Question

Space between two concentric spheres of radii $\mathrm { r } _ { 1 }$ $\mathrm { r } _ { 1 }$and ,such that $\mathrm { r } _ { 1 }$ < $\mathrm { r } _ { 2 }$ , is filled with a material of resistivity $\rho$ . Find the resistance between inner and outer surface of the material.

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

Answer 1

Answer: (B)

$\frac { \mathrm { r } _ { 2 } - \mathrm { r } _ { 1 } } { \mathrm { r } _ { 1 } \mathrm { r } _ { 2 } } \frac { \rho } { 4 \pi }$

Answer 2

: Since, $\mathrm { R } = \rho \frac { \mathrm { l } } { \mathrm { a } } \therefore \mathrm { R } = \rho \frac { \mathrm { dl } } { 4 \pi \mathrm { l } ^ { 2 } }$

(where l is any radius and dl is small element).

Total resistance,

$\mathrm { R } = \frac { \rho } { 4 \pi } \int _ { r _ { 1 } } ^ { r _ { 2 } } \frac { \mathrm { dl } } { 1 ^ { 2 } } = \frac { \rho } { 4 \pi } \left[ - \frac { 1 } { 1 } \right] _ { r _ { 1 } } ^ { r _ { 2 } } = \frac { \rho } { 4 \pi } \left[ \frac { 1 } { r _ { 1 } } - \frac { 1 } { r _ { 2 } } \right]$