$A\, 5\, ^oC$ rise in the temperature is observed in a conductor by passing some current. When the current is doubled,then the rise in temperature will be equal to ............. $^oC$.

The current density in a circular wire is given by $J(r) = (1 \times 10^5 \text{ A/m}^3) r$,where $r$ is the radial distance and the wire's radius is $2 \text{ mm}$. If the potential applied across the wire is $70 \text{ V}$,then the energy consumed by the wire in $1000 \text{ s}$ 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.

The resistance of a device component decreases as the current through it increases and it is described by the relation,$R = \frac{0.2 I}{I-4}$,where $I$ is the current. Determine the minimum power delivered. (Assume $I > 4$) (in $W$)

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

$A$ potential $V_0$ is applied across a uniform wire of resistance $R$. The power dissipation is $P_1$. The wire is then cut into two equal halves and a potential of $V_0$ is applied across the length of each half. The total power dissipation across two wires is $P_2$. The ratio $P_2: P_1$ is $\sqrt{x}: 1$. The value of $x$ is $.............$.