If the length of the filament of a heater is reduced by `10%`, the power of the heater will

Options

A. increase by about `9%`
B. increase by about `11%`
C. increase by about `19%`
D. decrease by about `10%`

Explanation

To solve the problem of how the power of a heater changes when the length of its filament is reduced by 10%, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between power, resistance, and length:**
The power \( P \) of a heater can be expressed using the formula:
\[ P = \frac{V^2}{R} \]
where \( V \) is the voltage across the heater and \( R \) is the resistance of the filament.

2. **Relate resistance to length:**
The resistance \( R \) of the filament can be expressed as:
\[ R = \rho \frac{L}{A} \]
where \( \rho \) is the resistivity of the material, \( L \) is the length of the filament, and \( A \) is the cross-sectional area of the filament.

3. **Substitute resistance into the power formula:**
By substituting the expression for resistance into the power formula, we get:
\[ P = \frac{V^2 A}{\rho L} \]

4. **Calculate the new length after reduction:**
If the length of the filament is reduced by 10%, the new length \( L' \) is:
\[ L' = L - 0.1L = 0.9L \]

5. **Determine the new power with the reduced length:**
Substitute \( L' \) into the power formula:
\[ P' = \frac{V^2 A}{\rho L'} = \frac{V^2 A}{\rho (0.9L)} = \frac{V^2 A}{0.9 \rho L} \]
This can be simplified to:
\[ P' = \frac{1}{0.9} \cdot \frac{V^2 A}{\rho L} = \frac{P}{0.9} \]

6. **Calculate the ratio of new power to old power:**
The ratio of the new power \( P' \) to the old power \( P \) is:
\[ \frac{P'}{P} = \frac{1}{0.9} \approx 1.111 \]

7. **Determine the percentage increase in power:**
The percentage increase in power can be calculated as:
\[ \text{Percentage Increase} = \left(\frac{P' - P}{P}\right) \times 100 = \left(\frac{1.111P - P}{P}\right) \times 100 = (1.111 - 1) \times 100 \approx 11.1\% \]

### Final Answer:
The power of the heater will increase by approximately **11%** when the length of the filament is reduced by 10%.