# Magnetic Field and Poynting Flux in a Charging Capacitor

• dmaling1
In summary, the conversation discusses the relationship between electric and magnetic fields in a charging capacitor. The first part of the question involves finding an expression for the magnitude of the Poynting vector on the surface between the plates, while the second part involves calculating the energy flow between the plates over time. The experts suggest integrating the Poynting vector over the surface and then over time, but the specifics of which variables to integrate with respect to are unclear.
dmaling1
This is a two part question. I completed the 1st part, but I am having a difficult time on the second part. I have one try remaining.

Magnetic Field and Poynting Flux in a Charging Capacitor-

When a circular capacitor with radius and plate separation is charged up, the electric field , and hence the electric flux , between the plates changes. According to Ampère's law as extended by Maxwell, this change in flux induces a magnetic field that can be found from

integral of B * dl = Mu 0 (i + Epsilon 0 (delta flux/delta t)) = ampere maxwell law

We can solve this equation to obtain the field inside a capacitor:

B(r) = Mu 0 (ir/2piR^2) theta

where r is the radial distance from the axis of the capacitor.

Part A.) Find an expression for the magnitude of the Poynting vector S on the surface that connects the edges of the two circular plates.

S = 1/Mu 0 (E X B) = S(t) = (i^2/(2pi^2R^3epsilon 0))t

Part B.) Calculate the the total amount of energy U that flows into the space between the capacitor plates from t= 0 to t= T, by first integrating the Poynting vector over the surface that connects the edges of the two circular plates, and then integrating over time.

Here is where I am unsure where to go.

I believe we will need to integrate S*Area, where the area is 2piRd, twice like they said, once respecting to the d distance, and again w/ respect to time. I am having a tough time with the setup, and which we are constants/variables in the integration.

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Is the current i supposed to be constant? If so, since S(t) = (i^2/(2pi^2R^3epsilon 0))t represents energy flow per unit area, multiplying it by 2piRd would give you the rate of energy flow. Integrating the resulting equation with respect to t would give you U.

dmaling1 said:
I believe we will need to integrate S*Area, where the area is 2piRd, twice like they said, once respecting to the d distance, and again w/ respect to time. I am having a tough time with the setup, and which we are constants/variables in the integration.

Yes, that's what they want us to do, but first integrate respect to distance, then integrate again with respect to time. I am confused how to go about this. What do i integrate with respect to for the two different integrations?

## 1. What is a magnetic field?

A magnetic field is an area surrounding a magnet or electric current where magnetic forces are exerted on other objects or particles.

## 2. What is a charging capacitor?

A charging capacitor is an electrical component that stores energy in the form of an electric charge by separating positive and negative charges on two conductive plates separated by an insulating material.

## 3. How does a magnetic field affect a charging capacitor?

A changing magnetic field can induce an electric field in a charging capacitor, which can cause the capacitor to discharge and release its stored energy.

## 4. What is Poynting flux?

Poynting flux is a measure of the flow of electromagnetic energy through a given area, and is calculated by multiplying the electric field strength by the magnetic field strength.

## 5. How does Poynting flux relate to a charging capacitor?

In a charging capacitor, Poynting flux represents the flow of energy into the capacitor as it charges and the flow of energy out of the capacitor as it discharges.

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