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A glass rod of radius r1 is placed symmetrically inside a vertical capillary tube of radius r2, with their lower ends at the same level, and the assembly is dipped into water (surface tension sigma, density rho, contact angle theta = 0 deg, with r2 only slightly greater than r1). Find the height to which water rises into the annular gap of the tube.

1. 2 sigma / ((r2 - r1) rho g)
2. sigma / ((r2 - r1) rho g)
3. 2 sigma / ((r2 + r1) rho g)
4. 2 sigma / ((r2² + r1²) rho g)

Correct answer: 2 sigma / ((r2 - r1) rho g)

Solution

Water fills the annulus between rod (r1) and tube (r2). Surface tension pulls up along both circumferences: F = sigma(2 pi r1 + 2 pi r2). Weight of column = rho g h * pi(r2² - r1²) = rho g h pi(r2 - r1)(r2 + r1). Equate: 2 pi sigma(r1 + r2) = rho g h pi(r2 - r1)(r2 + r1) => 2 sigma = rho g h(r2 - r1) => h = 2 sigma / ((r2 - r1) rho g).

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