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PHAS3201 EM Theory Introduction (UCL)

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					Electromagnetic Theory: PHAS3201, Winter 2008
1. 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) =        ϕ(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:
                                                         · Fdv =       F · nda                (4)
                                                    V              S

    • Stokes’ Theorem:
                                                        × F · nda =         F · dl            (5)
                                                S                       C

    • Notice the importance of    !

    TAKE NOTES




PHAS3201 Winter 2008                           Section I. Introduction                         1
PHAS3201: Electromagnetic Theory

2      Overview of PHAS2201
2.1      Fields
Basic Fields

      • In a vacuum, the basic fields are E and B
      • What are they ? What are their units ?
      • When they interact with matter, there are changes

      • Why should this happen ?
      • We have the the fields D and H
    • Be careful that you know what you’re dealing with !

      TAKE NOTES

2.2      Electrostatics
Electrostatics

      • For two charges, q1 and q2 at rest at points r1 and r2

                                                   Force                        Field
                                          F (r2 ) = 4π q1 q−r |2
                                                             2                         q
                                                                        E(r2 ) = 4π |r 1−r |2
                                                       0 |r1    2                   0 1     2
                                                                                                  (6)
                                                  Energy                      Potential
                                          E(r2 ) = − 4π 0q|rq2 2 |
                                                           1
                                                             1 −r
                                                                                       q
                                                                        ϕ(r2 ) = 4π 0 |r1 −r2 |
                                                                                         1



    • How are the force and field directed ?
    • What are the values at r1 ?
    • What is     0,   and what are its units ?

    TAKE NOTES

Gauss’ Law

    • For a charge density ρ(r)
                                                                          ρ(r)
                                                            · E(r) =                              (7)
                                                                              0

      • This can also be written for a collection of charges:

                                                                         1
                                                         E · nda =                 qi             (8)
                                                                          0   i


    • The integral and differential forms are linked by the divergence theorem

    Gauss’ law is our first Maxwell equation. Note that           i qi   =     V
                                                                                  ρdv




PHAS3201 Winter 2008                               Section I. Introduction                         2
PHAS3201: Electromagnetic Theory

2.3      Magnetostatics
Biot-Savart Law

      • For an element of a current loop, dl, carrying current I at r :

                                                              µ0 I dl × (r − r )
                                                 dB(r) =                                                      (9)
                                                              4π |r − r |3

      • We can perform a loop integral:
                                                       µ0 I         dl × (r − r )
                                                  B=                           3                            (10)
                                                       4π       C     |r − r |

      • We can show that B =          × A, so   ·B=0
   • What is µ0 , and what are its units ?

      TAKE NOTES

2.4      Electromagnetism
Ampère’s Law

      • For a surface S bounded by loop C,
                                                             B · dl = µ0 I,                                 (11)
                                                         c

      • where I is the current passing through the surface S
      • We can write I as    S
                                 J · nda
      • Using Stokes’ Theorem, we find:
                                                              × B = µ0 J                                    (12)

      • This is incomplete

    We will consider the detailed form of why Ampère’s 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.

Faraday’s Law of Induction

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

                                                       ΦC =          B · nda                                (13)
                                                                 S


      • The EMF induced around the circuit is
                                                              dΦ
                                                    E =−         =         E · dl                           (14)
                                                              dt       C


      • As before, we can use Stokes’ Theorem to derive:
                                                                        dB
                                                             ×E=−                                           (15)
                                                                        dt

      TAKE NOTES




PHAS3201 Winter 2008                             Section I. Introduction                                       3
PHAS3201: Electromagnetic Theory

Maxwell’s Equations

   • Ampère’s 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):
                                                            ρ
                                                 ·E =                                                        (16)
                                                               0
                                                 ·B =      0                                                 (17)
                                                                 dE
                                               × B = µ0 J + µ0 0                                             (18)
                                                                 dt
                                                       dB
                                               ×E = −                                                        (19)
                                                        dt

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

     Once Maxwell’s 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 matter
are still being understood.




PHAS3201 Winter 2008                         Section I. Introduction                                            4

				
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