# Magnetic Field and Poynting Flux in a Charging Capacitor

1. Dec 10, 2009

### dmaling1

This is a two part question. I completed the 1st part, but im 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 im 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. Im having a tough time with the setup, and which we are constants/variables in the integration.

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2. Dec 10, 2009

### ideasrule

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.

3. Dec 10, 2009

### dmaling1

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