The temperature difference between the ends of two cylindrical rods $A$ and $B$ of the same material is $2: 3$. In steady state, the ratio of the rates of flow of heat through the rods $A$ and $B$ is $5: 9$. If the radii of the rods $A$ and $B$ are in the ratio $1: 2$, then the ratio of lengths of the rods $A$ and $B$ is

Three metal rods made of copper, brass, and steel, each with a cross-sectional area of $4 \,cm^2$, are joined as shown in the figure. Their lengths are $46 \,cm, 13 \,cm$, and $12 \,cm$ respectively. Their coefficients of thermal conductivity are $0.92, 0.26$, and $0.12$ respectively, all in $CGS$ units. The rods are thermally insulated from the surroundings except at the ends. The rate of flow of heat through the copper rod, in $cal \,s^{-1}$, is:

Two plates of equal area are placed in contact. Their thicknesses are $2$ and $3$ units respectively. The temperature of the outer surface of the first plate is $-25^{\circ}C$ and the temperature of the outer surface of the second plate is $25^{\circ}C$. Find the temperature of the contact surface if: $(a)$ they are made of the same material,$(b)$ their thermal conductivities are in the ratio $2:3$.

One end of a metal rod of length $1.0 \ m$ and area of cross-section $100 \ cm^2$ is maintained at $100^{\circ}C$. If the other end of the rod is maintained at $0^{\circ}C$,the quantity of heat transmitted through the rod per minute is (Coefficient of thermal conductivity of the material of the rod = $100 \ W/m-K$).

Two rods with thermal conductivities $K$ and $3K$ and equal cross-sectional areas are joined as shown in the figure. Their lengths are $1 \ cm$ and $2 \ cm$,respectively. If the temperatures of the two ends of this composite rod are $0^{\circ}C$ and $100^{\circ}C$ (see figure),then the temperature $\phi$ of their interface is:

$A$ cylindrical rod with one end in a steam chamber and the other end in ice results in melting of $0.1 \ gm$ of ice per second. If the rod is replaced by another with half the length and double the radius of the first,and if the thermal conductivity of the material of the second rod is $\frac{1}{4}$ that of the first,the rate at which ice melts in $gm/sec$ will be: