Question

A liquid rises to a height of \[1.8\,{\text{cm}}\] in a glass capillary A. Another glass capillary B having diameter 90% of capillary A is immersed in the same liquid. The rise of liquid in capillary B is:
A.\[1.4\,{\text{cm}}\]
B.\[1.8\,{\text{cm}}\]
C.\[2.0\,{\text{cm}}\]
D.\[2.2\,{\text{cm}}\]

Answer 1

Hint: Use the formula for the height of rise in level of liquid in the capillary tube. Determine the relation between the radius of the capillary tube and the rise in the liquid level in the capillary tube.

Formula used:
The rise in the height \[h\] of liquid in capillary tube is
\[h = \dfrac{{2T\cos \theta }}{{r\rho g}}\] …… (1)
Here, \[T\] is the surface of the liquid, \[\theta \] is the angle of contact, \[r\] is the radius of the capillary tube, \[\rho \] is the density of the liquid and \[g\] is the acceleration due to gravity.

Complete step by step answer:
The rise in the liquid level for glass capillary A is \[1.8\,{\text{cm}}\].
Let is the radius \[{r_A}\] of the glass capillary A and \[{r_B}\] is the radius of the glass capillary B.
The radius \[{r_B}\] of the glass capillary B is 90% of the radius \[{r_A}\] of glass capillary B.
\[{r_B} = 0.9{r_A}\]
Suppose \[{h_A}\] is the rise in liquid level for glass capillary A and \[{h_B}\] is the rise in liquid level for glass capillary B.
Rewrite equation (1) for the rise in the capillary level.
\[h = \dfrac{{2T\cos \theta }}{{r\rho g}}\]
From the above equation, it can be concluded that the rise in liquid level for both the capillaries A and B depends on the radii of the glass capillaries A and B as the surface tension, angle of contact, density of liquid and acceleration due to gravity for both the capillaries A and B is the same.
\[h \propto \dfrac{1}{r}\]
\[hr = \operatorname{constant} \]
Rewrite the above relation for capillaries A and B.
\[{h_A}{r_A} = {h_B}{r_B}\]
Rearrange the above equation for \[{h_B}\].
\[{h_B} = \dfrac{{{h_A}{r_A}}}{{{r_B}}}\]
Substitute \[0.9{r_A}\] for \[{r_B}\] and \[1.8\,{\text{cm}}\] for \[{h_A}\] in the above equation.
\[{h_B} = \dfrac{{\left( {1.8\,{\text{cm}}} \right){r_A}}}{{0.9{r_A}}}\]
\[ \Rightarrow {h_B} = \dfrac{{1.8\,{\text{cm}}}}{{0.9}}\]
\[ \therefore {h_B} = 2.0\,{\text{cm}}\]
Therefore, the rise of liquid in capillary B is \[2.0\,{\text{cm}}\].

So, the correct answer is “Option C.

Note:
Since the capillary A and B are immersed in the same liquid, the surface tension, density of liquid and angle of contact are the same for both capillaries A and B. The ultimate answer of the rise of liquid is in centimeter; hence, the unit of rise of liquid in capillary A is not converted in the SI system of units.