Question

A capillary tube with inner cross-section in the form of a square of side a is dipped vertically in a liquid of density $\rho$ and surface tension $\sigma$ which wet the surface of capillary tube with angle of contact $\theta$. The approximate height to which liquid will be raised in the tube is: (Neglect the effect of surface tension at the corners of capillary tube)
A) $\dfrac {2\sigma \cos \theta} {a\rho g}$
B) $\dfrac {4\sigma \cos \theta} {a\rho g}$
C) $\dfrac {8\sigma \cos \theta} {a\rho g}$
D) None of these

Answer 1

In this question, we will apply the formula of upward force by capillary tube on top surface of liquid and then, we will calculate the equation by weight of liquid column of height h. Now, as we know that upward force by the capillary tube on top surface of liquid is equal to the weight of the liquid column of height h so, we will equate these two equations and get the result i.e. value of h.

Complete step-by-step answer:
Here we are going to find the value of liquid column of height h so, for that –
Firstly, We need to calculate the upward force by the capillary tube on top surface of liquid, we must calculate first the product of surface tension and the perimeter of the square of side a. Then, equate both of them as they are equal.
$F=4a\sigma \cos \theta$ ……………………………… (1)
Now, secondly, we will calculate the weight of the liquid column of height h.
$W= {{a} ^ {2}}\rho gh$ ………………………………………. (2)
As in equilibrium, the weight of the liquid column of height h is equal to the surface tension force by capillary tube on the liquid.
From (1) $\Rightarrow$ (2), we get-
$\because F=W$
$4a\sigma \cos \theta = {{a} ^ {2}}\rho gh$
$h=\dfrac {4\sigma \cos \theta} {a\rho g}$

So, the correct answer is “Option B”.

Note: We must know different forces applied by solid in different directions of liquid. By only having knowledge of the upward force by capillary tube on top of surface of liquid and weight of liquid column of height h and equating them by condition of equilibrium, we can tackle this type of question of fluid mechanics.