Two sheets of thickness $d$ and $3d$ are touching each other. The temperature just outside the thinner sheet side is $A$,and on the side of the thicker sheet is $C$. The interface temperature is $B$. If $A, B$,and $C$ are in arithmetic progression,the ratio of the thermal conductivity of the thinner sheet to the thicker sheet is:

Two plates $A$ and $B$ have thermal conductivities $84 \, W m^{-1} K^{-1}$ and $126 \, W m^{-1} K^{-1}$ respectively. They have the same surface area and same thickness. They are placed in contact along their surfaces. If the temperatures of the outer surfaces of $A$ and $B$ are kept at $100^{\circ} C$ and $0^{\circ} C$ respectively,then the temperature of the surface of contact in steady state is $.......... \, ^{\circ} C$.

In steady state heat conduction,the equations that determine the heat current $j(r)$ [heat flowing per unit time per unit area] and temperature $T(r)$ in space are exactly the same as those governing the electric field $E(r)$ and electrostatic potential $V(r)$ with the equivalence given in the table below.

Heat flow	Electrostatics
$T(r)$	$V(r)$
$j(r)$	$E(r)$

We exploit this equivalence to predict the rate $\dot{Q}$ of total heat flowing by conduction from the surfaces of spheres of varying radii,all maintained at the same temperature. If $\dot{Q} \propto R^{n}$,where $R$ is the radius,then the value of $n$ is

If the thermal conductivity of aluminum is $0.5 \ cal/cm \cdot s \cdot ^\circ C$,then the temperature gradient required to conduct $10 \ cal/s \cdot cm^2$ in the steady state is ...... $^\circ C/cm$.

The ratio of the diameters of two metallic rods of the same material is $2 : 1$ and their lengths are in the ratio $1 : 4$. If the temperature difference between their ends is equal,the rate of flow of heat in them will be in the ratio: (in $:1$)

Two identical rods $AC$ and $CB$ made of two different metals having thermal conductivities in the ratio $2:3$ are kept in contact with each other at the end $C$. $A$ is at $100^{\circ}C$ and $B$ is at $25^{\circ}C$. Then the junction $C$ is at: (in $^{\circ}C$)