Q.
Five rods of same dimensions are arranged as shown in figure. They have thermal conductivities $K_{1},\ K_{2},\ K_{3},\ K_{4}\ and\ \ K_{5}.$ When points $A\ and\ B$ are maintained at different temperature, no heat would flow through central rod, if
A) $K_{1}K_{4} = K_{2}\ K_{3}\ $
B) $K_{1} = K_{4}\ and\ \ K_{2} = K_{3}$
C) $\frac{K_{1}}{K_{4}} = \frac{K_{2}}{K_{3}}\ $
D) $K_{1}K_{2} = K_{3}K_{4}.$
Explanation
(a) The equivalent electrical circuit, figure in these cases is of Wheatstone bridge. No current would flow through central rod $CD$ when the bridge is balanced. The condition for balanced Wheatstone bridge is $\frac{P}{Q} = \frac{R}{S}$ (in terms of resistances)
$\frac{1/K_{1}}{1.K_{2}} = \frac{1/K_{3}}{1/K_{4}}$ or $\frac{K_{2}}{K_{1}} = \frac{K_{4}}{K_{3}}$
Or $K_{1}K_{4} = K_{2}K_{3}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.