Question

Three rods of same dimensions have thermal conductivities $3K$ , $2K$ ,and $K$ , with their ends at $100^\circ C$ , $50^\circ C$ and $0^\circ C$ respectively. They are arranged as shown in the diagram. The temperature of the junction $J$ in steady-state is

• A. $\frac{200}{3}^\circ C$
• B. $\frac{100}{3}^\circ C$
• C. $75^\circ C$
• D. $60^\circ C$

Answer 1

Answer: (A)

Let $T$ = temperature of the junction $J$
$L$ = length and $A$ = area of each rod
Heat conducted in first rod = heat conducted by second rod + heat conducted by third rod
$\left[\frac{3 K A \left(100 - T\right)}{L}\right]=\left[\frac{2 K A \left(T - 50\right)}{L}\right]+\left[\frac{A K \left(T - 0\right)}{L}\right]$
$3\left(100 - T\right)=2\left(T - 50\right)+T$
$400=6T$
$T=\frac{200}{3}^\circ C$