Question

Space between two concentric spheres of radii $r_1$ and $r_2$, such that $r_1 < r_2$, is filled with a material of resistivity $\rho$. Find the resistance between inner and outer surface of the material.

• A. $\frac{r_{1}}{r_{2}} \frac{\rho}{2}$
• B. $\frac{r_{2}-r_{1}}{r_{1}r_{2}} \frac{\rho}{4\pi}$
• C. $\frac{r_{1}r_{2}}{r_{2}-r_{1}} \frac{\rho}{4\pi}$
• D. None of these

Answer 1

Answer: (B)

Since $R=\rho \frac{l}{a}$
$\therefore R=\rho \frac{dl}{4\pi l^{2}}$.
(where $l$ is any radius and $dl$ is small element).
Total resistance,
$R=\frac{\rho}{4\pi} \,\int^ \limits{r_{2}}_{r_1} \frac{dl}{l^{2}}$
$=\frac{\rho}{4\pi}\left[-\frac{1}{l}\right]^{r_{2}}_{r_1}=\frac{\rho}{4\pi}\left[\frac{1}{r_{1}}-\frac{1}{r_{2}}\right]$.
$R=\left[\frac{r_{2}-r_{1}}{r_{1}r_{2}}\right] \frac{\rho}{4\pi}$.