What Causes Bubble Oscillation According to Gauss's Law?

[kenth]

Homework Statement

A bubble with radius $R$, mass $m$ and surface tension $S$ has initial pressure inside $P_i$ and outside pressure $P_0$ then charged with surface charge density $\sigma$, Find the period of the oscillation

Relevant Equations

Gauss's Law, Laplace equation, Newton's Law of Motion

From Gauss's Law

give $E=\dfrac{\sigma}{2\epsilon_0}$

$\therefore P_e=\dfrac{\sigma^2}{2\epsilon_0}$

Consider at equilibrium (before bubble being charged)

$P_i=P_0+\dfrac{4S}{R}$

Using Newton's 2^nd Law

$\Sigma F=m\ddot{R}$

Let $R+\delta R$ be the new radius

Give (after binomial approximation) $\dfrac{\sigma^2}{2\epsilon_0}+\dfrac{4S\delta R}{R^2}=\dfrac{m}{\pi R^2}\ddot{R}$
I think this should be SHM but the equation doesn't look like it, also do I need to consider the fact that $P_i$ decreases due to change in volume?

 

[haruspex]

You need to think of it as an oscillation around the new (i.e. charged) equilibrium. The only reason to bother with the old equilibrium would be to find the amplitude.
And yes, I would suppose the pressure changes adiabatically.