Question

Obtain the formula of maximum surface charge density for equilibrium of charge soap bubble.

Answer 1

Equilibrium Condition for Charged Soap Bubble

The pressure of air on the internal surface of the soap bubble is more than the atmospheric pressure present on the external surface. This pressure is balanced by the force of surface tension. If the radius of the bubble is $r$ and surface tension is $T$, then the excess pressure is

$P_{ex}=\frac{4T}{r}.....\left(1\right)$

Now, if the bubble is charged by the surface charge density of $\sigma$, then the electric pressure of $\frac{{\sigma }^2}{2{\epsilon }_0}$ acts outward on the surface. In this case,

$P_{ex}+\frac{{\sigma }^2}{2{\epsilon }_0}=\frac{4T}{r}$

or $P_{ex}=\frac{4T}{r}-\frac{{\sigma }^2}{2{\epsilon }_0}....\left(2\right)$

Thus a state will come when excess pressure will become zero on charging the bubble. After it, the bubble blasts. Thus for equilibrium,

$P_{ex}=\frac{4T}{r}-\frac{\sigma }{2{\epsilon }_0}=0$

or $\frac{4T}{r}=\frac{{\sigma }^2}{2{\epsilon }_0}...\left(3\right)$

Radius of bubble, $r=\frac{8T{\epsilon }_0}{{\sigma }^2}....\left(4\right)$

Surface charge density on bubble,

$\sigma =\sqrt{\frac{8T{\epsilon }_0}{r}}.....\left(5\right)$

and charge, $q=\sigma \times 4\pi r^2=4\pi \sqrt{8T{\epsilon }_0{r}^3}...\left(6\right)$