The figure shows a thick spherical shell made | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Consider a thin spherical shell of radius $r$ and thickness $dr$. The resistance of this thin shell is given by $dR = \frac{\rho dr}{4\pi r^2}$,wh

Vedclass · Accepted Answer

$\frac{5}{8}\ S/m$

Vedclass · Answer

(D) Consider a thin spherical shell of radius $r$ and thickness $dr$. The resistance of this thin shell is given by $dR = \frac{\rho dr}{4\pi r^2}$,where $\rho$ is the resistivity.The total resistance $R$ of the spherical shell is the integral of $dR$ from $r_1 = 0.1\ m$ to $r_2 = 0.2\ m$:$R = \int_{r_1}^{r_2} \frac{\rho dr}{4\pi r^2} = \frac{\rho}{4\pi} \left[ -\frac{1}{r} \right]_{r_1}^{r_2} = \frac{\rho}{4\pi} \left( \frac{1}{r_1} - \frac{1}{r_2} \right) = \frac{\rho(r_2 - r_1)}{4\pi r_1 r_2}$.The rate of heat production (power) in the shell is $P = I^2 R = \left( \frac{\epsilon}{R + r_{int}} \right)^2 R$. According to the maximum power transfer theorem,the power dissipated in the shell is maximum when its resistance $R$ equals the internal resistance of the battery $r_{int}$.Setting $R = r_{int} = \frac{2}{\pi}\ \Omega$:$\frac{\rho(0.2 - 0.1)}{4\pi(0.1)(0.2)} = \frac{2}{\pi} \Rightarrow \frac{\rho(0.1)}{4\pi(0.02)} = \frac{2}{\pi} \Rightarrow \frac{\rho}{0.8} = 2 \Rightarrow \rho = 1.6\ \Omega\cdot m$.Since conductivity $\sigma = \frac{1}{\rho}$,we have $\sigma = \frac{1}{1.6} = \frac{10}{16} = \frac{5}{8}\ S/m$.

Similar Questions

What is the maximum power output that can be obtained from a cell of $\operatorname{emf} E$ and internal resistance $r$?

Power generated across a uniform wire connected across a supply is $H$. If the wire is cut into $n$ equal parts and all the parts are connected in parallel across the same supply,the total power generated in the wire is

If $10$ bulbs of $50\,W$ are used for $10$ hours a day,how many units of electricity will be consumed in a month ($30$ days)?

How much heat (in $cal$) is produced when a $210 \, W$ bulb is kept on for $5 \, min$? (Given $J = 4.2 \, J/cal$)

The resistance of a bulb filament changes with temperature. If a bulb rated at $220 \ V$ and $100 \ W$ is connected to a supply of $(220 \times 0.8) \ V$,the actual power consumed will be:

Vedclass Test Series

Exam Paper Generator

Online Exam Module