A glass rod of diameter d1=1.5 mm is inserted symmetrically into a glass capillary of inside di

step 1 A glass rod of diameter 1 .5 meter, sorry, 1 .5 millimeter submerged in a capillary tube of diam

A glass rod of diameter d1=1.5 mm is inserted symmetrically...

A glass rod of diameter $d_{1}=1.5 \mathrm{~mm}$ is inserted symmetrically into a glass capillary of inside diameter $d_{2}=$ $2 \mathrm{~mm}$. Then the whole arrangement is vertically oriented and brought in contact with the surface of water. To what height (in $\mathrm{mm}$ ) will the water rise in the capillary (in $\mathrm{mm}$ )? (Density of water = $1 \mathrm{gm} / \mathrm{cc}$, surface tension $=0.07 \mathrm{~N} / \mathrm{m}$, angle of contact $=0^{\circ}, g=10 \mathrm{~m} / \mathrm{s}^{2}$ )

Key Concepts

Capillary Action

Capillary action refers to the ability of a liquid to flow in narrow spaces without the assistance of external forces, and often against gravity. This phenomenon is due to the interplay between adhesive forces between the liquid and the surrounding solid surfaces, and cohesive forces within the liquid itself. It is typically quantified by Jurin's Law, which relates the height of the liquid rise to surface tension, the contact angle, and the effective radius of the capillary gap.

Surface Tension

Surface tension is a property of liquids that arises from the cohesive forces between liquid molecules at the interface. It is responsible for the formation of curved surfaces, such as the meniscus in a capillary tube, and contributes to the upward force that lifts the liquid in narrow spaces. The magnitude of surface tension directly influences the extent of capillary rise.

Contact Angle

The contact angle is the angle at which a liquid interface meets a solid surface, and it represents the degree of wetting. A small contact angle (less than 90°) indicates that the liquid wets the surface well and that strong adhesive forces are present, facilitating capillary rise. When the contact angle is zero, as in complete wetting, the rise is maximized because the liquid spreads completely along the surface.

Effective Radius in Annular Capillarity

When an obstruction is inserted into a capillary, the liquid rises in the annular space between the obstruction and the tube wall, rather than in a simple cylindrical tube. In such configurations, the effective radius for calculating the capillary rise is determined by the width of the gap. This adjustment is critical because the height of the rise is inversely proportional to this effective radius.

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