I have seen this question asked before (https://forums.studentdoctor.net/threads/please-help-explain-this-concept.724066/) however, I don’t fully understand the explanation given. I am trying to understand why a mercury barometer will always rise to the same height irregardless of the surface area of the tube or container. I have seen it explained as P = F/A so F = ρhgA and P = ρhg. But I don’t understand why the last step is allowed. It seems that on the left side of the equation you are dividing by the area of the whole barometer dish but on the right hand side you are only dividing by the area of the tube.
Pascal's theorem states pressure is the same in all directions. This means that the pressure at the surface of the barometer dish is the same in all directions.
Bernoulli's theorem states that the sum of the inertial, potential and pressure components in a steadily flowing incompressible fluid is constant. Well, the fluid in a barometer isn't moving, so we just say that pressure + potential components are constant
So, now consider two barometers, one with a large area and one with a small area.
the pressure on the surface of both is P_atm. And using 1 & 2, we see that pressure (not force!) can be traced to the top of the barometer tube (no matter how narrow/wide the tube is!)