Two bulbs are rated $(40 \; W, 200 \; V)$ and $(100 \; W, 200 \; V)$. What is the correct relation for their resistances?

$A$ uniform heating wire of resistance $36 \, \Omega$ is connected across a potential difference of $240 \, V$. The wire is then cut into half and a potential difference of $240 \, V$ is applied across each half separately. The ratio of power dissipation in the first case to the total power dissipation in the second case would be $1:x$,where $x$ is........... .

If the electric current in a lamp decreases by $5 \ \%$,then the power output decreases by . . . . . . . (in $\%$)

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.

What amount of heat will be generated in a coil of resistance $R$ due to a charge $q$ passing through it if the current in the coil decreases to zero uniformly during a time interval $\Delta t$?

$A$ resistor develops $500 \, J$ of thermal energy in $20 \, s$ when a current of $1.5 \, A$ is passed through it. If the current is increased from $1.5 \, A$ to $3 \, A$,what will be the energy (in $J$) developed in $20 \, s$?