Question

A hollow metallic sphere of radius $20\text{cm}$ surrounds a concentric metallic sphere of radius $5\text{cm}$. The space between the spheres is filled with a material of thermal conductivity $k$. The inner and outer spheres are maintained at ${50}^{\circ }\text{C}$ and ${10}^{\circ }\text{C}$ respectively and it is found that $100\text{J}$ of heat passes from the inner to the outer surface per second. Find the value of $k$.

• A. $1{\text{Wm}}^{-1}{}^{\circ }{\text{C}}^{-1}$
• B. $0.5{\text{Wm}}^{-1}{}^{\circ }{\text{C}}^{-1}$
• C. $2{\text{Wm}}^{-1}{}^{\circ }{\text{C}}^{-1}$
• D. $3{\text{Wm}}^{-1}{}^{\circ }{\text{C}}^{-1}$

Answer 1

The correct option is D $3{\text{Wm}}^{-1}{}^{\circ }{\text{C}}^{-1}$

Given that,
$r_1=0.05\text{m}$
$r_1=0.20\text{m}$
Rate of heat flow $H=100\text{J}/s$

Rate of heat flow through a concentric shell is given by
$H=\frac{k4\pi r_1{r}_2({\theta }_1-{\theta }_2)}{r_2-r_1}$
$\Rightarrow 100=\frac{k\times 4\times 3.14\times 5\times 20\times 40\times {10}^{-4}}{15\times {10}^{-2}}$
$\Rightarrow k=2.985\approx 3{\text{Wm}}^{-1}{}^{\circ }{\text{C}}^{-1}$