A transistor, which may be approximated as a hemispherical heat source of radius r₀ = 0.1 mm. is embedded in a large silicon substrate (k = 125 W/m ∙ K) and dissipates heat at a rate q. All boundaries of the silicon are maintained at an ambient temperature of T_∞ = 27°C, except for the top surface, which is well insulated.

Obtain a general expression for the substrate temperature distribution and evaluate the surface temperature of the heat source for q = 4 W.

Silicon substrate To