Maxwell's Equations and a circular capacitor

[TheTourist]

Homework Statement

The current I=I₀exp(-t) is flowing into a capacitor with circular parallel plates of radius a. The electric field is uniform in space and parallel to the plates.

i) Calculate the displacement current I_D through a circular loop with radius r>a from the axis of the system

ii) Calculate an expression for the electric field between the capacitor plates.

The Attempt at a Solution

i) As the plates are being charged up by the current I, the displacement current I_D is increasing due to a growing electric field between the plates. I therefore reasoned that as the current I is decreasing with time, the displacement current is increasing with time by an equal amount so the answer would be I_D=I₀exp(t).
Though I don't know if this is correct.

ii) I used Gauss's law in integral form and a cylinder around the positive capacitor plate.
$\int$E.dS=Q(t)/ε₀

I performed the integration using the top surface of a cylinder, RdRd$\phi$$\hat{z}$

and ended with the equation E(t)=Q(t)/(ε₀πa²)$\hat{z}$
where a=R and I=dQ/dt

 

[Matterwave]

It is true that the Electric field is increasing, but why would that imply an increase in the displacement current? Remember that the displacement current is proportional to the time derivative of the Electric field, so that "the electric field increasing" only implies that the displacement current is positive. Whether the displacement current is increasing or decreasing depends on the SECOND derivative of the electric field.

Use Ampere's law with a circle as the boundary and try two different surfaces bound by that circle, one which intersects with the wire and one which goes through the gap in the capacitor. The answer (the strength of the magnetic field) has to be the same from the two surfaces. What does that tell you about the displacement current?

 

[TheTourist]

I think I have been misunderstanding the physical meaning of the displacement current. You have helped me to think about it in the correct way, thank you Matterwave.
The displacement current should be equal to the current in the wire in this case, i.e. amperes law gives the same result inside the capacitor as the wire, even though there is no charge flow in the capacitor.

 

[Matterwave]

Yep. =D

 

[paranormal]

I would like to commend the approach taken in this response. It is clear that the student has a good understanding of Maxwell's equations and their applications. However, there are a few points that I would like to clarify and expand upon.

Firstly, in part (i), the student correctly recognizes that the displacement current ID is increasing with time as the plates are being charged up. However, the expression given for ID, ID=I0exp(t), is not entirely correct. The displacement current is not simply equal to the current I, but rather it is equal to the rate of change of the electric flux through the surface. In this case, the electric flux is changing due to the increasing electric field between the plates, so the correct expression for ID would be ID=ε0πa2dE/dt.

Secondly, in part (ii), the student correctly uses Gauss's law to calculate the electric field between the plates. However, the expression given for E(t), E(t)=Q(t)/(ε0πa2), is missing a key factor. The electric field between the plates is actually given by E(t)=Q(t)/(ε0A), where A is the area of the plates. In this case, since the plates are circular, the area would be given by A=πa2. This would result in the correct expression for the electric field between the plates as E(t)=Q(t)/(ε0πa2).

Overall, the student has demonstrated a strong understanding of Maxwell's equations and their applications to the given scenario. To improve the response, I would suggest clarifying the expressions for the displacement current and the electric field, as well as providing more detail on the steps taken to arrive at the solutions. Additionally, it may be helpful to discuss the physical significance of the results obtained and how they relate to the given scenario.