options

A.
B.
C.
D.

Explanation

To solve the problem of how the current \( i \) varies with time \( t \) when a battery is connected across a series combination of self-inductance \( L \) and resistance \( R \), we can follow these steps:

### Step 1: Understand the Circuit
When a battery is connected to a series circuit consisting of a resistor \( R \) and an inductor \( L \), the current does not instantly reach its maximum value. Instead, it increases gradually over time due to the inductance.

### Step 2: Write the Current Equation
The current \( i(t) \) in an LR circuit can be described by the equation:
\[ i(t) = I_{\text{max}} \left(1 - e^{-\frac{t}{\tau}}\right) \]
where:
- \( I_{\text{max}} = \frac{E}{R} \) (the maximum current, where \( E \) is the emf of the battery),
- \( \tau = \frac{L}{R} \) (the time constant of the circuit).

### Step 3: Analyze the Current at Specific Times
- At \( t = 0 \):
\[ i(0) = I_{\text{max}} \left(1 - e^{0}\right) = I_{\text{max}} \cdot 0 = 0 \]
This indicates that the current starts at zero when the switch is first closed.

- At \( t \to \infty \):
\[ i(\infty) = I_{\text{max}} \left(1 - e^{-\infty}\right) = I_{\text{max}} \cdot 1 = I_{\text{max}} \]
This shows that the current approaches its maximum value as time goes on.

### Step 4: Graphical Representation
The equation \( i(t) = I_{\text{max}} \left(1 - e^{-\frac{t}{\tau}}\right) \) describes an exponential rise. The graph starts at zero when \( t = 0 \) and asymptotically approaches \( I_{\text{max}} \) as \( t \) increases.

### Step 5: Identify the Correct Option
Based on the description of the current's behavior over time, the correct graphical representation is an exponential curve that starts at zero and rises towards \( I_{\text{max}} \). This matches option B.

### Conclusion
The variation in the current \( i \) with time \( t \) when a battery is connected across a series combination of self-inductance \( L \) and resistance \( R \) is best represented by option B.

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