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
$\left[\frac{3KA\left(100-T\right)}{L}\right]=\left[\frac{2KA\left(T-50\right)}{L}\right]+\left[\frac{AK\left(T-0\right)}{L}\right]$
$3\left(100-T\right)=2\left(T-50\right)+T$
$400=6T$
$T={\frac{200}{3}}^{\circ }C$