Figure shows a thick shell made of electrical conductivity $\sigma$ 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 internal resistance $\frac{2}{\pi} \Omega \ \&\ \epsilon = 5V$. Find value of $\sigma$ for which ice melts at maximum possible rate if $25\%$ of heat generated by shell due to joule heating is used to melt ice.
Diffcult
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$\mathrm{dR}=\frac{\rho \mathrm{d} \mathrm{r}}{4 \pi \mathrm{r}^{2}}$
$\mathrm{R}=\int \mathrm{dR}=\frac{\rho}{4 \pi} \int_{\mathrm{r}_{1}}^{\mathrm{r}_{2}} \frac{\mathrm{dr}}{\mathrm{r}^{2}} \Rightarrow \mathrm{R}=\frac{\rho\left(\mathrm{r}_{2}-\mathrm{r}_{1}\right)}{4 \pi \mathrm{r}_{1} \mathrm{r}_{2}}$
Rate of melting is max when power dissipated in sphere is max. Using maximum power transfer theorem,
$\mathrm{R}=\mathrm{r} .$ of battery i.e. $\frac{\rho\left(\mathrm{r}_{2}-\mathrm{r}_{1}\right)}{4 \pi \mathrm{r}_{1} \mathrm{r}_{2}}=\frac{2}{\pi}$
$\rho = \frac{{8{r_1}{r_2}}}{{{r_2} - {r_1}}} = \frac{{8(200)}}{{10}} = \frac{{160}}{{100}}$ (in $SI$)
Also, $\sigma=\frac{1}{\rho}=\frac{10}{16}=\frac{5}{8}$
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