A capillary tube is partially dipped vertically into a vessel containing water. Because of capillarity, water rises in the tube. How can the height to which water rises in the capillary tube be increased?

Introduction / Context:
This question examines the concept of capillarity, which is the rise or fall of a liquid in a narrow tube due to surface tension and adhesive forces. Capillary action is important in many natural and practical situations, such as water rising in plant xylem, ink moving in a thin tube and oil lamps drawing fuel up a wick. Understanding which factors affect the height of capillary rise helps you apply the formula correctly in problems.

Given Data / Assumptions:

• A narrow capillary tube is partially dipped vertically in water.
• Water wets glass, so the liquid rises in the capillary.
• We want to increase the height h of the water column in the tube.
• Options involve changing the radius of the tube or the water level in the vessel.

Concept / Approach:
The height h of capillary rise for a liquid that wets the tube walls is given by the relation h = 2 * T * cosθ / (ρ * g * r), where T is surface tension, θ is the contact angle, ρ is the liquid density, g is acceleration due to gravity and r is the radius of the capillary. All other factors being constant, h is inversely proportional to r, the radius of the tube. This means that if you decrease the radius, the height of rise increases. Increasing the radius will reduce the height, and changing the water level in the vessel does not significantly affect the capillary rise formula.

Step-by-Step Solution:
Step 1: Write the formula for capillary rise: h = (2 * T * cosθ) / (ρ * g * r). Step 2: Observe that surface tension T, contact angle θ, liquid density ρ and g are properties of the liquid and environment and are not changed in the options. Step 3: Notice that the only variable mentioned in the options that appears in the formula is the radius r of the capillary tube. Step 4: Since h is inversely proportional to r, decreasing r will increase h, while increasing r will reduce h. Step 5: Realise that increasing the height of water in the vessel does not change r, T, θ, ρ or g, so it does not affect capillary rise. Step 6: Therefore, the way to increase the height of water rise is to use a capillary tube with a smaller radius.

Verification / Alternative check:
Capillary rise can be observed experimentally using glass tubes of different diameters dipped in the same liquid. You will see that the narrower tube shows a higher water column, while the wider tube shows a lower rise. This is a standard demonstration in school laboratories and matches the mathematical relationship h ∝ 1 / r. It confirms that decreasing the radius increases the height of the capillary rise.

Why Other Options Are Wrong:
Increasing the radius of the capillary tube increases r in the denominator, which decreases the height h of capillary rise. Increasing the height of water in the vessel changes the external water level but does not alter the capillary action formula, so it does not significantly change the additional rise inside the tube. None of these is incorrect because decreasing the radius is a valid and effective way to increase the height of capillary rise.

Common Pitfalls:
Students sometimes assume that pouring more water into the vessel will raise the level in the capillary tube, confusing the base water level with capillary rise. Another mistake is to think that wider tubes should allow more water to rise higher, which is opposite to what the formula shows. Remember the key relationship: for a given liquid and conditions, a narrower capillary means higher capillary rise. Writing down the formula and focusing on the dependence on r helps avoid these misconceptions.

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