The space between two coaxial cylinders, whose radii are $a$ and $b$ (where $a<b$), as shown in the figure, is filled with a conducting medium. The specific conductivity of the medium is $\sigma$, $\sigma =\frac{{\sigma }_{\mathit{0}}}{r}$, where $r$ is the radial distance from the common axis of both the cylinders and ${\sigma }_0$ is a constant. Assuming $L\gg b$, where $L$ is the length of cylinder, find the resistance considering flow of current in radial direction.

A carbon resistance has the following colour code. What is the value of the resistance?

Model a torch battery of length $l$ to be made up of a thin cylindrical bar of radius $a$ and a concentric thin cylindrical shell of radius $b$ filled in between with an electrolyte of resistivity $\rho$ (see figure). If the battery is connected to a resistance of value $R$, the maximum Joule heating in $R$ will take place for :

In a conductor, if the number of conduction electrons per unit volume is $8.5\times {10}^{28}{\mathrm{m}}^{-3}$ and mean free time is $25f\mathrm{s}$ (femtosecond), its approximate resistivity is: $\left({\mathrm{m}}_{\mathrm{e}}=9.1\times {10}^{-31}\mathrm{k}\mathrm{g}\right)$

The Wheatstone bridge shown in the figure below, gets balanced when the carbon resistor used as $R_1$ has the colour code (orange, red, brown). The resistors $R_2$ and $R_4$ are $80 \mathrm{\Omega }$ and $40 \mathrm{\Omega }$, respectively. Assuming that the colour code for the carbon resistors gives their accurate values, the colour code for the carbon resistor, used as $R_3$, would be