Two conductors made of the same material are connected across a common potential difference. Conductor $A$ has twice the diameter and twice the length of conductor $B$. The power delivered to the two conductors $P_A$ and $P_B$ respectively is such that $P_A/P_B$ equals to

$A$ $60\,W$ bulb operates on a $220\,V$ supply. The current flowing through the bulb is:

The figure shows a thick spherical shell made of a material with electrical conductivity $\sigma$. It has inner and outer radii of $10\ cm$ and $20\ cm$ respectively and is filled with ice inside. Its inner and outer surfaces are kept at different potentials by a battery with internal resistance $r = \frac{2}{\pi}\ \Omega$ and electromotive force $\epsilon = 5\ V$. Find the value of $\sigma$ for which the ice melts at the maximum possible rate,given that $25\%$ of the heat generated by the shell due to Joule heating is used to melt the ice.

$A$ current $i$ passes through a wire of length $l$, radius of cross-section $r$ and resistivity $\rho$. The rate of heat generation is

In an electrical circuit,a battery is connected to pass $20\, C$ of charge through it in a certain given time. The potential difference between two plates of the battery is maintained at $15\, V$. The work done by the battery is ........... $J$.

Incandescent bulbs are designed by keeping in mind that the resistance of their filament increases with the increase in temperature. If at room temperature,$100 \ W$,$60 \ W$ and $40 \ W$ bulbs have filament resistances $R_{100}$,$R_{60}$ and $R_{40}$,respectively,the relation between these resistances is