$A$ 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 $\frac{l_{ACB}}{l_{ADB}} = 3$.

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 $^{\circ}C$.

$A$ metallic prong consists of $4$ rods made of the same material,with the same cross-sections and the same lengths as shown below. The three forked ends are kept at $100^{\circ} C$ and the handle end is at $0^{\circ} C$. The temperature of the junction is ............. $^{\circ} C$.

$A$ cylinder of radius $R$ is made of a material with thermal conductivity $K_1$. It is surrounded by a cylindrical shell of a material with thermal conductivity $K_2$,having an inner radius $R$ and an outer radius $2R$. The two ends of this composite system are maintained at two different temperatures. There is no heat loss from the surface of the cylinder in the steady state. Find the equivalent thermal conductivity of the system.

For the shown figure,calculate the equivalent thermal resistance if the bricks are made of the same material of thermal conductivity $K$.

$ABCDE$ is a regular pentagon made of uniform wire. Heat enters at $A$ and leaves at $C$. $T_B$ and $T_D$ are the temperatures of points $B$ and $D$ respectively. Find the temperature $T_C$.