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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
$\frac{{{l_1} - {l_2}}}{{{l_2}{t_1} - {l_1}{t_2}}}$
$\frac{{{l_1} - {l_2}}}{{{l_1}{t_1} - {l_2}{t_2}}}$
$\frac{{{l_1} + {l_2}}}{{{l_2}{t_1} + {l_1}{t_2}}}$
$\frac{{{l_1} + {l_2}}}{{{l_1}{t_1} + {l_2}{t_2}}}$
Solution
(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 $l = \frac{{{l_1}}}{{1 + \gamma \,{t_1}}} = \frac{{{l_2}}}{{1 + \gamma \,{t_2}}}$
==> ${l_1} + \gamma \,{l_1}{t_2} = {l_2} + \gamma \,{l_2}{t_1}$ ==> $\gamma = \frac{{{l_1} – {l_2}}}{{{l_2}{t_1} – {l_1}{t_2}}}$
