Water rises in a capillary tube to a certain height such...

Water rises in a capillary tube to a certain height such that the upward force due to surface tension is balanced by $75 \times 10^{-4} \mathrm{~N}$, force due to the weight of the liquid. If the surface tension of water is $6 \times 10^{-2} \mathrm{~N} / \mathrm{m}$, the inner circumference of the capillary must be (A) $1.25 \times 10^{-2} \mathrm{~m}$ (B) $0.50 \times 10^{-2} \mathrm{~m}$ (C) $6.5 \times 10^{-2} \mathrm{~m}$ (D) $12.5 \times 10^{-2} \mathrm{~m}$

Water rises in a capillary tube to a certain height such that the upward force due to surface tension is balanced by $75 \times 10^{-4} \mathrm{~N}$, force due to the weight of the liquid. If the surface tension of water is $6 \times 10^{-2} \mathrm{~N} / \mathrm{m}$, the inner circumference of the capillary must be
(A) $1.25 \times 10^{-2} \mathrm{~m}$
(B) $0.50 \times 10^{-2} \mathrm{~m}$
(C) $6.5 \times 10^{-2} \mathrm{~m}$
(D) $12.5 \times 10^{-2} \mathrm{~m}$

Key Concepts

Surface Tension

Surface tension is the property of a liquid arising from unbalanced molecular cohesive forces at the surface, which causes the liquid to behave as if its surface were a stretched membrane. It is quantified as the force per unit length acting along the liquid’s surface and plays a critical role in phenomena where interfacial forces are significant.

Capillary Action

Capillary action is the ability of a liquid to flow in narrow spaces without the assistance of external forces, and it results from the interplay between cohesive forces within the liquid and adhesive forces between the liquid and a solid surface. This phenomenon causes liquids to rise or fall in capillary tubes, and is essential in understanding liquid transport in small channels.

Force Balance in Capillarity

In the context of capillary action, the rise or depression of liquid in a tube is determined by the balance between the upward force due to surface tension acting along the contact line (the circumference of the tube) and the downward gravitational force (the weight of the liquid). This equilibrium condition is crucial for calculating dimensions like the inner circumference of a capillary tube when provided with the net forces involved.

Geometric Relationship of Circumference

The inner circumference of a capillary tube is directly linked to the calculation of the net upward force exerted by surface tension, as the force is the product of the surface tension and the total length of the contact line. Understanding this geometric relationship is essential when connecting the physical properties of the liquid to the dimensions of the tube in capillary phenomena.

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