Question

Three metal rods made of copper, aluminium and brass, each 20 cm long and 4 cm in diameter, are maintained at 100 and $0^{\circ }$C respectively. Assume that the thermal conductivity of copper is twice that of aluminium and four times that of brass. The approximately equilibrium temperatures of the copper -aluminium and aluminium - brass junctions are respectively.

• A. ${68}^{\circ }$C and ${75}^{\circ }$C
• B. ${75}^{\circ }$C and ${68}^{\circ }$C
• C. ${57}^{\circ }$C and ${86}^{\circ }$C
• D. ${86}^{\circ }$C and ${57}^{\circ }$C

Answer 1

The correct option is D ${86}^{\circ }$C and ${57}^{\circ }$C

Heat current is same because connected in series.

$\frac{(100-T_1)2KA}{l}=\frac{(T_1-T_2)KA}{l}=\frac{(T_2-0)KA}{2l}$

Solving for $\frac{(T_1-T_2)KA}{l}=\frac{(T_2-0)KA}{2l}\Rightarrow (T_1-T_2)=\frac{T_2}{2}\Rightarrow T_1=T_2+\frac{T_2}{2}=\frac{{3T}_2}{2}\left(1\right)$

Solving for $\frac{(100-T_1)2KA}{l}=\frac{(T_1-T_2)KA}{l}\Rightarrow 200-2T_1=T_1-T_{2}From\left(1\right)\Rightarrow 200-2\frac{{3T}_2}{2}=\frac{{3T}_2}{2}-T_2\Rightarrow 200-3T_2=\frac{T_2}{2}\Rightarrow 200=\frac{7T_2}{2}\Rightarrow T_2=\frac{400}{7}\approx 57℃T_1=\frac{{3T}_2}{2}=\frac{3\times 57}{2}\approx 86℃$