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Consider a cylindrical conductor of length L and area of cross-section A . The specific conductivity varies as σ x = σ 0 L x where x is the distance along the axis of the cylinder from one of its ends. The resistance of the system along the cylindrical axis is

Options

  1. A2 L 3 Aσ 0
  2. B3 L 2 Aσ 0
  3. CL 3 Aσ 0
  4. D4 L 3 Aσ 0

Correct answer

A. 2 L 3 Aσ 0

Step-by-step solution

Given, σ x = σ 0 1 x ∵ Resistance of the system along the cylindrical axis, R = ∫ 0 L ρ x A dx = ∫ 0 L 1 σ 0 L x A dx ∵ ρ x = 1 σ x = ∫ 0 L x σ 0 AL dx = 1 σ 0 AL x 3 2 3 2 0 L = 2 3 1 σ 0 AL L 3 2 - 0 = 2 3 ⋅ 1 σ 0 AL × L 3 2 , R = 2 3 ⋅ L Aσ 0

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