A uniform capillary tube of inner radius r is dipped vertically into a beaker filled with water. The

Q: Why does water rise in a capillary tube but mercury gets depressed?

This depends on the relative strength of adhesive and cohesive forces. For water and glass, adhesive forces dominate, leading to a concave meniscus and capillary rise (contact angle $\theta 90^\circ$).

Q: What happens to the capillary rise if the tube is of insufficient length?

If the tube is shorter than the equilibrium height $h$, the water will rise to the very top of the tube but will not overflow. Instead, the radius of curvature of the meniscus will adjust (increase) so that the surface tension force matches the weight of the shorter water column.

Q: Does capillary rise happen in a freely falling lift or in zero gravity?

In a freely falling lift, the effective gravity $g_{\text{eff}} = 0$. According to Jurin's Law, as $g \to 0$, the theoretical height $h \to \infty$. In reality, the water will rise to fill the entire length of the capillary tube, limited only by the length of the tube itself.

Q: Why do we ignore the mass of the liquid in the meniscus?

For very narrow capillary tubes ($r \ll h$), the volume of the liquid in the curved meniscus is negligible compared to the volume of the cylindrical liquid column of height $h$. Ignoring it simplifies the calculations without introducing significant error.

Q: Is the contact angle $\theta$ constant for a given pair of solid and liquid?

Yes, for a clean, pure liquid and a uniform solid surface at a constant temperature, the contact angle $\theta$ is a characteristic thermodynamic property determined by the surface energies of the solid-gas, solid-liquid, and liquid-gas interfaces.

Question

A uniform capillary tube of inner radius $r$ is dipped vertically into a beaker filled with water. The water rises to a height $h$ in the capillary tube above the water surface in the beaker. The surface tension of water is $\sigma$. The angle of contact between water and the wall of the capillary tube is $	heta$. Ignore the mass of water in the meniscus. Which of the following statements is (are) true?

Options:

(A) For a given material of the capillary tube, $h$ decreases with increase in $r$

(B) For a given material of the capillary tube, $h$ is independent of $\sigma$

(C) If this experiment is performed in a lift going up with a constant acceleration, then $h$ decreases

(D) $h$ is proportional to contact angle $	heta$

Sigma Percentile · Accepted Answer

### Introduction: The Magic of Capillary Action

Have you ever wondered how tall trees manage to transport water from their deep roots all the way to their highest leaves, defying gravity?

Or how a paper towel quickly drinks up a spill on your kitchen counter?

This beautiful phenomenon is known as **capillary action**, and it is governed by a delicate balance of molecular forces at the interface of solids, liquids, and gases.

In this article, we will dive deep into a classic JEE Advanced problem from 2018 that tests our conceptual understanding of capillary rise under varying physical conditions.

---

### The Physics of the Meniscus: Cohesive vs. Adhesive Forces

When a narrow glass tube—called a **capillary tube**—is dipped vertically into a liquid like water, we observe that the liquid rises inside the tube.

Why does this happen?

It all comes down to a molecular tug-of-war between two types of forces:
1. **Adhesive Forces**: The attractive forces between unlike molecules (e.g., water molecules and glass molecules).
2. **Cohesive Forces**: The attractive forces between like molecules (e.g., water molecules attracting each other).

For water and glass, the adhesive forces are significantly stronger than the cohesive forces.

As a result, water molecules "climb" up the glass wall, creating a curved, concave surface called a **meniscus**.

This curvature creates a pressure difference across the liquid-gas interface, pulling the liquid column upward until the upward force is perfectly balanced by the downward weight of the raised liquid column.

---

### Deriving Jurin's Law: The Force Balance

To analyze the problem quantitatively, let us derive the governing equation for capillary rise, known as **Jurin's Law**.

Let:
- $r$ be the inner radius of the capillary tube.
- $\sigma$ be the surface tension of the liquid.
- $	heta$ be the contact angle between the liquid and the tube wall.
- $ho$ be the density of the liquid.
- $h$ be the height of the capillary rise.

The surface tension force acts along the perimeter of the contact line of length $2\pi r$.

The vertical component of this force pulls the liquid upward:

$$F_{	ext{up}} = (2\pi r) \sigma \cos	heta$$

This upward force is balanced by the weight of the liquid column of height $h$ (ignoring the small mass of the meniscus):

$$W = 	ext{Mass} 	imes g = (	ext{Volume} 	imes ho) 	imes g = (\pi r^2 h) ho g$$

At equilibrium, the upward force equals the downward weight:

$$(2\pi r) \sigma \cos	heta = \pi r^2 h ho g$$

Solving for $h$, we get the celebrated **Jurin's Law**:

$$h = \frac{2\sigma \cos	heta}{rho g}$$

Now, let us use this master equation to systematically evaluate each option of our problem.

---

### Deconstructing the Options

#### 1. The Radius Factor: Why Narrower is Mightier

Let us look at **Option (a)**: *"For a given material of the capillary tube, $h$ decreases with increase in $r$"*.

From Jurin's Law, if the liquid and tube material are kept constant, then $\sigma$, $	heta$, $ho$, and $g$ are constants. This gives us a simple inverse relationship:

$$h \propto \frac{1}{r}$$

This means that as the inner radius $r$ of the capillary tube increases, the height $h$ of the liquid column must decrease.

Physically, a wider tube contains a heavier volume of water for the same perimeter, so the surface tension force cannot lift it as high.

Therefore, **Option (a) is True**.

---

#### 2. The Surface Tension Factor: The Molecular Tug-of-War

Let us look at **Option (b)**: *"For a given material of the capillary tube, $h$ is independent of $\sigma$"*.

Looking back at our master equation:

$$h = \frac{2\sigma \cos	heta}{rho g}$$

We can clearly see that $h$ is directly proportional to the surface tension $\sigma$:

$$h \propto \sigma$$

If the surface tension of the liquid increases, the upward pulling force increases, causing the liquid to rise higher.

Thus, $h$ is highly dependent on $\sigma$.

Therefore, **Option (b) is False**.

---

#### 3. The Gravity Factor: Capillary Rise in an Accelerating Lift

Let us look at **Option (c)**: *"If this experiment is performed in a lift going up with a constant acceleration, then $h$ decreases"*.

When the entire setup is placed inside an elevator accelerating upward with a constant acceleration $a$, we must transition to the accelerated frame of reference.

In this non-inertial frame, a downward **pseudo force** of magnitude $ma$ acts on the water column.

This effectively increases the acceleration due to gravity to an effective value:

$$g_{	ext{eff}} = g + a$$

Substituting $g_{	ext{eff}}$ into Jurin's Law, the new height $h'$ becomes:

$$h' = \frac{2\sigma \cos	heta}{rho(g + a)}$$

Since the denominator has increased ($g + a > g$), the height of the capillary rise must decrease ($h' < h$).

Therefore, **Option (c) is True**.

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#### 4. The Angle Factor: The Subtle Difference Between $	heta$ and $\cos	heta$

Finally, let us look at **Option (d)**: *"$h$ is proportional to contact angle $	heta$"*.

This is a classic trap designed to test mathematical precision. Jurin's Law states:

$$h \propto \cos	heta$$

The height is proportional to the **cosine** of the contact angle, not the contact angle $	heta$ itself.

Since $\cos	heta$ is a non-linear function of $	heta$, doubling the angle $	heta$ does not double the height $h$.

Therefore, **Option (d) is False**.

---

### Conclusion: The Final Verdict

By systematically applying the principles of fluid mechanics and force balance, we have determined that:
- **Option (a)** is **True**.
- **Option (b)** is **False**.
- **Option (c)** is **True**.
- **Option (d)** is **False**.

Thus, the correct statements are **(a)** and **(c)**.

Select Answer:

Visualized Solution

Visualizing Capillary Rise

The Governing Equation (Jurin's Law)

Analyzing Option (a) - Radius Dependence

Analyzing Option (b) - Surface Tension Dependence

Analyzing Option (c) - Acceleration in a Lift

Analyzing Option (d) - Contact Angle Dependence

Final Verdict

The Sigma Insight: Surface Tension and Capillary Action

Introduction

The Physics of the Meniscus

Deriving Jurin's Law

Deconstructing the Options

# 1

# 2

# 3

# 4

Conclusion

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