Capillary Rise of Water in Truncated Conical Glass Tube

To determine the height $h$ to which water ascends in the conical capillary tube, we can analyze the equilibrium of the liquid column by considering the forces acting on it at the upper contact line.

1. Geometry of the Conical Tube

• The apex angle of the truncated cone is $\alpha$.
• The semi-vertical angle (the angle made by the inner wall of the tube with the vertical axis) is: $\phi = \frac{\alpha}{2}$
• Since the tube tapers upwards, the inner glass wall is inclined inwards at an angle $\frac{\alpha}{2}$ relative to the vertical.

2. Direction of Surface Tension Force

• The surface tension force acts along the liquid-gas interface at the line of contact (a circle of radius $b$).
• The contact angle $\theta$ is the angle between the glass surface (wall of the tube) and the liquid-gas interface.
• Since the wall itself is inclined at an angle $\frac{\alpha}{2}$ to the vertical, the angle made by the surface tension force vector with the vertical direction is: $\theta_v = \theta + \frac{\alpha}{2}$

3. Vertical Force Balance

At equilibrium, the upward component of the surface tension force balances the weight of the elevated liquid column.

• Upward force due to surface tension ($F_{\text{up}}$): The perimeter of the cross-section at height $h$ is $2\pi b$. The upward vertical component of the surface tension force is: $F_{\text{up}} = (2\pi b) \cdot S \cos\left(\theta + \frac{\alpha}{2}\right)$

• Downward force due to gravity ($W$): Neglecting the small volume of the meniscus, the weight of the liquid column of radius $b$ and height $h$ is: $W = \rho \cdot (\pi b^2 h) \cdot g$

Equating the upward surface tension force to the weight of the liquid column: $2\pi b S \cos\left(\theta + \frac{\alpha}{2}\right) = \pi b^2 \rho g h$

4. Solving for Height $h$

Canceling common terms on both sides: $h = \frac{2S}{b\rho g} \cos\left(\theta + \frac{\alpha}{2}\right)$

Thus, the correct option is D.