Animated Solution for Physics - Properties of Solids and Liquids: When two soap bubbles of radii $a$ and $b(b>a)$ coalesce, the radius of curvature of common surface is

Imagine two soap bubbles floating gently in the air. One is a smaller bubble with a radius $a$, and the other is a larger bubble with a radius $b$. When these two bubbles drift into each other and coalesce, they don't just merge into one giant sphere immediately. Instead, they form a shared boundary—a common surface—that separates the air inside the smaller bubble from the air inside the larger one.

Our mission is to find the exact radius of curvature of this common surface. To do this, we need to dive into the physics of excess pressure.

Before we analyze the intersection, let's recall a fundamental property of soap bubbles. The pressure inside a bubble is always greater than the atmospheric pressure outside. This is because the surface tension of the soap film acts like a stretched rubber balloon, constantly trying to compress the air inside.

For a soap bubble, which has two surfaces (an inner and an outer surface) in contact with air, the excess pressure is given by the formula:

where $T$ is the surface tension of the soap solution and $R$ is the radius of the bubble.

Applying this logic, we can write the absolute pressure for both bubbles. Let $p_0$ be the atmospheric pressure outside the bubbles.

For the smaller bubble (radius $a$), the internal pressure $p_1$ is:

Similarly, for the larger bubble (radius $b$), the internal pressure $p_2$ is:

Notice a crucial detail here: because the radius $a$ is smaller than $b$, the fraction $\frac{4T}{a}$ is larger than $\frac{4T}{b}$. Therefore, the pressure inside the smaller bubble is strictly greater than the pressure inside the larger bubble ($p_1>p_2$). This is why the common surface bulges into the larger bubble!

Now, let's focus on the common surface separating the two bubbles. This surface acts just like a piece of a new bubble. The net pressure acting across this surface is simply the difference between the two internal pressures, $p_1-p_2$.

If we assume the radius of curvature of this common surface is $r$, then the excess pressure across it must satisfy the same fundamental law:

Let's substitute the expressions we found earlier into this master equation:

When we open the brackets, the atmospheric pressure term $p_0$ beautifully cancels out. This makes physical sense because the atmospheric pressure acts equally on the outside of both bubbles and doesn't affect the pressure difference between them.

Look closely at the equation. The term $4T$ is common across all the numerators. We can elegantly divide the entire equation by $4T$, reducing our relation to a purely geometric one:

Finally, let's take the common denominator on the right side:

To find $r$, we simply invert both sides. And there we have it! The radius of curvature of the common surface is:

This elegant result shows that the curvature of the common boundary depends purely on the radii of the two interacting bubbles.