Question

The resistance of a wire of length \( L \) and radius \( r \) is \( R \). Which one of the following would provide a wire of the same material with resistance \( \frac{R}{2} \)?

• A. Using a wire of same radius and twice the length
• B. Using a wire of same radius and half length
• C. Using a wire of same length and twice the radius
• D. Using a wire of same length and half the radius

Hint:

Remember, the resistance of a wire is inversely proportional to the square of its radius. So, reducing the radius reduces the resistance.

Answer 1

The Correct Option is D

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. The area of a wire with radius \( r \) is: \[ A = \pi r^2 \] So the resistance becomes: \[ R = \rho \frac{L}{\pi r^2} \] If the radius is halved, the new radius is \( \frac{r}{2} \), and the new area will be: \[ A' = \pi \left( \frac{r}{2} \right)^2 = \frac{\pi r^2}{4} \] Thus, the new resistance will be: \[ R' = \rho \frac{L}{A'} = \rho \frac{L}{\frac{\pi r^2}{4}} = 4 \rho \frac{L}{\pi r^2} = 4R \] So halving the radius will increase the resistance by a factor of 4. To decrease the resistance by half, we need to reduce the radius, which will happen if the radius is halved. Therefore, the correct option is (D).