Question

The radii of two soap bubbles are \(\mathrm{r}_{1}\) and \(\mathrm{r}_{2} .\) In isothermal conditions two meet together is vacuum Then the radius of the resultant bubble is given by (A) \(\mathrm{R}=\left[\left(\mathrm{r}_{1}+\mathrm{r}_{2}\right) / 2\right]\) (B) \(\mathrm{R}=\mathrm{r}_{1}\left(\mathrm{r}_{1}+\mathrm{r}_{2}+\mathrm{r}_{3}\right)\) (C) \(\mathrm{R}^{2}=\mathrm{r}_{1}^{2}+\mathrm{r}_{2}^{2}\) (D) \(\mathrm{R}=\mathrm{r}_{1}+\mathrm{r}_{2}\)

Short answer

The radius of the resultant soap bubble is given by the formula \(R = \sqrt{r_1^2 + r_2^2}\). The correct answer is (C) \(R^2 = r_1^2 + r_2^2\).

Step-by-step solution

Given information

We are given the radii of two soap bubbles, \(r_1\) and \(r_2\), and need to find the radius of the resultant soap bubble, R, when they merge. Step 2: Understand the concept of surface area conservation

Surface area conservation

Under isothermal conditions and in a vacuum, the surface area of the two soap bubbles (assuming they are spherical) will be conserved during the merging process. The surface area of a soap bubble is given by the formula: \(A = 4\pi r^2\) Step 3: Write the equation for the conservation of surface area

Conservation of surface area equation

Using the surface area formula, we can set up the equation for the conservation of surface area: \(A_1 + A_2 = A_R\) Substituting the respective radii and the surface area formula, we get: \(4\pi r_1^2 + 4\pi r_2^2 = 4\pi R^2\) Step 4: Simplify and solve for R

Solve for R

Now, we need to solve for R. First, simplify the equation by dividing both sides by \(4\pi\): \(r_1^2 + r_2^2 = R^2\) Now, take the square root of both sides of the equation: \(\sqrt{r_1^2 + r_2^2} = R\) The radius of the resultant soap bubble, R, is given by the formula: \(R = \sqrt{r_1^2 + r_2^2}\) Comparing with the given options, the correct answer is: (C) \(R^2 = r_1^2 + r_2^2\)

Key concepts

Soap Bubbles Physics

Soap bubbles are not just beautiful but also a fascinating study of physics. These thin films of soapy water take such spherical shapes because of the surface tension of the liquid. Surface tension strives to minimize the bubble's shape to conserve energy, resulting in a sphere, which has the smallest possible surface area for a given volume.
Soap bubbles display intriguing properties when they come into contact or merge. This is because they aim to minimize the total surface area, conserving energy. When two bubbles meet, they try to form a new structure where the total surface area is as small as possible, producing a resultant bubble with a radius that is determined by the surfaces of the original bubbles. This phenomena makes bubbles an exciting demonstration of concepts such as surface area conservation.

Isothermal Conditions

In the world of soap bubbles, isothermal conditions play a crucial role. "Isothermal" means consistent temperature throughout a process. When bubbles merge under such conditions, it means there's no temperature change.
At a constant temperature, the energy within the soap solution remains stable, which is vital for maintaining the surface tension balance during the merging process. This ensures conservation laws, like surface area, remain applicable. Thus, in isothermal conditions, the rules derived from surface area conservation are kept intact.

• This stability helps in precisely calculating the new radius of the resultant bubble after merging.
• It also means the physics governing the bubble's surface tensions are easier to predict and explore.

Vacuum Interactions

Understanding how soap bubbles interact in a vacuum is key. A vacuum lacks air and other forms of matter, which alters how bubbles behave.
When bubbles merge in a vacuum, there is an absence of external air pressure. This means the bubble is solely influenced by its own surface tension and internal pressures.
The unique conditions of a vacuum allow for the easy application of conservation laws, as they exclude complicating factors present in atmospheric conditions.

• In a vacuum, we focus purely on the geometry and physics of the bubbles without external interference.
• This makes it easier to apply theories like the conservation of surface area.

Sphere Geometry

Sphere geometry is foundational to understanding soap bubbles. The physics behind why bubbles form spheres is simple yet profound.
A sphere is the shape with the smallest surface area to volume ratio, making it energetically favorable for a soap film. For this reason, when you see a soap bubble, you're looking at nature's way of achieving energy efficiency.
The formula for the surface area of a sphere, given by \[A = 4\pi r^2\] is central to bubble physics. When considering the merging of bubbles, we calculate the combined surface area using this formula.

• It involves using the geometry to account for changes in surface area when two or more bubbles combine.
• This geometric understanding aids in comprehending how surface area conservation affects the final radius of the resultant bubble.

Sphere geometry thus allows us to predict and understand bubble behavior with precise mathematical clarity.