Question

In steady state heat conduction, the equations that determine the heat current $j (r)$ [heat flowing per un it time per unit area] and temperature $T(r)$ in space are exactly the same as those governing the electric field $E (r)$ and electrostatic potential $V (r)$ with the equivalence given in the table below.

Heat flow	Electrostatics
T( r)	V(r)
j(r)	E(r)

We exploit this equivalence to predict the rate $Q$ of total he at flowing by conduction from the surfaces of spheres of varying radii, all maintained at the same temperature . If $Q\propto R^{n} ,$ where $R$ is the radius, then the value of $n$ is

• A. 2
• B. 1
• C. -1
• D. -2

Answer 1

Answer: (A)

Temperature difference causes heat to flow.
So, potential difference is equivalent to temperature difference.
As, $\frac{d V}{d R}=-E=\frac{-k Q}{R^{2}}$
We can write for heat flux,
$\frac{d T}{d R}=\frac{-k Q}{R^{2}}$
Here, $Q=$ heat transferred.
$\Rightarrow Q \propto R^{2}$
when $\frac{d T}{d R}=$ constant.
$\therefore n=2$