Question

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

• A. Increase by about 9%
• B. Decrease by about 10%
• C. Increase by 11%
• D. Decrease by 19%

Answer 1

Answer: (C)

Increase by 11%

Answer 2

To determine how reducing the length of a filament by 10% affects the power of a heater, we need to consider how the resistance of the filament changes and how that impacts the power output.
Step 1: Understanding Resistance and Power Relationship
The resistance ($R$) of a filament is given by:
$R = \rho \frac{L}{A}$
where:
- $\rho$ is the resistivity of the material,
- $L$ is the length of the filament,
- $A$ is the cross-sectional area.
If the length ($L$) is reduced by 10%, the new length $L'$ is:
$L' = 0.9L$
Step 2: Effect on Resistance
The new resistance $R'$ with the reduced length can be written as:
$R' = \rho \frac{L'}{A} = \rho \frac{0.9L}{A} = 0.9 \left( \rho \frac{L}{A} \right) = 0.9R$
Step 3: Power Relationship
The power ($P$) of the heater is related to the voltage ($V$) and resistance ($R$) by:
$P = \frac{V^2}{R}$
With the new resistance $R'$, the new power $P'$ is:
$P' = \frac{V^2}{R'} = \frac{V^2}{0.9R}$
Step 4: Comparing Powers
The ratio of the new power to the original power is:
$\frac{P'}{P} = \frac{V^2 / 0.9R}{V^2 / R} = \frac{1}{0.9} = \frac{10}{9} \approx 1.11$
Conclusion:
If the length of the filament of a heater is reduced by 10%, the power of the heater will increase by approximately 11%.
The correct answer isoption (C): Increase by 11%