Question

Liquid medicine is to be put in the eye, with the help of a dropper. As the bulb on the top of the dropper is pressed, a drop is formed at the opening of the dropper. Assume that the drop formed at the opening is spherical. When, the net vertical force due to the surface tension\[T~=~0.11~N/m\], becomes smaller than the weight of the drop, the drop gets detached from the dropper. Then, after the drop detaches, its surface energy is: (Given, radius of the drop is \[r~=~1.4\times {{10}^{-3}}m\])
\[\begin{align}
  & A)1.4\times {{10}^{-6}}J \\
 & B)2.7\times {{10}^{-6}}J \\
 & C)5.4\times {{10}^{-6}}J \\
 & D)8.1\times {{10}^{-6}}J \\
\end{align}\]

Answer 1

Hint: The radius and surface tension of the liquid drop is given in the question. Surface energy of a sphere is the product of its surface tension and area of the sphere. Hence, surface energy of the drop after detaching from the bulb can be found using the above relation.
Formula used:
\[U=T\times A\]
\[A=4\pi {{r}^{2}}\]

Complete answer:
We have,
Surface energy, \[U=T\times A\] ---------- 1
Where,
\[T\]is the surface tension
\[A\]is the area of the drop
Give that,
\[T~=~0.11~N/m\] ------------- 2
We have,
Area of the sphere, \[A=4\pi {{r}^{2}}\]
Given,
\[r~=~1.4\times {{10}^{-3}}m\]
Then,
\[A=4\pi {{\left( 1.4\times {{10}^{-3}} \right)}^{2}}=7.84\pi \times {{10}^{-6}}{{m}^{2}}\] ----------3
Substitute 2 and 3 in equation 1. Then,
Surface energy, \[U=0.11\times 7.84\pi \times {{10}^{-6}}=2.7\times {{10}^{-6}}J\]

Answer is option B.

Additional information:
Surface tension is an impact on a liquid’s surface layer. It causes the layer to possess characteristics like elasticity and it's the result of the cohesive forces that attract liquid molecules to one another. This surface force keeps dense objects from sinking into the liquid. Molecules within a volume of liquid pull one another equally in all directions. However molecules below and to its sides pull a molecule on the liquid’s surface. Hence, the net force on this surface molecule is inward. So the surface behaves as if it was beneath tension and obtains a spherical shape.

Note:
As a result of inward pull of molecules, the surface of the liquid tends to be within the smallest possible space for a given volume of the liquid. This leads to the lowest energy level of the liquid. For a given volume, a sphere has a minimum area. The work done to increase the surface area is stored in the droplet as potential energy. This work done to extend the area is known as surface energy.