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. l1 > l and l2 > l  Also \[l=\frac{{{l}_{1}}}{1+\gamma \,{{t}_{1}}}=\frac{{{l}_{2}}}{1+\gamma \,{{t}_{2}}}\] Þ \[{{l}_{1}}+\gamma \,{{l}_{1}}{{t}_{2}}={{l}_{2}}+\gamma \,{{l}_{2}}{{t}_{1}}\] Þ \[\gamma =\frac{{{l}_{1}}-{{l}_{2}}}{{{l}_{2}}{{t}_{1}}-{{l}_{1}}{{t}_{2}}}\]