question_answer

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 \[{{l}_{2}}\] 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) \[\frac{{{l}_{1}}-{{l}_{2}}}{{{l}_{1}}{{t}_{1}}-{{l}_{1}}{{t}_{1}}}\]

C) \[\frac{{{l}_{1}}-{{l}_{2}}}{{{l}_{2}}{{t}_{1}}-{{l}_{1}}{{t}_{2}}}\]

D) \[\frac{{{l}_{1}}-{{l}_{2}}}{{{l}_{1}}{{t}_{1}}-{{l}_{1}}{{t}_{1}}}\]

Correct Answer: A

Solution :

[a] Suppose, height of liquid in each arm before With temperature rise height of liquid in each With the temperature rise height of liquid in each arm increases i.e \[{{l}_{1}}>l\] and \[{{l}_{2}}>l\] Also \[l=\frac{{{l}_{1}}}{1+\gamma {{t}_{1}}}=\frac{{{l}_{2}}}{1+\gamma {{t}_{2}}}\] \[\Rightarrow {{l}_{1}}+\gamma {{l}_{2}}{{t}_{2}}={{l}_{2}}+\gamma {{l}_{2}}{{t}_{1}}\Rightarrow \gamma =\frac{{{l}_{1}}-{{l}_{2}}}{{{l}_{2}}{{t}_{1}}-{{l}_{1}}{{t}_{2}}}\]