## Rise of Liquid in a Capillary Tube:

Consider a glass capillary of radius R dipped in water as shown in **Figure a**. The pressure just above the liquid surface in the capillary is equal to atmospheric pressure p_{o}. As the meniscus is concave and nearly spherical, the pressure below the meniscus will be (p_{o} – 2T/r) where r is the radius of the meniscus. This is the condition of unstable equilibrium because the pressure in the liquid at this level just outside the capillary is p_{o}. Therefore, the liquid from the surroundings of the capillary will rush into the capillary till the pressure at that level in the capillary is equal to the pressure outside **Figure b**. As a result, the liquid level in the capillary rises to a height till the hydrostatic pressure of the liquid compensates for the decrease in pressure, i.e.

p_{o} = (p_{o} – 2T/r) + hρgor h = 2T/rρg …………………….(a) |

Where ‘ρ’ is the density of the liquid.

From the simple geometry, the radius of the meniscus r is related to the radius of the capillary through the relationship Figure (II).

R/r = cos θ or r = R/cos θ |

Substitute the value of r in equation (a), we get-

h = 2Tcos θ/Rρg |

From this result we note-

(i) The capillarity depends on the nature of both the liquid and the solid. If θ > 90**°**, i.e., the meniscus is convex, h will be negative, i.e., the liquid will descend in the capillary (mercury in glass tube). However, if θ = 90**°**, i.e. the meniscus is plane, h = 0, i.e., no capillarity.

(ii) For a given liquid and solid at a given place as T, ρ and g are constants,

∴ h ∝ 1/R |

i.e., the liquid will rise more in the narrow capillary and vice-versa.

(iii) The height h to which a liquid rises is independent of the shape of the capillary tube till the radius of the meniscus remains the same. It is because the difference in pressure (2T/r) is compensated by the hydrostatic pressure (hρg) so h depends only on r and the shape of the tube is of no significance.

(iv) In case of capillary of insufficient length, i.e., L < h, the liquid will rise in the capillary and on reaching the upper end, will increase the radius r’ of its meniscus **Figure c** such that hr = Lr’.

Thus, the liquid neither overflows from the upper end like a fountain nor does it trickles down along the vertical sides of the tube.

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