Question

A metallic sphere having an inner radius $a$ and outer radius $b$ has thermal conductivity $k=\frac{K_0}{r^2},(a\le r\le b).$ Thermal resistance between the inner and outer surface for radial heat flow is:

• A. $\frac{b-a}{4\pi K_o}$
• B. $\frac{b^2-a^2}{4\pi K_o}$
• C. $\frac{4\pi k_0}{(b-a)}$
• D. $Zero$

Answer 1

The correct option is A $\frac{b-a}{4\pi K_o}$

As the heat is flow radially,

Considering a differential shell

of thickness dr

we know

$R=\frac{fl}{A}$

Also, $f=\frac{1}{k}$

Here, $f=\frac{r^2}{k_0},l-dr$ and $A=4\pi r^2$

So, $dR=\frac{r^2}{k_0}\frac{dr}{4\pi r^2}=\frac{dr}{4\pi k_0}$

For finding total resistance of the shell

$R_{net}={\int }_a^{b}dR={\int }_a^b\frac{dr}{4\pi k_0}$

$={\left[\frac{r}{4\pi k_0}\right]}_a^b=\frac{(b-a)}{4\pi k_0}$

Option - A is correct.