A vacuum parallel-plate capacitor has sinusoidal applied voltage $V(t)=V_0\cos\omega t$. Ignore edge effects. The induced emf within the capacitor can be expanded in powers of ω. Find its lowest nonvanishing exponent.
The time-dependent electric field creates a magnetic field through displacement current. Since $\partial E/\partial t\propto\omega$, that magnetic field is first order in ω. Its changing flux then induces an electric circulation by Faraday’s law, introducing a second time derivative and another factor of ω. Thus the leading induced emf is proportional to $\omega^2$, with lowest exponent 2.
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