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 Rn , where R is the radius, then the value of n is

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 Rn , where R is the radius, then the value of n is (A) 2 (B) 1 (C) -1 (D) -2

Tardigrade Admin · Accepted Answer

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.

∴ n = 2

2

1

-1

-2

Temperature difference causes heat to flow. So, potential difference is equivalent to temperature difference. As, dRdV​=−E=R2−kQ​ We can write for heat flux, dRdT​=R2−kQ​ Here, Q= heat transferred. ⇒Q∝R2 when dRdT​= constant. ∴n=2