The rate of flow of heat through a copper rod with a temperature difference of $28^{\circ} C$ is $1400 \ cal s^{-1}$. The thermal resistance of the copper rod will be:

The temperature $\theta$ at the junction of two insulating sheets,having thermal resistances $R_{1}$ and $R_{2}$ as well as top and bottom temperatures $\theta_{1}$ and $\theta_{2}$ (as shown in figure) is given by

Three rods made of the same material and having the same cross-section have been joined as shown in the figure. Each rod is of the same length. The left and right ends are kept at $0^{\circ}C$ and $90^{\circ}C$ respectively. The temperature of the junction of the three rods will be ...... $^{\circ}C$.

Four rods of the same material and having the same cross-section and length have been joined as shown in the figure. The temperature of the junction of the four rods will be ............... $^{\circ}C$.

An insulated container is filled with ice at $0\,^{\circ}\text{C}$,and another container is filled with water that is continuously boiling at $100\,^{\circ}\text{C}$. In a series of experiments,the containers are connected by various thick metal rods that pass through the walls of the container as shown in the figure.
In experiment $I$: a copper rod is used and all ice melts in $20$ minutes.
In experiment $II$: a steel rod of identical dimensions is used and all ice melts in $80$ minutes.
In experiment $III$: both the rods are used in series and all ice melts in $t_{10}$ minutes.
In experiment $IV$: both rods are used in parallel and all ice melts in $t_{20}$ minutes.

Three conducting rods of the same material and cross-section are shown in the figure. The temperatures at $A$, $D$, and $C$ are maintained at $20 ^oC$, $90 ^oC$, and $0 ^oC$ respectively. If there is no heat flow in $AB$, find the ratio of the lengths of $BD$ and $BC$.