A boy blows a soap bubble of radius \(R\) which floats in the air a few moments before breaking. What is the difference in pressure between the air inside the bubble and the air outside the bubble when (a) \(R=1 \mathrm{~cm}\) and (b) \(R=1 \mathrm{~mm}\) ? The surface tension of the soap solution is \(\sigma=25\) \(\mathrm{dyn} / \mathrm{cm}\). (Note that soap bubbles have two surfaces.)

Surface tension is a property of the liquid surface which makes it behave like a stretched elastic membrane. It occurs due to the cohesive forces between liquid molecules. At the surface, molecules experience an inward pull, creating a minimal surface area for the liquid. For soap bubbles, surface tension is crucial because it provides the force that maintains the bubble's shape.

In the given problem, the surface tension of the soap solution is given as \(\text{25 dyn/cm}\). This value indicates the force per unit length required to stretch the surface of the liquid. The higher the surface tension, the more energy is needed to increase the surface area of the bubble. This concept helps explain why bubbles form and how they maintain their shape against external pressures.

Bubble physics involves understanding how bubbles form, stabilize, and eventually burst. A soap bubble consists of a thin film of liquid with air trapped inside. This film has two surfaces – one facing inside and one facing outside, each contributing to the overall surface tension.

When a boy blows a bubble, the air pressure inside increases, expanding the bubble until the internal and external pressures balance, maintained by surface tension. This balance can be represented by the relationship between pressure, surface tension, and radius:
\[\triangle P = \frac{4 \times \text{surface tension}}{\text{radius}} \]

In our scenario, we analyze the pressure difference between the inside and the outside of a bubble with different radii. The interplay of forces within the bubble is complex but follows straightforward principles of fluid dynamics and surface tension.

To calculate the pressure difference \(\triangle P\) between the inside and outside of a soap bubble, we use the formula:
\[\triangle P = \frac{4 \times \text{surface tension}}{\text{radius}} \]

For a bubble with a radius of 1 cm, plugging in values gives:
\[\triangle P = \frac{4 \times 25 \text{ dyn/cm}}{1 \text{ cm}} = 100 \text{ dyn/cm}^2 \]

This means the pressure inside is 100 dyn/cm\(^2\) greater than outside.

For a smaller radius of 1 mm (or 0.1 cm), the calculation is:
\[\triangle P = \frac{4 \times 25 \text{ dyn/cm}}{0.1 \text{ cm}} = 1000 \text{ dyn/cm}^2 \]

Here, the pressure difference is a much larger 1000 dyn/cm\(^2\). This shows that smaller bubbles have a higher internal pressure, explaining why they tend to burst more readily.