Two wires A and B of the same material and same mass have radii 2r and r, respectively. If the resistance of wire A is 4 Ω, then resistance of B will be
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 64 Ω
Given: Two wires of same material and same mass.
Wire A: radius $2r$, resistance $R_A = 4~\Omega$
Wire B: radius $r$, resistance ,$R_B $
Volume of each wire (same mass): $V = A L = \pi r^2 L$
Wire A: $A_A L_A = \pi (2r)^2 L_A = 4 \pi r^2 L_A$
Wire B: $A_B L_B = \pi r^2 L_B$
Equate volumes: $4 \pi r^2 L_A = \pi r^2 L_B \Rightarrow L_B = 4 L_A$
Resistance formula: $R = \rho \frac{L}{A}$
Wire A: $R_A = \rho \frac{L_A}{4 \pi r^2} = 4~\Omega$
Wire B: $R_B = \rho \frac{L_B}{A_B} = \rho \frac{4 L_A}{\pi r^2} = 16 \frac{\rho L_A}{4 \pi r^2} = 64~\Omega$
Answer: $R_B = 64~\Omega$