Question

A soap bubble of radius $3cm$ is charged with $9nC$. Find the excess pressure inside the bubble (in SI units). Consider surface tension of soap solution $=3\times 10^{-3}N{m}^{-1}$.

Answer 1

Answer: 0.36

Excess pressure due to surface tension $=\frac{4T}{r}$.
(This pressure tends to contract the bubble.)
Outward pull due to electric charge per unit area $=\frac{{\sigma }^2}{2{\epsilon }_0}$
$\therefore$ Excess pressure inside a charged soap bubble
$=\frac{4T}{r}-\frac{{\sigma }^2}{2{\epsilon }_0}$
Now, $q$ (total charge) $=4\pi r^2\sigma$
$\therefore \sigma =\frac{q}{4\pi r^2}$
$\therefore$ Excess pressure
$\Delta P=\frac{4T}{r}-\frac{q^2}{32{\pi }^2{r}^4{\epsilon }_0}$
$=\frac{4\times 3\times 10^{-3}}{3\times 10^{-2}}-\frac{q^2{\epsilon }_0}{2\times 4\pi {\epsilon }_0\times 4\pi {\epsilon }_0\times r^4}$
$=0.4-\frac{{\left(9\times 10^{-9}\right)}^2\times {\left(9\times 10^9\right)}^2\times 8.85\times 10^{-12}}{2\times {\left(3\times 10^{-2}\right)}^4}$
$=0.4-\frac{9\times 9\times 8.85\times 10^{-12}}{2\times 10^{-8}}$
$=0.4-\frac{716.85}{2}\times 10^{-4}$
$=0.4-0.036$
$\approx 0.36N{m}^{-2}$