Problem 58 - Master Resource Book in JEE Main Physics

Question

By inserting a capillary tube upto a depth \(l\) in water, the water rises to a height \(h\). If the lower end of the capillary tube is closed inside water and the capillary is taken out and closed end opened, to what height the water will remain in the tube, when \(l>h\) ? (a) zero (b) \(l+h\) (c) \(2 h\) (d) \(h\)

Step-by-Step Solution

Answer

(d) The water will remain at height \( h \) in the tube.

Step 1: Understanding Capillary Action

Capillary action is the phenomenon where liquid rises or falls in a thin tube due to the adhesive forces between the liquid and the tube's surface. When a capillary tube is dipped in water, water rises to a certain height \( h \) due to this action.

Step 2: Analyze the Situation Inside Water

The capillary tube is inserted up to a depth \( l \). Because of capillary action, water rises inside the tube to a height \( h \) above the water's surface, as long as \( l > h \). The atmospheric pressure acts both inside and outside, balancing the pressure at height \( h \).

Step 3: Consider the Situation When the Tube is Removed

When the capillary tube is removed from the water with its lower end closed, the water column inside the tube is still \( h \). This is because no air can enter to disrupt the existing balance, maintaining the height of the water column at \( h \).

Step 4: What Happens After Opening the Closed End

When the closed end of the tube is opened to the atmosphere, external air pressure no longer supports the water column's height. The water column seeks to reach equilibrium with atmospheric pressure by adjusting its height to the capillary rise height \( h \).

Step 5: Conclusion on Water Height in Tube

Once the tube is open to the air, the only force maintaining water in the tube is capillary action, identical to when the tube was initially dipped. Thus, the water will remain at height \( h \) in the tube.

Key Concepts

Surface Tension

Surface tension is a key concept in understanding the phenomenon of capillary action. It refers to the elastic tendency of a liquid surface, which makes it acquire the least surface area possible. This effect is a result of the cohesive forces between the molecules in the liquid. When a liquid is in contact with a solid, like the water in the capillary tube, surface tension acts along the surface and pulls the liquid into the tube, raising it to a higher level.
The stronger the surface tension, the more significant the rise or fall of the liquid within the tube. Mathematically, surface tension is represented by the symbol \( \sigma \). In capillary action, surface tension works together with adhesive forces to allow liquids to resist gravity and climb upwards.

Adhesive Forces

Adhesive forces are the attractive forces between different molecules, such as those between a liquid and a solid surface. These forces play an essential role in capillary action. When a capillary tube is placed in water, the adhesive forces between the water molecules and the inner walls of the tube cause the water to "stick" to the walls. As a result, the water begins to climb the tube.
This adhesive interaction is typically stronger than the cohesive forces within the liquid itself, making the liquid move against gravity's pull. This movement is because the molecules at the surface attempt to attach themselves to the tube, dragging the bulk of the liquid with them. The balance of these forces tends to maintain the water at a stable height within the tube.

Atmospheric Pressure

Atmospheric pressure is the force exerted on a surface by the weight of the air above that surface in the Earth's atmosphere. It plays a vital role in determining the height to which water can rise in a capillary tube. In the initial situation described in the exercise, atmospheric pressure acts equally inside and outside the tube when it is submerged. This equilibrium helps maintain the water column at height \( h \).
When the tube is removed and the closed end is opened, atmospheric pressure adjusts to the new configuration, becoming the dominant force outside the tube. This adjustment causes the capillary action to reach a new equilibrium, aligning the water column's height with the capillary rise. Atmospheric pressure thus serves as a balancing factor against the liquid's tendency to spill out of the tube.

Hydrostatic Equilibrium

Hydrostatic equilibrium refers to the state where the pressure within a fluid, including a fluid subject to gravitational forces, is balanced. In the context of the capillary action exercise, hydrostatic equilibrium is achieved when the forces acting on the column of water in the tube are balanced. This includes the upward adhesive forces and the downward gravitational force on the water column, combined with the atmospheric pressure exerting its influence.
Once the tube's closed end is opened, the system reestablishes this equilibrium by adjusting the water column to the capillary rise height \( h \). Hydrostatic equilibrium ensures that no additional water is drawn up the tube or forced back down, maintaining a steady state as dictated by capillary action in the presence of atmospheric pressure.