A 2 m long wire of resistance 4 ohm and diameter 0 . 64 mm is coated with plastic insulation of thickness 0 . 06 mm . When a current of 5 ampere flows through the wire, find the temperature difference across the insulation in steady-state if K = 0 . 16 × 10 - 2 cal cm - 1 ° C - 1 s - 1 .

Step-by-step solution

Consider a coaxial cylindrical shell of radius r and thickness dr as shown in diagram. The radial rate of flow of heat through this shell in steady state will be H = d Q dt = - K A d θ dr [Negative sign is used as with increase in r, θ decreases.] Now as for cylindrical shell A = 2πrL, H = - 2 π r L K d θ dr or ∫ a b dr r = - 2 π LK H ∫ θ 1 θ 2 d θ which on integration and simplification gives: H = d Q d t = 2 π KL θ 1 - θ 2 log e b / a ....(i) Here H = I 2 R 4.2 = 5 2 × 4 4.2 = 24 cal/s, L = 2 m = 200 cm r 1 = (0.