Question

When two soap bubbles of radius \(\mathrm{r}_{1}\) and \(\mathrm{r}_{2}\left(\mathrm{r}_{2}>\mathrm{r}_{1}\right)\) coalesce, the radius of curvature of common surface is........... (A) \(r_{2}-r_{1}\) (B) \(\left[\left(r_{2}-r_{1}\right) /\left(r_{1} r_{2}\right)\right]\) (C) \(\left[\left(\mathrm{r}_{1} \mathrm{r}_{2}\right) /\left(\mathrm{r}_{2}-\mathrm{r}_{1}\right)\right]\) (D) \(\mathrm{r}_{2}+\mathrm{r}_{1}\)

Short answer

The radius of curvature of the common surface when two soap bubbles of radius \(r_1\) and \(r_2\) (with \(r_2>r_1\)) coalesce is given by: \[r = \frac{r_1r_2}{r_2-r_1}\] Hence, the correct answer is option (C).

Step-by-step solution

Understand the Basic Concept of Soap Bubbles Coalescing

Before solving the problem, it is important to understand the basic concept of the coalescing of soap bubbles. When two soap bubbles of unequal radii stick together, they form a common film. According to Plateau's rules about soap bubbles shapes and structures, the common wall between two soap bubbles is always curved towards the bubble with the smaller radius. This curvature has a specific radius which this problem asks to find.

Identify the Parameter to be Found and the Given Parameters

It is asked to find the radius of curvature of the common surface after the two soap bubbles have coalesced. The given parameters are the radii of the individual soap bubbles before coalescing, i.e., \( r_1 \) and \( r_2 \) where \( r_2 > r_1 \).

Apply the Concept of Pressure Difference across the Soap Film

According to the physics of surface tension, the pressure inside a soap bubble is more than the pressure outside. The pressure difference across a soap film is given by \( P = 4T/r \), where \( P \) is the pressure difference, \( T \) is the surface tension (a property of the soap solution), and \( r \) is the radius of the bubble. For the common surface when the bubbles coalesce, the pressure differences on both sides must be the same, hence \( 4T/r_1 = 4T/r \) and \( 4T/r_2 = 4T/r \), where \( r \) is the radius of curvature of the common surface.

Solve the Equations

From the above equations, we can equate the two expressions for pressure difference to find the parameter \( r \): \[ \frac{4T}{r_1} = \frac{4T}{r_2} \] Solving this equation gives: \[ \frac{1}{r} = \frac{1}{r_1} - \frac{1}{r_2} \] Or, rearranged: \[ r = \frac{r_1r_2}{r_2-r_1} \] So, the correct answer is option (C).

Key concepts

Radius of Curvature

When two soap bubbles merge, a common surface is formed where the curvature faces the smaller bubble. The radius of curvature of this common surface is crucial to understanding the bubble dynamics. This radius of curvature acts like the bending of the surface and depends on both bubbles' radii. To find this radius for the coalesced bubble, we use this formula: \[ r = \frac{r_1r_2}{r_2-r_1} \]Here, \( r_1 \) and \( r_2 \) are the radii of the smaller and larger bubbles, respectively. This equation tells us how tightly curved the common interface is based on the size differences between the original bubbles. The calculation shows that the final radius of curvature is larger when the difference between the two original radii is small, meaning less bending.

Surface Tension

Surface tension is a crucial property in studying soap bubbles as it governs the behavior of bubbles and their stability. Surface tension arises due to the attractive forces among liquid molecules at the bubble's surface, leading to a minimized surface area. In the context of coalesced soap bubbles, surface tension must balance the pressures inside and outside the bubbles. When bubbles merge, the surface tension helps form the new common surface, and it ensures the film is minimally stable but durable enough to hold the shape. The key thing to remember is that surface tension

• Makes the bubble walls contract.
• Influences the pressure inside the bubble.
• Is a constant in calculating pressures within and external to the bubble.

Understanding these properties helps in solving problems regarding bubbles and predicting their behavior when external conditions change.

Pressure Difference

Pressure difference is a driving factor in the stability and shape of soap bubbles. Within a bubble, the pressure is always higher than the outside to balance the surface tension, which is expressed as \( P = \frac{4T}{r} \), where \( P \) stands for pressure difference, \( T \) indicates surface tension, and \( r \) symbolizes radius.When bubbles of differing sizes merge, pressure inside each bubble is different owing to their size variation. The smaller bubble has higher internal pressure. When the common surface forms, it evens out these pressure differences according to the principle that the pressures must be equal on both sides of the new film. Pressure difference influences

• How bubbles deform when they merge.
• The overall stability of the soap film.
• The force required to maintain the bubble's shape.

Comprehending how pressure difference works is essential in explaining why bubbles behave the way they do when they merge.

Plateau's Rules

Plateau's Rules guide the shape and structure of soap films and bubbles. They determine how bubbles and their networks form and stabilize. According to these rules:

• Soap films meet at a standard angle of 120° to minimize energy.
• Three films meet at a line, and four lines meet at a point.
• Common surfaces between bubbles curve toward the smaller bubble.

These rules are essential in understanding how the common surface forms when soap bubbles coalesce. Plateau's laws predict not only how bubbles will connect but also how they will hold their shape under equilibrium. Comprehending these rules helps anticipate the behavior of soapy interfaces and refine calculations related to bubble dynamics.