Question

Water rises in a capillary up to a height h. If now this capillary is tilted by an angle of ${45^ \circ }$, then the length of the water column in the capillary becomes
A. $2h$
B. $\dfrac{h}{2}$
C. $\dfrac{h}{{\sqrt 2 }}$
D. $h\sqrt 2 $

Answer 1

Hint: In order to understand the answer to this question, it is important to understand the concept of capillary action. However, it is very important to understand that the capillary action occurs in very small tubes and inversely proportional to the diameter of the tube.

Complete step by step answer:
The capillary action is the phenomenon by which a liquid fills the space in very narrow diameters, without the assistance and even with the opposition of forces like gravity etc.
It mainly occurs, due to the presence of intermolecular forces in the liquid. This is represented by the quantity known as surface tension. Capillary action is a consequence of the quantity surface tension.
Surface tension is defined as the tendency of the liquid to occupy the most minimum surface area. The surface of the liquid tends to draw inwards as much as possible, to the highest extent so as to occupy the lowest value of surface area. This is because of the domination of cohesion of liquid molecules compared to adhesion of the molecules of air and liquid at the air-liquid interface. This results in the liquid appearing like a stretched membrane.
The correlation between the height achieved due to capillary action and the surface tension is given by Jurin’s law as:
$h = \dfrac{{2T\cos \theta }}{{\rho gr}}$
where,
T = surface tension of the liquid at the air-liquid interface
$\theta $ = contact angle or the angle made by the air-liquid interface with the surface, where it is present.
$\rho $= density of the liquid
g = acceleration due to gravity and,
r = radius of the tube.
When the tube is tilted by an angle, the contact angle changes to that angle. Now, if the tube is tilted by ${45^ \circ }$, the angle of contact moves from $\theta = {0^ \circ }$ to $\theta = {45^ \circ }$.
Now, the new value of $\cos \theta = \cos {45^ \circ } = \dfrac{1}{{\sqrt 2 }}$
Substituting the value of $\cos \theta $ in the expression, we get –
$h' = \dfrac{{2T}}{{\rho gr}} \times \dfrac{1}{{\sqrt 2 }}$
Hence, the new height will be equal to, $h' = \dfrac{h}{{\sqrt 2 }}$
Hence, the correct option is Option C.

Note:
The capillary action plays a very important role in our lives. The water is transported through the xylem from the roots in the soil to the whole tree by capillary action. Other examples include wetting of paint brush when dipped in paint and wetting of wick in an oil lamp.