The following image shows capillary tubes placed in beakers containing water and mercury:
We know that the rise or fall in the level of liquid in a capillary tube is given by Jurin's law:
$$h=\frac{2S\cos\theta}{r\rho g}$$
where $h$ is the rise or fall in height accordingly as it's positive or negative, $S$ is the surface tension, $\theta$ is the contact angle of the liquid on the tube wall, $r$ is the radius of the capillary tube, $\rho$ is the mass density and $g$ is the local acceleration due to gravity. Contact angle for water with glass is $0^\circ$ and it is $140^\circ$ for mercury with glass. So $\cos\theta$ term is positive for water and negative for mercury, and so, water rises and mercury falls in a capillary tube.
I understood the mechanism due to which the level rises or falls in a capillary tube. But, when I tried to find the pressure variation within the fluid in the tube, I faced some problems as discussed below:
In figure $(a)$ the pressure at $A$ and $B$ is equal to the atmospheric pressure $P_{atm}$. From fluid statics, we know that pressure at a particular level is same and it differs only if there is any variation in vertical height. So, we can say that pressure inside the capillary tube in the horizontal level of $B$ is also $P_{atm}$. From this, we see that pressure at both $A$ and the point below it in the horizontal level of $B$ are same and is equal to $P_{atm}$. But from fluid statics we must expect there must a pressure difference due to the difference in the vertical height given by $\Delta P=\rho g \Delta h$. Why is there an inconsistency in the results obtained? I feel both methods are equally reasonable.
The case becomes even more interesting in $(b)$. Pressures at $A'$ and $B'$ are equal to $P_{atm}$. From fluid statics, pressure at the depth $h'$ must be same. We know pressure at $A'$ is $P_{atm}$. Now if we conclude pressure at all the points in this level is $P_{atm}$, we see pressure at two different vertical levels - one at the free surface in the beaker and the other at a depth $h'$ are same. But this result is counterintuitive and I think there must be at least some pressure difference. At the same time, I don't think my first argument is incorrect. Then why do we get contradictory results?
In short, I don't understand how pressure varies in a fluid within the capillary tube? Further, it would be great if you could explain why do we get contradictory results when we apply our familiar results from fluid statics - pressure at the same horizontal level is same and pressure difference due to difference in vertical heights is $\Delta P=\rho g \Delta h$?
Image Courtesy: My own work :)