A copper wire of radius 0.5 mm is insulated uniformly with plastic (k = 0.5 W/m K) sheathing 1mm thick. The wire is exposed to atmosphere at $30^0 C$ and the outside surface coefficient is $8 W/m^2 K$. Find the maximum safe current carried by the wire so that no part of the insulated plastic is above $75^0 C$. Also calculate critical thickness of insulation.

For copper: Thermal conductivity = 400 W/m K, specific electrical resistance = $2×10^{-8} ohm-m$.

Given,

$r1 = 0.5mm = 0.5 × 10^{-3}m$ (Radius of wire)

$r2 = 1.5mm = 1.5 × 10^{-3}m$ (Outer radius of insulation)

$Kins = 0.5W/mK$ (Thermal conductivity of insulation material)

$t3 = 30℃ = 303K$ (Atmospheric temperature)

$h = 8W/m2K$ (Convective heat transfer coefficient for surrounding atmosphere)

$t1 = 75℃ …