Question

Four identical rods AB, CD, CF and DE are joined as shown in figure. The length, cross -sectional area and thermal conductivity of each rod are l, A and K respectively. The ends A, E and F are maintained at temperatures $T_1,T_2$ and $T_3$ respectively. Assuming no loss of heat to the atmosphere, find the temperature at B.

Answer 1

Let the temperature at B be T

$\frac{Q_A}{t}=\frac{Q_B}{t}+\frac{Q_C}{t}$

$\Rightarrow \frac{KA(T-T_1)}{l}=\frac{KA(T-T_2)}{l+\frac{1}{2}}$

+$\frac{KA(T-T_3)}{\frac{3l}{2}}$

$\Rightarrow \frac{(T-T_1)}{l}=\frac{(T-T_2)}{l+\frac{1}{2}}+\frac{(T-T_3)}{\frac{3l}{2}}$

$\Rightarrow T-T_1=\frac{2}{3}(2T-T_2-T_3)$

$\Rightarrow 3T-3T_1=4T-2(T_2+T_3)$

$\Rightarrow -7T=+3T_1-2(T_2+T_3)$

$\Rightarrow T=\frac{-3T_1+2(T_2+T_3)}{7}$