Question

Compare the resistance of 100 watt and 400 watt of two bulbs if their voltage is same.
OR
What will be the resistance of a wire if their length changes to half of their original length and cross-sectional area changes two times of original?

Hint:

The resistance of a wire is inversely proportional to its cross-sectional area and directly proportional to its length.

Answer 1

a. Resistance of 100 watt and 400 watt Bulbs:
The power \( P \) consumed by an electrical device is related to the voltage \( V \) and resistance \( R \) by the formula:
\[ P = \frac{V^2}{R} \]
Rearranging the equation to solve for \( R \):
\[ R = \frac{V^2}{P} \]
Since both bulbs have the same voltage, we can compare their resistances using their respective power ratings.
For the 100 watt bulb:
\[ R_1 = \frac{V^2}{100} \]
For the 400 watt bulb:
\[ R_2 = \frac{V^2}{400} \]
Thus, the resistance of the 100 watt bulb is four times the resistance of the 400 watt bulb. Therefore:
\[ R_1 = 4 \times R_2 \]
So, the resistance of the 100-watt bulb is four times greater than the resistance of the 400-watt bulb.
b. Resistance Change with Length and Area:
The resistance of a wire is given by the formula:
\[ R = \rho \frac{L}{A} \]
where:
- \( \rho \) is the resistivity of the material,
- \( L \) is the length of the wire,
- \( A \) is the cross-sectional area of the wire.
When the length of the wire is halved and the area is doubled, we can substitute these changes into the formula.
New length = \( \frac{L}{2} \)
New area = \( 2A \)
The new resistance \( R_{\text{new}} \) will be:
\[ R_{\text{new}} = \rho \frac{\frac{L}{2}}{2A} = \frac{1}{4} \times R \]
Thus, the new resistance will be one-fourth of the original resistance.