Problem 27 - Master Resource Book in JEE Main Physics

Question

Two soap bubbles \(A\) and \(B\) are formed at the two open ends of a tube. The bubble \(A\) is smaller than bubble \(B\). Valve and air can flow freely between the bubbles, then (a] there is no change in the size of the bubbles (b) the two bubbles will become of equal size (c) \(A\) will become smaller and \(B\) will become larget (d) \(B\) will become smaller and \(A\) will become larger

Step-by-Step Solution

Answer

(c) \(A\) will become smaller and \(B\) will become larger.

Step 1: Understand the Surface Tension Principle

Bubbles are formed due to the surface tension of the liquid film. Surface tension acts to minimize the surface area, causing different pressure inside bubbles of different sizes.

Step 2: Recall Pressure Difference in Bubbles

The pressure inside a soap bubble is given by the formula: \( P = P_0 + \frac{4\gamma}{r} \), where \( P_0 \) is the atmospheric pressure, \( \gamma \) is the surface tension, and \( r \) is the radius of the bubble. Smaller bubbles have a higher internal pressure because their radius is smaller.

Step 3: Apply Air Flow Between Bubbles

When the valve is open, air flows from high-pressure regions to low-pressure regions. Since Bubble \( A \) is smaller, its internal pressure is higher than that of Bubble \( B \). Therefore, air will flow from Bubble \( A \) to Bubble \( B \).

Step 4: Determine Bubble Size Outcome

As air flows from Bubble \( A \) to Bubble \( B \), Bubble \( A \) will lose air and become smaller. Conversely, Bubble \( B \) will gain air and become larger.

Key Concepts

Pressure Difference in Bubbles

Inside a bubble, there’s an intriguing balance involving pressure and surface tension. The pressure difference in bubbles is a core aspect that explains their dynamics. This phenomenon is described by the formula:\[ P = P_0 + \frac{4\gamma}{r} \]Here,:

• \( P \) represents the pressure inside the bubble.
• \( P_0 \) is the atmospheric pressure.
• \( \gamma \) is the surface tension of the bubble’s surface.
• \( r \) is the radius of the bubble.

In simple terms, the smaller the bubble, the higher its internal pressure, because the radius \( r \) appears in the denominator. This means a smaller radius leads to a larger pressure term being added onto the atmospheric pressure. Therefore, smaller bubbles exert a higher pressure compared to larger bubbles.
As the pressure increases exponentially in smaller bubbles, the air tends to migrate from areas of high pressure to areas of lower pressure. This concept is crucial in understanding the exchange dynamics between two soap bubbles of different sizes, as we’ll explore further.

Soap Bubble Dynamics

Soap bubbles are fascinating in their behavior due to the principles of physics. When two soap bubbles are connected, like in our exercise, they don’t remain static but instead interact dynamically.
**Why Do Size Changes Occur?** In the case of bubbles connecting through a tube, the size difference arises from the pressure differential between them. The bubble with smaller volume has higher internal pressure due to its smaller radius, leading the air to flow from the smaller to the larger bubble.
**Characteristics of Soap Bubbles**

• Soap bubbles have a flexible structure due to liquid surface tension, which minimizes their surface area.
• They compress easily when the pressure inside them increases.
• Colorful due to thin-film interference, showing a sheen of rainbow hues depending on thickness.

The physics behind bubble dynamics ensures that when two bubbles of different sizes connect, the smaller bubble shrinks, and the larger bubble grows, continuing until equilibrium is achieved or one bubble vanishes.

Air Flow Between Bubbles

The flow of air between bubbles is dictated by differences in pressure. This flow continues until pressure equilibrium is reached.
**How Does Air Move?** Air always moves from a region of higher pressure to a region of lower pressure. In our scenario with two connected soap bubbles:

• The smaller bubble (A) has a greater internal pressure due to its smaller size.
• The larger bubble (B) has a lower internal pressure.

This pressure gradient causes air to flow from Bubble A to Bubble B.
**Outcome of Air Transfer** As air flows from the smaller bubble to the larger one, the smaller bubble's size diminishes, while the larger bubble expands. Thus, Bubble A shrinks, and Bubble B enlarges. The air transfer stops when the pressure equalizes or when one bubble contains all the air.
Understanding this concept of air flow between bubbles offers valuable insights into the broader study of thermodynamics and fluid dynamics. Recognizing how substances move in response to pressure differences is an essential aspect of physical science exploration.