Q. Two rods $P$ and $Q$ have equal lengths. Their thermal conductivities are $K_{1}\ and\ K_{2}$ and cross sectional areas are $A_{1}\ and\ A_{2}.$ When the temperature at ends of each rod are $T_{1}\ and\ T_{2}$ respectively, the rate of flow of heat through $P\ and\ Q$ will be equal, if
A) $\frac{A_{1}}{A_{2}} = \frac{K_{2}}{K_{1}}$
B) $\frac{A_{1}}{A_{2}} = \frac{K_{2}}{K_{1}} \times \frac{T_{2}}{T_{1}}\ $
C) $\frac{A_{1}}{A_{2}} = \ \sqrt{\frac{K_{1}}{K_{2}}\ }$
D) $\frac{A_{1}}{A_{2}} = \left( \ \frac{K_{2}}{K_{1}} \right)^{2}$
Explanation
(a) $\left( \frac{\mathrm{\Delta}Q}{\mathrm{\Delta}t} \right)_{P} = \left( \frac{\mathrm{\Delta}Q}{\mathrm{\Delta}t} \right)_{Q}$
$K_{1}A_{1}\frac{\left( T_{1} - T_{2} \right)}{l} = K_{2}A_{2}\frac{\left( T_{1} - T_{2} \right)}{l}$
Or $K_{1}A_{1} = K_{2}A_{2}\ \ \ or\ \ \ \frac{A_{1}}{A_{2}} = \frac{K_{2}}{K_{1}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.