To find the maximum possible amplitude of vibration (a) for a parallel plate capacitor with one plate oscillating while connected to a battery of emf V, we can start by examining the relationship between current, charge, and the properties of the capacitor. When one plate of the capacitor oscillates, the distance between the plates changes, which in turn affects the capacitance and the current flowing in the circuit.

Understanding Capacitance and Current Relationship

The capacitance (C) of a parallel plate capacitor is given by the formula:

• C = ε₀ * (A / d)

where:

• ε₀ = permittivity of free space (approximately 8.85 x 10^-12 F/m)
• A = area of the plates
• d = distance between the plates

When one plate oscillates, the distance (d) changes as a function of time. If we let d = d₀ + a * sin(ωt), where d₀ is the average distance, a is the amplitude of oscillation, and ω is the angular frequency of oscillation, we can observe how this affects the capacitance:

• C(t) = ε₀ * (A / (d₀ + a * sin(ωt)))

Current in the Circuit

The current (I) flowing through the capacitor can be expressed as the rate of change of charge (Q) over time:

• I = dQ/dt

Since the charge on the capacitor is related to the capacitance and voltage by the equation Q = C * V, we can express the current as:

• I = V * (dC/dt)

By differentiating C(t) with respect to time, we obtain:

• dC/dt = ε₀ * A * (-a * ω * cos(ωt)) / (d₀ + a * sin(ωt))²

Finding Maximum Current

The maximum current (I_max) occurs when cos(ωt) = ±1. Thus:

• I_max = V * ε₀ * A * (a * ω) / (d₀ + a)

Relating Current to Amplitude

Now, we can rearrange the equation for maximum current to solve for the amplitude of vibration (a):

• a = (I_max * (d₀ + a)) / (V * ε₀ * A * ω)

To simplify this, we can isolate 'a' on one side:

• a(V * ε₀ * A * ω) = I_max * d₀ + I_max * a

From here, we can factor 'a' out:

• a(V * ε₀ * A * ω - I_max) = I_max * d₀

Finally, we can express 'a' as:

• a = (I_max * d₀) / (V * ε₀ * A * ω - I_max)

Conclusion

This relationship shows how the amplitude of vibration depends on the maximum current observed, the average distance between the plates, and the properties of the capacitor. By plugging in the values for I_max, V, ε₀, A, and ω, you can calculate the maximum possible amplitude of vibration (a).