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Electromagnetic Theory: PHAS3201, Winter 20081. Introduction

Administration

Office Hours

Attendance Sheets

Problem Sheets: four during term; one more for vacation

Handouts

Moodle: enrolment key

1 Mathematical Tools

The easy use of mathematical tools is vital to understanding electromagnetic theory.

Differential

The differential operators transform vectors and scalars

Grad : scalar to vector F(r) = (r) (1)

Div : vector to scalar q(r) = F(r) (2)

Curl : vector to vector G(r) = F(r) (3)

These are all given in the Preliminaries handout

They should be reasonably familiar

Integral

Integrals of vectors can produce scalars or vectors

There are 1-, 2- and 3-D integrals (line, surface and volume)

These are all important in Electromagnetic theory !

There are important theorems relating integrals of the differential operators

Integral Theorems

Divergence Theorem: V

Fdv =

S

F nda (4)

Stokes Theorem: S

F nda =

C

F dl (5)

Notice the importance of

!

TAKE NOTES

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PHAS3201: Electromagnetic Theory

2.3 Magnetostatics

Biot-Savart Law

For an elementof a current loop,

dl, carrying current I at r:

dB(r) =0I

4

dl (r r)

|r r|3

(9)

We can perform a loop integral:

B =0I

4

C

dl (r r)

|r r|3

(10)

We can show that B = A, so B = 0

What is 0, and what are its units ?

TAKE NOTES

2.4 Electromagnetism

Ampres Law

For a surface S bounded by loop C, c

B dl = 0I, (11)

where I is the current passing through the surface S

We can write I asSJ nda

Using Stokes Theorem, we find: B = 0J (12)

This is incomplete

We will consider the detailed form ofwhy Ampres law is incomplete later in the lectures, though you should

already have seen this and understood it at some level. This will form our third Maxwell equation when complete.

Faradays Law of Induction

If a conducting circuit, C, is intersected by a B field, then the flux is given by:

C = S

B nda (13)

The EMF induced around the circuit is

E = d

dt=

C

E dl (14)

As before, we can use Stokes Theorem to derive:

E = dB

dt(15)

TAKE NOTES

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PHAS3201: Electromagnetic Theory

Maxwells Equations

Ampres law as described above is incomplete: it needs to account for time-varying electric fields

When we do this, we can write (in a vacuum):

E =

0(16)

B = 0 (17)

B = 0J+ 00dE

dt(18)

E = dB

dt(19)

Force on a moving charge: F = q (E+ v B)

Once Maxwells equations and the Lorentz force law have been specified, classical electromagnetism is essen-

tially complete: the basic physics has not changed, though the details of the interaction of the fields with matterare still being understood.

PHAS3201 Winter 2008 Section I. Introduction 4