1. Capacitance Expression

The capacitance of a parallel plate capacitor is C = ε₀A / d. With vibrating plates, d = d₀ + a cos(ωt). Using approximation for a << d₀:

C(t) ≈ (ε₀A / d₀) * [1 – (a/d₀)cos(ωt)]

2. Current Calculation

Charge Q = CV. Current I = dQ/dt.

I(t) = d/dt [ V * (ε₀A / d₀) * (1 – (a/d₀)cos(ωt)) ]

I(t) = [ V ε₀ A a ω / d₀² ] sin(ωt)

3. Maximum Amplitude

The maximum current I₀ is the coefficient of the sine term:

I₀ = (V ε₀ A a ω) / d₀²

Solving for a:

a = (I₀ d₀²) / (V A ε₀ ω)