An aluminium rod of length $1 \,m$ and a steel rod of length $2 \,m$, both having the same cross-sectional area, are soldered together end-to-end. The thermal conductivity of the aluminium rod and the steel rod is $200 \,Js^{-1} \,m^{-1} \,K^{-1}$ and $50 \,Js^{-1} \,m^{-1} \,K^{-1}$ respectively. The temperatures of the free ends are maintained at $300 \,K$ and $500 \,K$. What is the temperature of the junction (in $\,K$)?

Wires $A$ and $B$ have identical lengths and have circular cross-sections. The radius of $A$ is twice the radius of $B$,$i.e.$,${r_A} = 2{r_B}$. For a given temperature difference between the two ends,both wires conduct heat at the same rate. The relation between the thermal conductivities is given by:

The temperature of the water of a pond is $0^{\circ} C$ while that of the surrounding atmosphere is $-20^{\circ} C$. If the density of ice is $\rho$, the coefficient of thermal conductivity is $k$, and the latent heat of melting is $L$, then the thickness $Z$ of the ice layer formed increases as a function of time $t$ as:

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 $2:3$.

Two different rods $A$ and $B$ are kept as shown in the figure. The ratio of thermal conductivities of $A$ and $B$ is

Two conducting cylinders of equal length but different radii are connected in series between two heat baths kept at temperatures $T_1 = 300 \ K$ and $T_2 = 100 \ K$,as shown in the figure. The radius of the bigger cylinder is twice that of the smaller one and the thermal conductivities of the materials of the smaller and the larger cylinders are $K_1$ and $K_2$ respectively. If the temperature at the junction of the two cylinders in the steady state is $200 \ K$,then $K_1 / K_2 = \dots$