A narrow capillary tube is dipped vertically into water and water rises in the tube due to capillarity. How can the height to which water rises be increased?

Introduction / Context:
Capillarity is the phenomenon in which a liquid rises or falls in a very thin tube relative to the surrounding liquid level. This effect is important in many natural and technological processes, such as the movement of water in soil and plants, and ink in capillary pens. In the case of water in a clean glass capillary tube, the liquid rises above the outside water level. This question asks how we can increase the height of that capillary rise by changing the geometry of the tube or the vessel.

Given Data / Assumptions:

• A clean, narrow glass capillary tube is dipped vertically into pure water.
• The water wets the glass, so the contact angle is acute.
• Capillary rise occurs due to surface tension and adhesion between water and glass.
• Other conditions such as temperature and liquid properties are kept constant.

Concept / Approach:
For a wetting liquid like water in a glass capillary tube, the height h of capillary rise is given by the formula h = (2 * T * cos(theta)) / (rho * g * r), where T is surface tension, theta is the contact angle, rho is density of the liquid, g is acceleration due to gravity and r is the radius of the capillary tube. All other quantities being constant, the formula shows that h is inversely proportional to r. That means if the radius of the tube is made smaller, the height of the liquid column increases.

Step-by-Step Solution:
Step 1: Write the capillary rise formula for a wetting liquid: h = (2 * T * cos(theta)) / (rho * g * r). Step 2: Assume T, theta, rho and g remain unchanged for the same water and glass at the same temperature. Step 3: Observe that h is inversely proportional to r, the radius of the capillary tube. Step 4: Conclude that if r decreases, h must increase, and if r increases, h must decrease. Step 5: Changing the depth of water in the vessel does not appear in the formula and therefore does not directly affect capillary rise height. Step 6: Therefore, the height of water rise can be increased by decreasing the radius of the capillary tube.

Verification / Alternative check:
This prediction matches everyday observations. Very fine tubes or narrow gaps between plates show a noticeable rise of water, whereas wider tubes show little capillary effect. In plant xylem vessels, which are extremely narrow, significant capillary rise combines with transpiration pull to lift water to great heights. Demonstrations with tubes of different internal diameters dipped into the same water clearly show higher columns in the thinner tubes, confirming the inverse relationship between h and r.

Why Other Options Are Wrong:
By increasing the radius of the capillary tube: Increasing r reduces h according to the formula, so the rise becomes smaller, not larger.

By increasing the depth of water in the vessel only: The formula does not involve the depth of water in the reservoir; the external level only serves as a reference for measuring the rise.

By no practical change in radius or vessel height: If no relevant parameter is changed, the height of capillary rise will remain the same under identical conditions.

Common Pitfalls:
Some learners think that more water in the vessel will somehow push more water up the tube, forgetting that capillary rise is driven by surface tension and adhesion, not by hydrostatic pressure from depth for the small heights considered. Others may misremember the formula and think h is proportional to r instead of inversely proportional. Always look at the correct formula and focus on which variables are in the numerator and which are in the denominator.

Final Answer:
The height of capillary rise of water can be increased by decreasing the radius of the capillary tube.