A resistor is made from a hollow cylinder of length $L$, inner radius $a$, and outer radius $b$. The interior of the cylinder is filled with a different materials. Let $r$ be the distance from the axis of the cylinder. The region $a<r<b$ is filled with material of resistivity ${\rho }_1$. The region $r<a$ is filled with material of resistivity ${\rho }_2$ where ${\rho }_2=\frac{{\rho }_1}{2}$. Find the resistance $R$ across both ends of this resistor.

 

Current $I$ is flowing through the two materials having electrical conductivities ${\sigma }_1$ and ${\sigma }_2$ rest $\left({\sigma }_1>{\sigma }_2\right)$ as shown in the figure. The total amount of charge piled at the junction of the materials is

The equivalent resistance between the points $A$ and $B$ is

The current $i$ in the circuit ( see figure ) is

A battery of internal resistance $4 \mathrm{\Omega }$ is connected to the network of resistances as shown in figure. In order that the maximum power can be delivered to network, the value of R in $\mathrm{\Omega }$ should be

In the circuit, the electric current through the battery is Ampere(s).

If $X,Y$ and $Z$ in figure are identical lamps, which of the following changes to the brightness of the lamps occur when switch $S$ is closed?