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I have seen derivations of capillary height using force equations:

$$2\pi RT \cos{\theta} = \rho \pi R^{2}hg$$

which gives,

$$h = \frac{2T\cos{\theta}}{\rho gR}$$

Now, if we go about this another way, the pressure at the free surface $P_{\text{atm}}$ should be equal to the pressure inside the capillary tube, at the same level.

Or,

$$P_{\text{atm}} - \frac{2T}{R} + ρgh = P_{\text{atm}}$$

This will give:

$$h = \frac{2T}{\rho gR},$$ which is wrong.

What is the mistake in this alternate derivation?

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2 Answers 2

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I figured it out. In the derivation using force equations, we use the radius of the capillary tube, while the pressure equation uses the radius of curvature of the meniscus.

As cosθ is the ratio of these two radii, the formula holds.

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In h= 2Tcos@/rgd , r represents radius of capillary tube

h= 2T/Rdg is also right but here R represents radius of meniscus

R and r are related as r=Rcos@ where @ is angle of contact.

So you can substitute for R in one expression to get the other expression

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