Question

If one end of a capillary tube is dipped into water, then the water rises up to 3 cm. If the surface tension of water is $ 75 \times {10^{ - 3}} $ N/m, then the diameter of the capillary will be?
A. $ 0.1 $ mm
B. $ 0.5 $ mm
C. 1 mm
D. 2 mm

Answer 1

Hint
Capillary action is caused by the surface tension of liquid, which makes it move upwards in a tube without the assistance of external forces like gravity. This question is a formula based exercise to find the unknown diameter.
$ h = \dfrac{{2T\cos \theta }}{{\rho Rg}} $
where $ {{\rho }} $ is the density of water, $ {{\theta }} $ is the angle of contact made by water with the capillary’s surface which is $ {\text{0}}^\circ $ and $ {\text{g}} $ is the gravitational acceleration given as $ 9.81m/{s^2} $ .

Complete step by step answer
In the given question we are told that the capillary tube is dipped in water which rises to a certain height. We are asked to find the diameter of the said tube. The following data is provided to us:
Surface tension of water $ T = 75 \times {10^{ - 3}} $ N/m
Height to which the water rises $ h = 3cm = 0.03m $ [1 m = 100 cm]
We also know the following properties of water:
Angle of contact for water $ \theta = {0^\circ } $
Density of water $ \rho = 1000kg/{m^3} $
Acceleration due to gravity $ g = 9.81m/{s^2} $
We know that the rise of liquid in a capillary is given as:
 $\Rightarrow h = \dfrac{{2T\cos \theta }}{{\rho Rg}} $
We are aware of all the values except $ R $ . So, we put values in this equation to find the unknown as:
$\Rightarrow 0.03 = \dfrac{{2 \times 75 \times {{10}^{ - 3}} \times \cos 0}}{{1000 \times R \times 9.81}} $
We take $ R $ on the LHS:
$\Rightarrow R = \dfrac{{2 \times 75 \times {{10}^{ - 3}} \times \cos 0}}{{1000 \times 0.03 \times 9.81}} $
$\Rightarrow \Rightarrow R = \dfrac{{150 \times {{10}^{ - 6}} \times 1}}{{0.03 \times 9.81}} $ [As $ \cos 0 = 1 $ ]
Solving further gives us
$\Rightarrow R = 5.09 \times {10^{ - 4}} $ m
Converting this into mm:
$\Rightarrow R = 0.5 $ mm [ $ 1m = {10^3}mm $ ]
We calculate the diameter as:
$\Rightarrow D = 2R = 2 \times 0.5 = 1 $ mm
Hence, the correct answer is option (C) i.e. 1 mm.

Note
As you can see, the rise of water in a capillary tube is inversely proportional to its radius. Hence, the smaller the radius of the tube, the more is the rise.