Answer :
Height of liquid [tex](\(h\))[/tex] in capillary tube is inversely proportional to tube radius [tex](\(r\)), \(h \propto \frac{1}{r}\).[/tex]
The rise of liquid in a capillary tube is governed by the balance between cohesive forces (the attraction between molecules of the liquid) and adhesive forces (the attraction between the liquid molecules and the tube material), as well as the pressure difference between the inside and outside of the tube.
Let's denote:
- [tex]\(h\)[/tex] as the height of the liquid column in the capillary tube.
- [tex]\(r\)[/tex] as the radius of the capillary tube.
- [tex]\(\gamma\)[/tex] as the surface tension of the liquid.
- [tex]\(\theta\)[/tex] as the contact angle between the liquid and the tube material.
- [tex]\(\rho\)[/tex] as the density of the liquid.
- [tex]\(g\)[/tex] as the acceleration due to gravity.
The upward force due to surface tension acting on the circumference of the meniscus of the liquid column is given by[tex]\(2\pi r \gamma \cos(\theta)\),[/tex] where [tex]\(2\pi r\)[/tex] is the circumference of the tube and [tex]\(\gamma \cos(\theta)\)[/tex] represents the component of the surface tension acting parallel to the tube surface.
The downward force due to the weight of the liquid column is given by [tex]\(\rho \pi r^2 h g\).[/tex]
For the system to be in equilibrium, the upward force due to surface tension must balance the downward force due to gravity:
[tex]\[2\pi r \gamma \cos(\theta) = \rho \pi r^2 h g\][/tex]
From this equation, we can derive the expression for the height \(h\) of the liquid column:
[tex]\[h = \frac{2\gamma \cos(\theta)}{\rho g r}\][/tex]
Now, let's analyze the relationship between the height of the liquid column \(h\) and the radius of the capillary tube [tex]\(r\):[/tex]
[tex]\[h \propto \frac{1}{r}\][/tex]
This shows that the height of the liquid column supported in the capillary tube is inversely proportional to the radius of the tube, as required.
