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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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PHAS3201 EM Theory Introduction (UCL), astronomy, astrophysics, cosmology, general relativity, quantum mechanics, physics, university degree, lecture notes, physical sciences

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