In a mercury barometer, if we make a hole in the portion above the level of mercury, then the mercury level in the column drops and ultimately all the mercury in the column goes into the reservoir below or if it's capacity is not large enough it overflows.
The following image shows a different situation, when we make a hole below the level of mercury in the column:
The pressure at $A$ is zero and it gradually increases as we move down the column. The pressure at $C$ and $D$ are same as they are in the same horizontal level, and it is equal to the atmospheric pressure $P_{atm}$ From this reasoning, we can say that the pressure at $B$ is lesser than $P_{atm}$.
So, I concluded that the mercury doesn't come out through this hole.
But, what will happen after this step? The reason, why I got this doubt is described below:
We know from Pascal's Law that a pressure applied to a particular section of a fluid gets transmitted to all regions without any loss. Before we made the hole, the force exerted by the mercury column at $B$ was balanced by the contact force provided by the walls of the column. But, after we make a hole, the force exerted on the hole by the atmosphere is larger than the internal force. So, the pressure at all the points within the fluid must increase by $P_{atm}$.
Will the level of mercury in the column rise above $A$? Will it eventually touch the topmost part of the vertical column, as pressure at the topmost level of mercury must increase by $P_{atm}$ which was initially $0$? What will happen to the mercury in the reservoir? Will all the mercury in it will be sucked in by the column till the mercury level touches the upper part of the column? Doesn't this seem counterintuitive?
Kindly explain what will happen to this system. Is it correct to apply Pascal's Law in this way? Are my reasonings correct?
Image Courtesy: My Own Work :)
Answer
The simple force diagram that we rely on to prove that we can measure air pressure with this device gets far more complicated when you start drilling holes.
Consider doing this with water. Water behaves reasonably intuitively. If you drill a hole in the end of a water barometer like this, what you expect will happen. Water will pour out of the hole, and then all the water will drain from the tube.
The reason for this has less to do with the top and bottom of the barometer and far more to do with the top and bottom of the hole. The pressure differences there are sufficient to permit a water/air exchange, letting air into the tube.
Mercury, however, has an extraordinarily high surface tension. It loves to attach to itself and cares much less to interact with air or the edges of the tube (for most tubing materials).
Now our model is more complex. What you will find is that the forces on the mercury near the hole are affected not just by air pressure but by the surface tension as well. The mercury will try to keep the surface area of the mercury/air interface as small as possible, preventing air from going in or mercury from leaking out.
The result? The barometer will probably remain intact. The surface tension effects will provide the additional forces needed which the air pressure does not, and you'll see no significant change.
This, of course, depends on the size of the hole. Make the hole too big, and the surface tension will no longer be able to hold the mercury in.
By the way, mercury spills are bad news. I highly recommend not drilling into mercury barometers. Its really nasty stuff and needs to be disposed of properly.
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