A, B and C are the three identical conductors but made from different materials. They are kept in contact as shown.

Their thermal conductivities are $\mathrm{K}, 2 \mathrm{~K}$ and $\frac{\mathrm{K}}{2}$. The free end of $\mathrm{A}$ is at $100^{\circ} \mathrm{C}$ and the free end of $\mathrm{C}$ is at $0^{\circ} \mathrm{C}$. During steady state, the temperature of the junction of $\mathrm{A}$ and $\mathrm{B}$ is nearly

• A

  $37^{\circ} \mathrm{C}$
• Correct answer
  B

  $71^{\circ} \mathrm{C}$
• C

  $29^{\circ} \mathrm{C}$
• D

  $63^{\circ} \mathrm{C}$

Solution

Let $\mathrm{R}$ be the thermal conductivity of conductor, then thermal conductivity of conductor $B=\frac{R}{2}$ and thermal conductivity of conductor $C=2 R$.

$\therefore$ Heat current, $\mathrm{H}=\frac{100^{\circ}-0^{\circ}}{\mathrm{R}+\frac{\mathrm{R}}{2}+2 \mathrm{R}}=\frac{200}{7 \mathrm{R}}$ If $\mathrm{T}^{\prime}$ be the temperature of the junction of $\mathrm{A}$ and B, then $$ \mathrm{H}=\frac{100-\mathrm{T}^{\prime}}{\mathrm{R}} \text { or } \frac{200}{7 \mathrm{R}}=\frac{100-\mathrm{T}^{\prime}}{\mathrm{R}} $$ or $$ \mathrm{T}^{\prime}=\frac{500}{7}=71^{\circ} \mathrm{C} $$

Hence, the correct answer is option: 'B'.