9702/May Jun/11/2011/Q34

The resistance of a metal cube is measured by placing it between two parallel plates, as shown.

The cube has volume V and is made of a material with resistivity ρ. The connections to the cube
have negligible resistance.

Which expression gives the electrical resistance of the metal cube between X and Y?

Solution:
Answer: C

Resistance R of a wire = ρL / A

Where L is the length of the wire and A is the cross-sectional area of the wire.

For a cube, all the sides are of equal lengths. Let the length be L/

Volume V = L³

Length = L and cross-sectional area (in contact with the plates) = L²

Resistance R = ρL / L² = ρ / L

But since V = L³, length L = V^1/3

Resistance R = ρ / V^1/3

Alternatively, if we consider the units of the quantities involved, only choice C gives the unit of resistance (Ω).

Unit of resistivity = Ωm

Unit of volume V = m³

Unit of ρ / V^1/3 = [Ωm] / [m³]^1/3 = [Ωm] / m = Ω 

Reference: PYQ - May/Jun 2011 Paper 11 Q34