Mechanical Properties of Fluids: Capillarity | Physics for JEE Main & Advanced PDF Download

A glass tube with a very fine bore throughout its length is called a capillary tube. When a capillary tube is dipped in water, the water wets the inner side of the tube and rises inside it, as shown in figure (a). Conversely, if the same capillary tube is dipped in mercury, the mercury is depressed, as shown in figure (b). This phenomenon, where liquids rise or fall in a capillary tube, is known as capillarity.

Practical Applications of Capillarity

1. Oil in a Lamp: The oil rises in the wick by capillary action.
2. Pen Nib: The tip of the nib is split to form a narrow capillary, allowing ink to rise continuously.
3. Sap in Trees: Sap and water rise to the top of tree leaves by capillary action.
4. Wet Towels: If one end of a towel is dipped in a bucket of water and the other end hangs outside, the towel becomes wet throughout due to capillary action.
5. Blotting Paper: Ink is absorbed by blotting paper due to capillary action.
6. Soil Moisture: Sandy soil is drier than clay because the capillaries between sand particles are too large to draw water up effectively.
7. Agricultural Practices: Moisture rises in soil capillaries to the surface, where it evaporates. Ploughing and leveling fields break these capillaries to preserve soil moisture.
8. Porous Bricks: Bricks are porous and act like capillaries.

Capillary Rise (Height of a Liquid in a Capillary Tube)

Consider a liquid that wets the wall of the tube, forming a concave meniscus as shown in the figure. Let a capillary tube of radius $rr$r be dipped in a liquid with surface tension $TT$T and density $\rho \backslash rho$ρ. Let $hh$h be the height the liquid rises in the tube. The excess pressure $(p-pa)(p\; -\; p\_a)$(p−pa) is given by:

where R is the radius of the meniscus. This excess pressure causes the liquid to rise in the capillary tube until it balances the hydrostatic pressure $h\backslash rho\; g$hρg. Thus, in equilibrium:

From triangle DOAC:

Therefore,

and

This expression is called the Ascent Formula.

Discussion

(i) Wetting Liquids: For liquids that wet the glass tube, θ<90∘ and $\cos \theta \backslash cos\; \backslash theta$cosθ is positive, hence $hh$h is positive. These liquids rise in the capillary tube. Examples include water, milk, kerosene oil, and petrol.

(ii) Non-wetting Liquids: For liquids that do not wet the glass tube, $\theta >90\circ \backslash theta\; >\; 90^\backslash circ$θ>90∘ and $\cos \theta \backslash cos\; \backslash theta$cosθ is negative, hence $hh$h is negative. These liquids are depressed in the capillary tube, such as mercury.

(iii) Jurin’s Law: Since $T,\theta ,r$ and $g$g are constants, $h\propto \frac{1}{r}h\; \backslash propto\; \backslash frac\{1\}\{r\}$ . Thus, the liquid rises more in a narrow tube and less in a wider tube.

(iv) Parallel Plates in Water: If two parallel plates with spacing d are placed in a water reservoir, the height of the rise $h$h is:

(v) Concentric Tubes: If two concentric tubes of radii $r1r\_1$r1 and $r2r\_2$r2 (inner one solid) are placed in water, the height of the rise $hh$h is:

(vi) Considering Liquid Weight in Meniscus:

(vii) Meniscus Shape in Capillary Tube: When the capillary tube is vertical, the upper meniscus is concave, and the pressure due to surface tension is directed vertically upward.

(a) If $p1>p2p\_1\; >\; p\_2$, the lower meniscus is concave downward. 

(b) If , the lower meniscus is convex upward. (c) If $p1=p2p\_1\; =\; p\_2$, the lower surface will be flat.

(viii) Liquid Between Two Plates: When a small drop of water is placed between two glass plates, it forms a thin film concave outward along its boundary.

The pressure inside the film is:

If $\theta =0\backslash theta\; =$(water and glass), the upper plate is pressed downward by: