Question

Two thin metallic spherical shells of radius $a$ and $2a$ are placed with their centers coinciding. A material of thermal conductivity $k$ is filled in the space between them. Determine the expression for thermal resistance of the system.

• A. $\frac{1}{8\pi ka}$
• B. $\frac{8a^2}{\pi k}$
• C. $\frac{1}{4\pi ka}$
• D. $\frac{1}{2\pi ka}$

Answer 1

The correct option is A $\frac{1}{8\pi ka}$

Given that,
Length of inner radius $r_1=a$
Length of outer radius $r_2=2a$
We know that thermal current
$i_T=\frac{\Delta \theta }{R_T}$ ... $\left(i\right)$
where $R_T=\text{Thermal resistance}$
Let the inner and outer sphere be maintained at ${\theta }_1$ and ${\theta }_2$ temperature respectively.
For the system of concentric shells, we know
$i_T=\frac{4\pi k{r}_1{r}_2({\theta }_2-{\theta }_1)}{r_2-r_1}...\left(ii\right)$
Comparing (i) and (ii),
$R_T=\frac{r_2-r_1}{4\pi k{r}_1{r}_2}$
$\Rightarrow R_T=\frac{2a-a}{4\pi ka\times 2a}$
$\Rightarrow R_T=\frac{a}{4\pi k\left(2a^2\right)}$
or $R_T=\frac{1}{8\pi ka}$