The height h to which a liquid rises in a capillary tube of radius r depends, in addition to r, on i)surface tension σ of the liquid, ii)density ρ of the liquid & iii)the acceleration due to gravity g. Is it possible to obtain dimensionally a relation for h without the experimental information that h is inversely proportional to r? Obtain the relation using this information.

Dear Student,

Please find below the solution to the asked query:

The height h of a liquid column is given by:

where  is the liquid-air surface tension (force/unit length), θ is the contact angle, ρ is the density of liquid  (mass/volume), g is local acceleration due to gravity (length/square of time), and r is radius of tube (length). Thus the thinner the space in which the water can travel, the further up it goes.

We can verify the relation by using the dimensional analysis. We can't be able to derive it from dimensional analysis.

Here the dimensional equation for RHS,

$$\begin{gathered} \left[\frac{2\gamma \cos \theta }{\rho gr}\right]=\frac{\left[\gamma \right]}{\left[\rho \right]\left[g\right]\left[r\right]}=\left[M{L}^0{T}^{-2}\right]\left[M^{-1}{L}^3\right]\left[L^{-1}{T}^2\right]\left[L^{-1}\right]=\left[M^{0}L{T}^0\right] \\ \Rightarrow \left[\frac{2\gamma cos \theta }{\rho gr}\right]=\left[h\right] \\ Hence Proved \end{gathered}$$

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