Suppose the liquid rises to a height h. Then the total...

Suppose the liquid rises to a height $h$. Then the total energy of the liquid in the capillary is $$ E(h)=\frac{\pi}{4}\left(d_{2}^{2}-d_{1}^{2}\right) h \times \rho g \times \frac{h}{2}-\pi\left(d_{2}-d_{1}\right) \alpha h $$ Minimising $E$ we get $$ h=\frac{4 \alpha}{\rho g\left(d_{2}-d_{1}\right)}=6 \mathrm{~cm} $$

Suppose the liquid rises to a height $h$. Then the total energy of the liquid in the capillary is
$$
E(h)=\frac{\pi}{4}\left(d_{2}^{2}-d_{1}^{2}\right) h \times \rho g \times \frac{h}{2}-\pi\left(d_{2}-d_{1}\right) \alpha h
$$
Minimising $E$ we get
$$
h=\frac{4 \alpha}{\rho g\left(d_{2}-d_{1}\right)}=6 \mathrm{~cm}
$$

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Key Concepts

Capillarity

Capillarity refers to the phenomenon where liquid rises or falls in a narrow tube due to the interplay between adhesive forces (between the liquid and the tube material) and cohesive forces (within the liquid). This behavior is crucial in understanding how liquids behave in confined geometries.

Surface Tension

Surface tension is the energy per unit area at the interface between a liquid and another phase, typically air. It plays a critical role in capillary phenomena by contributing a surface energy term that influences the shape and behavior of the liquid surface in contact with solid boundaries.

Gravitational Potential Energy

Gravitational potential energy in a liquid column represents the energy stored due to the liquid’s elevated position in a gravitational field. As the liquid rises in a capillary tube, it gains gravitational potential energy, which must be accounted for when determining the equilibrium height of the liquid.

Energy Minimization

Energy minimization is a fundamental principle in physics stating that systems will settle into a configuration that minimizes their total energy. In capillary problems, the equilibrium height of the liquid is found by balancing the gravitational energy against the surface energy, resulting in a minimal total energy configuration.

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