Question

A spherical soap bubble of radius R has uniformly distributed charge over its surface with surface charge density $\sigma$ then [T = surface tension of the soap solution]

• A. excess pressure inside the bubble is $\frac{4T}{R} - \frac{\sigma^2}{2\epsilon_0}$
• B. excess pressure inside the bubble is $\frac{4T}{R} + \frac{\sigma^2}{2\epsilon_0}$
• C. excess pressure inside the bubble is $\frac{4T}{R}$
• D. electrostatic pressure is $\frac{\sigma^2}{2\epsilon_0}$

Answer 1

Answer: (A), (D)

The correct answers are:

• Option (A): excess pressure inside the bubble is $\frac{4T}{R} - \frac{\sigma^2}{2\epsilon_0}$
• Option (D): electrostatic pressure is $\frac{\sigma^2}{2\epsilon_0}$

Answer 2

Solution:

1. A soap bubble (with two surfaces) has the Laplace (mechanical) excess pressure:

   $$\Delta P_{Laplace}=\frac{4T}{R}.$$

2. A uniformly charged spherical surface produces an outward (electrostatic) pressure given by:

   $$P_{electrostatic}=\frac{\sigma^2}{2\epsilon_0}.$$

3. Thus, the net excess pressure inside the bubble (i.e. the pressure difference between the inside and the outside) becomes the Laplace pressure reduced by the outward electric pressure:

   $$\Delta P = \frac{4T}{R} - \frac{\sigma^2}{2\epsilon_0}.$$

Also, note that option (D) correctly states that the electrostatic pressure is $\frac{\sigma^2}{2\epsilon_0}$.

Summary:

• Explanation: The Laplace pressure for a bubble is $4T/R$. The electric field due to the surface charge gives an outward Maxwell stress (pressure) of $\frac{\sigma^2}{2\epsilon_0}$. Hence, the net inward excess pressure is reduced by the charge effect, i.e. $\frac{4T}{R} - \frac{\sigma^2}{2\epsilon_0}$.
• Answer: Options (A) and (D)