Question

Resistance of a wire is \( 2 \, \Omega \). The radius of the wire is halved on stretching it. Find out the new resistance of the wire.

Hint:

When a wire is stretched, its resistance increases due to the increase in length and the decrease in cross-sectional area.

Answer 1

Step 1: Resistance of a Wire.
The resistance \( R \) 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, and \( A \) is the cross-sectional area of the wire.
Step 2: Effect of Stretching the Wire.
When the wire is stretched, its length increases, and its radius decreases. If the radius is halved, the cross-sectional area \( A \), which is proportional to the square of the radius (\( A \propto r^2 \)), will decrease by a factor of 4. The resistance is directly proportional to the length and inversely proportional to the area. If the length increases by a factor of \( k \) (since stretching the wire increases the length), and the area decreases by a factor of 4, the new resistance \( R_{\text{new}} \) can be calculated as: \[ R_{\text{new}} = R \times \left( \frac{L_{\text{new}}}{L} \right) \times \left( \frac{A}{A_{\text{new}}} \right) = 2 \times 4 = 8 \, \Omega \]
Step 3: Conclusion.
The new resistance of the wire is \( 8 \, \Omega \).