Question

Two hollow spheres of different materials, one with double the radius and one-fourth wall thickness of the other, are filled with ice. If the times taken for complete melting of ice in the larger to the smaller one are in the ration of 25:16, then their corresponding thermal conductivities are in the ratio

• A. 4:5
• B. 5:4
• C. 8:25
• D. 25:8

Answer 1

The correct option is C

8:25

$\frac{\Delta Q}{\Delta T}$ = $\frac{KA\Delta T}{\Delta x}$ $\Rightarrow$ $\Delta Q$ = $KA\left(\frac{\Delta T}{\Delta x}\right)\Delta t$

Assuming the thickness of the spheres to be small, we have

For smaller sphere:

(rate of heat flow)(time) = (volume of ice melted)$\rho L$

i.e., $K_1\left[4\pi \right(2r{)}^2]\frac{\Delta \theta }{d}.16$ = $\frac{4}{3}\pi r^3\rho L$................(i)

For larger sphere:

$K_2\left[4\pi \right(2r{)}^2]\frac{\Delta \theta }{d}.25$ = $\frac{4}{3}\pi (2r{)}^3\rho L$................(ii)

Dividing Eq. (ii) by Eq.(i),

$\frac{K_2}{K_1}$ = $\frac{8}{25}$