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Question: A liquid of density and surface tension T rises to a height h in a capillary tube of radius r. The p...

A liquid of density and surface tension T rises to a height h in a capillary tube of radius r. The potential energy of the liquid in the capillary is
A. πT2ρg\dfrac{\pi {{T}^{2}}}{\rho g}
B. 2T2πρg\dfrac{2{{T}^{2}}\pi }{\rho g}
C. π2T2ρg\dfrac{{{\pi }^{2}}{{T}^{2}}}{\rho g}
D. 4π2T2ρg\dfrac{4{{\pi }^{2}}{{T}^{2}}}{\rho g}

Explanation

Solution

We will apply the formula of potential energy and then consider the value of height in terms of surface tension, radius of the tube and density of liquid. Then, balance the force acting on liquid and substitute the values.
Formula used:
Potential energy =mgh=mgh
Force =mg=mg

Complete answer:
Let us consider a capillary tube filled with liquid. The liquid rises up to a height ‘H’. Let T be surface tension on the liquid and is the density of the liquid.

Now we will calculate potential energy. For calculating potential energy we need to consider centre of mass and mg force will act from centre of mass. Then potential energy
=mgh2=mg\dfrac{h}{2}

The centre of mass is taken as half of the length of liquid filled. That’s why we take h/2.

We know h=2Tcos⁡θrρgh=\dfrac{2T\cos \theta }{r\rho g}

Where ‘r’ is the radius of the capillary tube.

The force acting downward i.e. ‘mg’ is balanced by an upward force i.e. T(2πr)T(2\pi r) then potential energy will be

& =T(2\pi r)\dfrac{1}{2}\left( \dfrac{2T\cos \theta }{r\rho g} \right) \\\ & =\dfrac{2{{T}^{2}}\pi \cos \theta }{\rho g} \\\ \end{aligned}$$ If we assume the angle of contact to be $$0{}^\circ $$ then potential energy will be $$\dfrac{2{{T}^{2}}\pi }{\rho g}$$. **So, the correct answer is “Option C”.** **Additional Information:** When a liquid or solid is in contact with any surface then the angle it makes with the surface is called the angle of contact. Angle of contact depends on the surface of material. **Note:** Surface tension is a force exerted on the liquid surface. It has the property to occupy minimum possible surface area. Surface tension depends on the radius of the test tube, density of the liquid and height of the column. In our daily life the principle of surface tension can be seen in thermometer, tooth paste, the detergent we use etc.