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 \( 4\,\Omega \), then resistance of B will be
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( 64\,\Omega \)
The wires have the same resistivity and, having the same material and mass, the same volume. So the wire with the smaller radius is also the longer one, and both changes raise its resistance.
\( \displaystyle V = A L \Rightarrow L = \frac{V}{A}, \qquad A = \pi r^{2} \propto r^{2} \Rightarrow L \propto \frac{1}{r^{2}} \)
\( \displaystyle R = \frac{\rho L}{A} = \frac{\rho V}{A^{2}} \propto \frac{1}{r^{4}} \)
The radius of B is half that of A:
\( \displaystyle \frac{R_B}{R_A} = \left(\frac{2r}{r}\right)^{4} = 16 \)
\( \displaystyle R_B = 16 \times 4\,\Omega = 64\,\Omega \)