A film of a liquid is held on a circular ring of radius r....

A film of a liquid is held on a circular ring of radius $r$. If the surface tension of the liquid is $T$, the surface energy of liquid is :
(a) $\pi r^{2} T$
(b) $2 \pi r^{2} T$
(c) $4 \pi r^{2} T$
(d) none of these

Key Concepts

Surface Tension

Surface tension is a physical property of liquids that represents the force per unit length acting along the liquid's surface. It arises because molecules at the surface experience a net inward force due to the imbalance of intermolecular forces compared to molecules in the bulk, causing the surface to behave like a stretched elastic membrane.

Surface Energy

Surface energy is the energy required to increase the surface area of a liquid. It is directly related to surface tension because, as new surface area is created, work must be done against the cohesive forces within the liquid. Typically, the surface energy equals the product of the surface tension and the area of the surface created.

Double Surface Contribution in Films

When dealing with thin films of liquids, it is important to account for the fact that such films generally have two surfaces (exposed on both sides). This means that any calculation of surface energy must include the contribution from both the inner and outer surfaces, effectively doubling the energy compared to that of a single surface.

Geometrical Considerations

Geometrical analysis is essential when applying the concepts of surface energy and tension to real-world shapes. For surfaces with symmetry, like those formed by circular or spherical shapes, the calculation of the total surface area, and thereby the total surface energy, involves applying the appropriate geometric formulas. This ensures that all surfaces contributing to the overall energy are correctly accounted for.

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