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

Correct Answer: A

Solution :

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  \[l=\frac{{{l}_{1}}}{1+\gamma \,{{t}_{1}}}=\frac{{{l}_{2}}}{1+\gamma {{t}_{2}}}\]

\[\Rightarrow \,\,{{l}_{1}}+\gamma {{l}_{1}}{{t}_{2}}={{l}_{2}}+\gamma {{l}_{2}}{{t}_{1}}\Rightarrow \gamma =\frac{{{l}_{1}}-{{l}_{2}}}{{{l}_{2}}{{t}_{1}}-{{l}_{1}}{{t}_{2}}}\]