What is thermal steady state ?

If $K_{1}$ and $K_{2}$ are the thermal conductivities $L_{1}$ and $L _{2}$ are the lengths and $A _{1}$ and $A _{2}$ are the cross sectional areas of steel and copper rods respectively such that $\frac{K_{2}}{K_{1}}=9, \frac{A_{1}}{A_{2}}=2, \frac{L_{1}}{L_{2}}=2$.

Then, for the arrangement as shown in the figure. The value of temperature $T$ of the steel - copper junction in the steady state will be ........... $^{\circ} C$

Four rods of identical cross-sectional area and made from the same metal form the sides of square. The temperature of two diagonally opposite points and $T$ and $\sqrt 2 $ $T$ respective in the steady state. Assuming that only heat conduction takes place, what will be the temperature difference between other two points

An iron bar $\left(L_{1}=0.1\; m , A_{1}\right.$ $\left.=0.02 \;m ^{2}, K_{1}=79 \;W m ^{-1} K ^{-1}\right)$ and a brass bar $\left(L_{2}=0.1\; m , A_{2}=0.02\; m ^{2}\right.$ $K_{2}=109 \;Wm ^{-1} K ^{-1}$ are soldered end to end as shown in Figure. The free ends of the iron bar and brass bar are maintained at $373 \;K$ and $273\; K$ respectively. Obtain expressions for and hence compute

$(i)$ the temperature of the junction of the two bars,

$(ii)$ the equivalent thermal conductivity of the compound bar, and

$(iii)$ the heat current through the compound bar.

A brass boiler has a base area of $0.15\; m ^{2}$ and thickness $1.0\; cm .$ It boils water at the rate of $6.0\; kg / min$ when placed on a gas stove. Estimate the temperature (in $^oC$) of the part of the flame in contact with the boiler. Thermal conductivity of brass $=109 \;J s ^{-1} m ^{-1} K ^{-1} ;$ Heat of vaporisation of water $=2256 \times 10^{3}\; J kg ^{-1}$

Two materials having coefficients of thermal conductivity $3K$ and $K$ and thickness $d$ and $3d$, respectively, are joined to form a slab as shown in the figure. The temperatures of the outer surfaces are  $\theta_2$ and $\theta_1$ respectively  $\left( {\theta _2} > {\theta _1} \right)$ . The temperature at the interface is