Question

26. Figure shows a thick shell made of electrical conductivity,Ơ and has inner & outer radii of 10 cm & 20 cm respectively and is filled with ice inside it. Its inside and outside surface are kept at different potentials by a battery of kept at different potentials by a battery of internal resistance 2 & -5, Find value of Ơ for w maximum possible rate if 25% of heat generated by shell due to joule heating is used to melt ice. hich ice melts at (a) siemen /m (c) siemen / m (d) 8 siemen I m (b) 2 siemen Im

Answer 1

First find the resistance of cylinder. The thin shells that make up the cylinder are in series.

$dR=\frac{dr}{\sigma\cdot2\pi rL}$

where L is the length of cylinder. So,

$\int dR=\int_a^b\frac{dr}{\sigma\cdot2\pi rL}$

for inner radius a and outer radius b.

$R=\frac{1}{2\pi\sigma L}\log\left ( \frac{b}{a} \right )$

This is like a variable load resistance. The maximum power is generated in the load when its resistance becomes equal to internal resistance of the battery.

$\frac{1}{2\pi\sigma L}\log\left ( \frac{b}{a} \right )=\frac{2}{\pi}$

$\frac{1}{4L}\log\left ( \frac{b}{a} \right )=\sigma$

Take L=1

$\sigma=\frac{\log2}{4}=0.17$