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 '\(\theta_1\)' and '\(\theta_2\)' respectively, (\(\theta_2>\theta_1\)${}_{}$). The temperature at the interface is:

                 
1. \( \frac{\theta_2+\theta_1}{2} \)
2. \( \frac{\theta_1}{2}+\frac{2 \theta_2}{3} \)
3. \( \frac{\theta_1}{10}+\frac{9 \theta_2}{10} \)
4. \(\frac{\theta_1}{6}+\frac{5 \theta_2}{6}\)

Three rods of identical cross-sections and lengths are made of three different materials of thermal conductivity \({K_1,K_2,K_3,}\) respectively. They are joined together at their ends to make a long rod (see figure). One end of the long rod is maintained at \(100^\circ \mathrm{C}\) and the other at \(0^\circ \mathrm{C}\) (see figure). If the joints of the rod are at \(70^\circ \mathrm{C}\) and \(20^\circ \mathrm{C}\) in a steady state and there is no loss of energy from the surface of the rod, the correct relationship between \({K_1,K_2,}\) and \({K_3}\) is:

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 \(\theta\) (as shown in the figure) is given by :

  
1. \( \dfrac{\theta_2 R_2-\theta_1 R_1}{R_2-R_1} \)

2. \( \dfrac{\theta_1 R_2-\theta_2 R_1}{R_2-R_1} \)

3. \( \dfrac{\theta_1 R_2+\theta_2 R_1}{R_1+R_2} \)

4. \( \dfrac{\theta_1 R_1+\theta_2 R_2}{R_1+R_2} \)