Question

As shown in figure, a metal rod of radius $r_1$ is concentric with a metal cylindrical shell of radius $r_2$ and length $L$. The space between rod and cylinder is tightly packed with material of resistivity $\rho .A$ battery having terminal voltage $V_0$ is connected as shown. Neglecting the resistance of rod and cylinder

• A. Effective resistance R between rod and cylinder will be $\frac{\rho \ln \left(\frac{r_2}{r_1}\right)}{2\pi L}$
• B. Electric field at point $P$ between rod and cylinder will be $E=\frac{V_0}{r\ln \left(\frac{r_2}{r_1}\right)}$
• C. Current density at $P$ will be $J$= $J=\frac{V_0}{\rho r\ln \left(\frac{r_2}{r_1}\right)}$
• D. Total current $I$ will be $\frac{2\pi L{V}_0}{\rho \ln \left(\frac{r_2}{r_1}\right)}$

Answer 1

Answer: (A), (B), (C), (D)

Total current $I$ will be $\frac{2\pi L{V}_0}{\rho \ln \left(\frac{r_2}{r_1}\right)}$

$dR=\rho \frac{dr}{2\pi rL}$

$R=\frac{\rho }{2\pi L}\underset{r_1}{\overset{r_2}{\int }}\frac{dr}{r}$

$R=\frac{\rho }{2\pi L}\ln \left(\frac{r_2}{r_1}\right)$

$I=\frac{V_0}{R}=\frac{2\pi L{V}_0}{\rho \ln \left(\frac{r_2}{r_1}\right)}$