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Eddy Currents

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I have decided on a change in the grading policy. I did not want to take up class time with extra quizzes so we only had 4 of the planned 6-7 quizzes. I feel that having 10% of the grade based on just 4 (really 2 since I drop the 2 lowest) quiz grades is too much. Therefore the homework will count for 30% of the grade instead of 25% and the quizzes will only count for 5%

I will be skipping Chapters 30 and 31 so that I may cover electromagnetic waves and optics as these will be covered on the MCAT. In Chapters 32 -34, I will be focusing only on those topics covered in the MCAT.

I have put some practice problems on line.

I will not be here next week. The last lecture will be given by Professor Rasmussen. However, you may email me with questions.

The final is Dec 13 at 2 PM. Remember that it will be 40% comprehensive and 60% on material covered since the last exam. You will be allowed two 8 ½ X 11 sheets of paper for notes (both sides) and it is open book.

Your grades will be available by Dec 16. You may email me or come by my office in WSTC if you want to know your grade on your final. I leave Dec 18, so you must contact me by Dec 16 if you want to talk about your grade.

I will have a review session in FN 2.212 the day before the final starting at noon and going until ????. You need to have studied for the exam prior to the session for it to do any good as you need to know what you don’t understand so I can review it.


Eddy Currents

When magnetic field is on, currents (eddy currents) are induced in conductors so that the pendulum slows down or stops


The Reality of Displacement Current

Field in the region outside of the capacitor exists

as if the wire were continuous within the capacitor


Maxwell equations in all their consistency and beauty

Gauss’s Law for E

Faraday’s Law

Ampere’s Law

Gauss’s Law for B

differential form

integral form

How are these equivalent?


Use Divergence and Curl Theorems

  • Special cases of the more general Stokes' theorem
  • The Divergence theorem relates the flow (flux) of a vector field through a surface to the behavior of the vector field inside the surface.
  • More precisely: the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence of the region inside the surface.
    • the sum of all sources minus the sum of all sinks gives the net flow out of a region.

The left side is a volume integral over the volume V, the right side is the surface integral over the boundary of the volume V.

The Curl Theorem relates the surface integral of the curl of a vector field over a surface S to the line integral of the vector field over its boundary,

The left side is a surface integral and the right side is a line integral


Equivalence of integral and differential forms of Gauss’s law for electric fields

If r is the charge density (C/m3), the total charge in a volume is the integral over that volume of r

But from the divergence theorem:

It is often written as

where D=ε0E




Something very fundamental


Maxwell equations and electromagnetic waves

Gauss’s Law for E

Faraday’s Law

Ampere’s Law

Gauss’s Law for B

differential form

integral form


No charges and no currents!

Electromagnetic disturbances in free space

With a complete set of Maxwell equations, a remarkable new phenomenon occurs:

Fields can leave the sources and travel alone through space.

The bundle of electric and magnetic fields maintains itself:

If B were to disappear, this would produce E; if E tries to go away, this would create B.

So they propagate onward in space.


Generating Electromagnetic Radiation

Heinrich Hertz was the first person to produce

electromagnetic waves intentionally in the lab

Oscillating charges in the LC circuit were sources of electromagnetic waves

Marconi – first radio communication.

Radio transmitter- electric charges oscillate along the antennae

and produce EM waves. Radio receiver – incoming EM waves induce

charge oscillations and those are detected


Plane EM waves

We will first show that such a plane

EM wave satisfies Maxwell equations

A simple plane EM wave

First, we will see if it satisfies Gauss’s laws for E and B fields

Wavefront – boundary plane between the regions with and without EM disturbance


Consider Faraday’s Law

Circulation of vector E around loop efgh equals to -Ea

Rate of change of flux through the surface

bounded by efgh is dF=(Ba)(cdt)

Hence –Ea=-Bac, and


Now consider Ampere’s Law

Circulation of vector B around loop efgh equals to Ba

Rate of change of flux through the surface

bounded by efgh is dF=(Ea)(cdt)


Key Properties of EM Waves

The EM wave in vacuum is transverse; both E and B are perpendicular

to the direction of propagation of the wave, and to each other.

Direction of propagation and fields are related by

There is definite ratio between E and B; E=cB

The wave travels in vacuum with definite and unchanging speed c

Unlike mechanical waves, which need oscillating particles of a medium

to transmit a disturbance, EM waves require no medium.


EM Wave Equation

Plane waves


Plane waves

sinusoidal waves