$A$ ring consisting of two parts $ADB$ and $ACB$ of same conductivity $k$ carries an amount of heat $H$. The $ADB$ part is now replaced with another metal keeping the temperatures $T_1$ and $T_2$ constant. The heat carried increases to $2H$. What should be the conductivity of the new $ADB$ part? Given $\frac{l_{ACB}}{l_{ADB}} = 3$.

Three rods $AB, BC$ and $AC$ having thermal resistances of $10 \text{ units}, 10 \text{ units}$ and $20 \text{ units}$ respectively,are connected as shown in the figure. Ends $A$ and $C$ are maintained at constant temperatures of $100^{\circ}C$ and $0^{\circ}C$ respectively. The rate at which the heat is crossing junction $B$ is . . . . . . $units$.

The ratio of the coefficient of thermal conductivity of two different materials is $5 : 3$. If the thermal resistance of the rods and the cross-sectional area of these materials are the same,then the ratio of the lengths of these rods will be

$A$ cylinder of radius $R$ made of a material of thermal conductivity $k_1$ is surrounded by a cylindrical shell of inner radius $R$ and outer radius $2R$ made of a material of thermal conductivity $k_2$. The two ends of the combined system are maintained at different temperatures. There is no loss of heat from the cylindrical surface and the system is in steady state. The effective thermal conductivity of the system is

The dimensional formula for thermal resistance is

Twelve identical conducting rods form the edges of a uniform cube of side length $l$. In the steady state,the junctions $B$ and $H$ are maintained at $100^{\circ}C$ and $0^{\circ}C$ respectively. Find the temperature of the junction $A$ in $^{\circ}C$.