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What is the derivation for the capillary water pressure equation? somehow I need to wind up...

What is the derivation for the capillary water pressure equation?

somehow I need to wind up at -4T/d.

If don't consider weight, the summation of forces in the vertical direction should only consider the vertical component of the surface tension of the water, and the pressure itself I believe.

The pressure is applied over the surface area; P_atm*4pi*r^2 or pi*d^2

Equating the forces (summation = 0), and assuming alpha = 1 such that T acts entirely in the vertical direction, I end up with the pressure P_atm = u_c = T/d (by magnitude at least, I guess i have a sign error somewhere).

Where is the 4 coming from in u_c = -4T/d?

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