Q. In a vertical U-tube containing a liquid, the two arms are maintained at different temperatures $t_{1}$ and $t_{2}.$ The liquid columns in the two arms have heights $l_{1}$ and $l_{2}$ respectively. The coefficient of volume expansion of the liquid is equal to
A) $\frac{l_{1} - l_{2}}{l_{2}t_{1} - l_{1}t_{2}}$
B) $\frac{l_{1} - l_{2}}{l_{1}t_{1} - l_{2}t_{2}}$
C) $\frac{l_{1} + l_{2}}{l_{2}t_{1} + l_{1}t_{2}}$
D) $\frac{l_{1} + l_{2}}{l_{1}t_{1} + l_{2}t_{2}}$
Explanation
(a) Suppose, height of liquid in each arm before rising the temperature is $l$.
With temperature rise height of liquid in each arm increases $i.e.\ l_{1} > l\ and\ l_{2} > l$
Also $l = \frac{l_{1}}{1 + \gamma t_{1}} = \frac{l_{2}}{1 + \gamma t_{2}}$
$\Rightarrow l_{1} + \gamma l_{1}t_{2} = l_{2} + \gamma l_{2}t_{1} \Rightarrow \gamma = \frac{l_{1} - l_{2}}{l_{2}t_{1} - l_{1}t_{2}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.