Question

A uniform sphere is supplied heat electrically at the centre at a constant rate. In the steady temperatures are established at all radial locations $r$; heat flows outwards radial and is ultimately radiated out by the outer surface isotropically. In this steady state, the temperature gradient varies with radial distance $r$ according to

• A. $r^{-1}$
• B. $r^{-2}$
• C. $r^{-3}$
• D. $r^{-3/2}$

Answer 1

Answer: (B)

$r^{-2}$

As we know that $Q=-KA\frac{dT}{dX}$

where $\frac{dT}{dX}=temperaturegradient$

$Q=heatflow$

now $\frac{Q}{A}=-K\frac{dT}{dX}$

hence $\frac{dT}{dX}\propto \frac{K}{A}$ and for sphere area $A=4\pi r^2$

hence $\frac{dT}{dX}\propto r^{-2}$