To solve this problem, we will use the relationship between the displacement current ($I_d$) and the rate of change of the electric flux ($\frac{d\Phi_E}{dt}$) in a capacitor.

Given Information:

1. Radius of the capacitor plates ($R$): $2.0 \, \text{cm} = 0.02 \, \text{m}$
2. Separation distance ($d$): $0.60 \, \text{mm} = 0.0006 \, \text{m}$
3. Frequency ($f$): $75.0 \, \text{Hz}$
4. Maximum displacement current ($I_d$): $38 \, \mu\text{A} = 38 \times 10^{-6} \, \text{A}$

Objective:

Determine the maximum value of $\frac{d\Phi_E}{dt}$ between the plates, with fringing effects neglected.

Solution:

1. Displacement Current and Electric Flux Relationship: The displacement current $I_d$ is related to the rate of change of electric flux $\frac{d\Phi_E}{dt}$ by the equation: $I_d = \epsilon_0 \frac{d\Phi_E}{dt}$ where $\epsilon_0$ is the permittivity of free space, approximately $8.85 \times 10^{-12} \, \text{F/m}$.

2. Solving for $\frac{d\Phi_E}{dt}$: Rearranging the equation to solve for $\frac{d\Phi_E}{dt}$: $\frac{d\Phi_E}{dt} = \frac{I_d}{\epsilon_0}$

   Substituting the given values: $\frac{d\Phi_E}{dt} = \frac{38 \times 10^{-6} \, \text{A}}{8.85 \times 10^{-12} \, \text{F/m}}$

3. Calculating the Result: $\frac{d\Phi_E}{dt} \approx \frac{38 \times 10^{-6}}{8.85 \times 10^{-12}} = 4.3 \times 10^6 \, \text{V}\cdot\text{m/s}$

Answer:

The maximum value of $\frac{d\Phi_E}{dt}$ between the plates is approximately $4.3 \times 10^6 \, \text{V}\cdot\text{m/s}$, expressed with two significant figures.

Would you like more details, or do you have any questions?

Here are some related questions:

1. How does the displacement current relate to the magnetic field around a capacitor?
2. What would happen to $\frac{d\Phi_E}{dt}$ if the separation $d$ between plates increased?
3. How does frequency affect the maximum displacement current in a capacitor?
4. How would the electric field between the plates change if the radius of the plates increased?
5. What assumptions are made by neglecting fringing effects?

Tip: When dealing with time-varying electric fields, the displacement current plays a role similar to conduction current in generating a magnetic field.