Very confused about surface tension

[dEdt]

See link: http://www3.interscience.wiley.com:8100/legacy/college/cutnell/0471713988/ste/ste.pdf. When defining surface tension, the article makes reference to a C-shaped apparatus. Why is γ = F/2l and not F/l? It says something about there being two surfaces, but it seems to me that because there's only one surface touching the wire, it should be F/l. Second question: In example one, surface tension is applying a force outwards on the needle. But based on the introductory explanation of surface tension, I thought surface tension was only inwards. I can see why compressing the liquid will produce some outward force against the needle, but why would that equal γL? It seems like those are two different phenomena.

I am obviously well aware that I probably have a deep missunderstanding of what's actually going on. The way I see it, any molecule on the surface will be 'sucked' in, and that's the force which is pulling the wire in in the first question. Is this at all correct?

Thanks a lot!

 

[alxm]

It says something about there being two surfaces, but it seems to me that because there's only one surface touching the wire, it should be F/l.

There are two sides.

I can see why compressing the liquid will produce some outward force against the needle, but why would that equal γL? It seems like those are two different phenomena.

Think in terms of surface area, rather than directions. The surface tension is the force working to minimize the surface area. When you place a needle on the surface, it's "stretching" the surface, increasing the surface area. The force isn't from compressing the liquid- Pour soap in it to disrupt the surface tension and the needle will sink, even though the compressibility of the water doesn't change at all.

 

[Doc Al]

Why is γ = F/2l and not F/l? It says something about there being two surfaces, but it seems to me that because there's only one surface touching the wire, it should be F/l.

The sheet of liquid has two surfaces. See Fig 3, which calls them the upper and lower surfaces.

Second question: In example one, surface tension is applying a force outwards on the needle. But based on the introductory explanation of surface tension, I thought surface tension was only inwards. I can see why compressing the liquid will produce some outward force against the needle, but why would that equal γL? It seems like those are two different phenomena.

The surface is always under tension, just like a piece of taut rope. (Imagine it as a stretched rubber sheet.) The tension is always tangential to the surface. The needle rests on the surface (which bends around it), and the surface exerts a tangential force, which in this case has a vertical component that balances the weight of the needle.

Part of the confusion may be the diagrams in Fig 2, which are inaccurate. Fig 2a implies that the inside of the liquid is under tension, when it's really under compression; Fig 2b implies that there's a net inward force at the surface, which would produce an inward acceleration.

 

[dEdt]

The tension is always tangential to the surface.

I didn't realize that. Fig. 2 made me believe that force was perpendicular to the surface, but now I see why that's wrong. Thanks a lot.