Question

Two wires P and Q are made of the same material. Wire Q has twice the diameter and half the length of wire P. If the resistance of wire P is \( R \), the resistance of wire Q will be:

• A. \( R \)
• B. \( \frac{R}{2} \)
• C. \( \frac{R}{8} \)
• D. \( 2R \)

Hint:

Remember that resistance is inversely proportional to the cross-sectional area and directly proportional to the length of the wire.

Answer 1

The Correct Option is C

To find the resistance of wire Q, we begin by using the formula for resistance: \[ R = \frac{\rho L}{A} \] where \( \rho \) is the resistivity, \( L \) is the length, and \( A \) is the cross-sectional area. 

For wire P, the resistance is given as \( R = \frac{\rho L}{A} \).

Since wire Q has twice the diameter of wire P, its cross-sectional area \( A_Q \) is four times that of P (because area \( A \propto d^2 \)). Thus, \[ A_Q = 4A \].

Wire Q also has half the length of wire P, so \( L_Q = \frac{L}{2} \).

The resistance of wire Q, \( R_Q \), is: \[ R_Q = \frac{\rho L_Q}{A_Q} = \frac{\rho \left( \frac{L}{2} \right)}{4A} = \frac{\rho L}{8A} = \frac{R}{8} \].

Therefore, the resistance of wire Q is \( \frac{R}{8} \). The correct answer is \( \frac{R}{8} \).