Question

Three wires A, B, and C of the same material have lengths and area of cross-sections as \( (2l, \frac{A}{2}) \), \( (l, A) \) and \( (\frac{l}{2}, 2A) \), respectively. If the resistances of these wires are \( R_A, R_B, R_C \) respectively, then:

• A. \( R_A>R_B>R_C \)
• B. \( R_B>R_C>R_A \)
• C. \( R_C>R_A>R_B \)
• D. \( R_A>R_C>R_B \)

Hint:

The resistance of a wire depends directly on its length and inversely on its cross-sectional area. Larger lengths and smaller cross-sections result in higher resistance.

Answer 1

The Correct Option is A

The resistance 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, - \( A \) is the cross-sectional area of the wire. Now, applying this formula to the three wires: For wire A: \[ R_A = \rho \frac{2l}{\frac{A}{2}} = \rho \frac{4l}{A} \] For wire B: \[ R_B = \rho \frac{l}{A} \] For wire C: \[ R_C = \rho \frac{\frac{l}{2}}{2A} = \rho \frac{l}{4A} \] Now, comparing the resistances: \[ R_A = \frac{4 \rho l}{A}, \quad R_B = \frac{\rho l}{A}, \quad R_C = \frac{\rho l}{4A} \] Clearly, \( R_A>R_B>R_C \). Hence, the correct answer is: \[ R_A>R_B>R_C \]