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Chapter 7: Problem 999

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

Expert verified
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

01

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.
02

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 \( r_2 > r_1 \).
03

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.
04

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).

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