Three conducting rods $AB, BC$ and $BD$ made of the same material and same cross-section are arranged as shown in the figure. Temperatures at the points $A, D$ and $C$ are maintained at $20^{\circ} C, 90^{\circ} C$ and $0^{\circ} C$ respectively. The ratio of lengths of $BD$ to $BC$ when there is no heat flow in $AB$ is:

The figure shows three different arrangements of materials $1, 2$,and $3$ to form a wall. The thermal conductivities are $k_1 > k_2 > k_3$. The left side of the wall is $20\,^{\circ}\text{C}$ higher than the right side. The temperature difference $\Delta T$ across material $1$ has the following relation in the three cases:

In a composite slab, there are two materials having coefficients of thermal conductivity $K$ and $2K$, and thicknesses $x$ and $4x$ respectively. The temperatures of the two outer surfaces of the composite slab are $T_2$ and $T_1$ $(T_2 > T_1)$. $T_2$ is on the side with conductivity $K$ and $T_1$ is on the side with conductivity $2K$. The rate of heat transfer through the slab in a steady state is $\left[ \frac{A(T_2 - T_1)K}{x} \right] \cdot f$, where $f$ is equal to:

Two identical metal rods are welded end-to-end as shown in figure $(1)$. It takes $4 \ min$ for $20 \ cal$ of heat to flow through them. If these rods are now welded side-by-side as shown in figure $(2)$,the time taken for the same amount of heat to flow through them is .......... $\min$.

Two identical metal wires of thermal conductivities $K_{1}$ and $K_{2}$ respectively are connected in series. The effective thermal conductivity of the combination is

Four identical conducting rods are joined as shown in the figure. Points $A$ and $D$ are maintained at temperatures of $200^{\circ}C$ and $20^{\circ}C$ respectively. The temperature of junction $B$ will be ....... $^{\circ}C$.