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It is written in my book that when a capillary tube is of insufficient length, the liquid does not overflow and the shape of the liquid meniscus changes, but the angle of contact remains the same. I did not understand that how it is possible because, $$r=R\cos\theta,$$ where

  • $R$ is the radius of curvature of liquid meniscus
  • $r$ is the radius of capillary tube.

As the shape is changing, hence $R$ is changing; but $r$ is a constant so $\cos\theta$ must change.

So why does angle of contact not change when the radius of curvature is changing?

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When a capillary tube is of insufficient length, once the liquid rises up and reaches the top of the tube it adjusts its radius of curvature instead of overflowing.Due to this the contact angle will change. Consider the image:enter image description here

r denotes radius of tube.

R denotes the radius of curvature of the liquid meniscus.

θ denotes the normal contact angle that the liquid has.

θ' denotes the contact angle of the liquid when the tube has insufficient length.

Let h is the sufficient height that the capillary tube must have so that the contact angle remains same

h=$\frac{2Tcos θ}{\rho gr}$

Let h' is the height of the tube which is of insufficient length.

h'=$\frac{2Tcos θ'}{\rho gr}$

Hence h and h' can be related by the following relation.

$\frac{h}{h'}$=$\frac{cos\theta}{cos\theta'}$

In conclusion, yes the contact angle will change when the tube is of insufficient length and clearly, it increases. .

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