Question

Three rods of same dimensions have thermal conductivities $k,\, 2 k$, and $3 k .$ They are arranged as shown in figure with their ends at $100^{\circ} C ,\, 40^{\circ} C$ and $0^{\circ} C$. The temperature of their junction is

• A. $40^{\circ} C$
• B. $50^{\circ} C$
• C. $60^{\circ} C$
• D. $30^{\circ} C$

Answer 1

Answer: (D)

$R_{1}=\frac{l}{K A}$
$R_{2}=\frac{l}{2 K A}$
$R_{3}=\frac{l}{3 K A}$
$R_{1}: R_{2}: R_{3}:: 1: \frac{1}{2}: \frac{1}{3}$
$\frac{d H_{1}}{d t}=\frac{d H_{2}}{d t}+\frac{d H_{3}}{d t}$
$\frac{100-T}{R_{1}}=\frac{T-40}{R_{2}}+\frac{T-0}{R_{3}}$
$(100-T)=2(T-40)+3(T-0)$
$100-T=2 T-80+3 T$
$100+80=3 T+2 T+T$
$T=30^{\circ} C$