A solid glass rod of radius r is placed inside a glass...

A solid glass rod of radius $r$ is placed inside a glass cylinder of internal radius $R$. This glass rod is placed coaxially with the cylinder and in contact with the botom of cylinder and perpendicular to the surface of liquid of density $\rho$ in the glass cylinder. The height to which water will rise in the region between the rod and the cylinder due to surface tension is (The angle of contact is $0^{\circ}$ and surface tension is $T$ ) (a) $\frac{2 T}{(R-r) \rho g}$ (b) $\frac{T}{(R+r) \rho g}$ (c) $\frac{2 T}{\left(R^{2} r^{2}\right) \rho g}$ (d) $\frac{2 T}{\left(R^{2} \mid r^{2}\right) \rho g}$

A solid glass rod of radius $r$ is placed inside a glass cylinder of internal radius $R$. This glass rod is placed coaxially with the cylinder and in contact with the botom of cylinder and perpendicular to the surface of liquid of density $\rho$ in the glass cylinder. The height to which water will rise in the region between the rod and the cylinder due to surface tension is (The angle of contact is $0^{\circ}$ and surface tension is $T$ )
(a) $\frac{2 T}{(R-r) \rho g}$
(b) $\frac{T}{(R+r) \rho g}$
(c) $\frac{2 T}{\left(R^{2} r^{2}\right) \rho g}$
(d) $\frac{2 T}{\left(R^{2} \mid r^{2}\right) \rho g}$

Key Concepts

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. In the context of capillary rise, the upward force due to surface tension is balanced by the downward gravitational pull represented by the hydrostatic pressure of the liquid column. This balance determines the height to which the liquid can rise in narrow spaces.

Contact Angle

The contact angle is the angle at which a liquid interface meets a solid surface, indicating the degree of wetting. It plays a significant role in capillary action because it influences the shape and curvature of the meniscus. In cases where the contact angle is zero degrees, the liquid completely wets the surface, which affects the balance between surface tension and gravitational forces.

Effective Radius in Capillary Rise

In capillary phenomena involving non-standard geometries, such as the annular space between a rod and a cylindrical container, the effective radius of curvature becomes crucial. The effective radius is determined by the gap between the surfaces, and it directly influences the curvature of the meniscus and, consequently, the capillary pressure that drives the liquid rise. Understanding how to correctly identify this effective radius is key to solving such problems.

Capillary Action

Capillary action is the phenomenon where a liquid rises or falls in a narrow space due to the combined effects of cohesive forces within the liquid and adhesive forces between the liquid and the surrounding solid surfaces. It is a key concept in understanding how fluids behave in confined geometries, such as tubes or gaps, and is fundamentally driven by the surface tension of the liquid and the nature of its interaction with the material surfaces.

Surface Tension

Surface tension is the property of a liquid's surface that makes it behave like a stretched elastic membrane. It is defined as the force per unit length along the surface and arises due to the imbalance of molecular interactions at the interface. This concept is crucial as it is the driving force behind capillary rise, determining how much the liquid will rise or fall in a narrow space.

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