The operating temperature of the filament of lamp is `2000^(@) C`. The temperature coefficient of the material of the filament is `0.005^(@) C^(-1)`. If the atmospheric temperature is `0^(@) C`, then the current in the `100 W - 200 V` lamp when it is switched on is nearest to

Options

A. `2.5 A`
B. `3.5 A`
C. `4.5 A`
D. `5.5 A`

Explanation

To solve the problem step by step, we will follow the reasoning presented in the video transcript.

### Step 1: Calculate the resistance of the filament at operating temperature
The power \( P \) of the lamp is given as \( 100 \, W \) and the voltage \( V \) is \( 200 \, V \). We can use the formula for power to find the resistance \( R \) at the operating temperature:

\[ P = \frac{V^2}{R} \]

Rearranging the formula to find \( R \):

\[ R = \frac{V^2}{P} \]

Substituting the given values:

\[ R = \frac{200^2}{100} = \frac{40000}{100} = 400 \, \Omega \]

### Step 2: Use the temperature coefficient to find the resistance at room temperature
The resistance of the filament at a lower temperature (0°C) can be calculated using the formula:

\[ R_T = R_0 \left(1 + \alpha \Delta T\right) \]

Where:
- \( R_T \) is the resistance at the operating temperature (400 Ω),
- \( R_0 \) is the resistance at 0°C (which we need to find),
- \( \alpha \) is the temperature coefficient (0.005 °C⁻¹),
- \( \Delta T \) is the change in temperature (2000 °C - 0 °C = 2000 °C).

Rearranging the formula to solve for \( R_0 \):

\[ R_0 = \frac{R_T}{1 + \alpha \Delta T} \]

Substituting the known values:

\[ R_0 = \frac{400}{1 + 0.005 \times 2000} \]

Calculating \( 0.005 \times 2000 = 10 \):

\[ R_0 = \frac{400}{1 + 10} = \frac{400}{11} \approx 36.36 \, \Omega \]

### Step 3: Calculate the current when the lamp is switched on
Now that we have the resistance at 0°C, we can calculate the current \( I \) using Ohm's law:

\[ I = \frac{V}{R_0} \]

Substituting the values:

\[ I = \frac{200}{36.36} \approx 5.5 \, A \]

### Final Answer
The current in the lamp when it is switched on is approximately \( 5.5 \, A \).