Two plates of the same area are placed in contact. Their thicknesses as well as their thermal conductivities are in the ratio $2:3$. The outer surface of one plate is maintained at $10^{\circ} C$ and that of the other at $0^{\circ} C$. The temperature at the common surface is (in $^{\circ} C$)

What properties should a material used for making a cooking pot possess? ($K =$ thermal conductivity,$S =$ specific heat capacity)

One end of a $2.35\,m$ long and $2.0\,cm$ radius aluminium rod $(K = 235\,W\cdot m^{-1}K^{-1})$ is held at $20^{\circ}C$. The other end of the rod is in contact with a block of ice at its melting point. The rate in $kg\cdot s^{-1}$ at which ice melts is (Take latent heat of fusion for ice as $\frac{10}{3} \times 10^5\,J\cdot kg^{-1}$)

Three rods of the same dimensions have thermal conductivities $3k, 2k$ and $k$. They are arranged as shown,with their ends at $100\,^{\circ}C, 50\,^{\circ}C$ and $0\,^{\circ}C$. The temperature of their junction is

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 $\,K$)?

For a spherical shell with inner and outer radii $r_1$ and $r_2$ respectively,the formula for the heat flow rate is = ......... where $T_1$ and $T_2$ are the temperatures of the inner and outer surfaces of the shell $(T_1 > T_2)$. The thermal conductivity of the material of the shell is $k$.