Electrodynamics Part 1 12 Lectures

Transcription

1 NASSP Honour - Electrodynamic Firt Semeter 2014 Electrodynamic Part 1 12 Lecture Prof. J.P.S. Rah Univerity of KwaZulu-Natal rah@ukzn.ac.za 1

2 Coure Summary Aim: To provide a foundation in electrodynamic, primarily electromagnetic wave. Content: 12 Lecture and 3 Tutorial Maxwell equation: Electric field, Gau law, Poion equation, continuity equation, magnetic field, Ampere law, dielectric and magnetic material, Faraday law, potential, gauge tranformation, Maxwell equation Electromagnetic wave plane wave in vacuum and in media, reflection and tranmiion, polarization Relativity and electromagnetim 2

3 Syllabu 1. Reviion: Electrotatic 2. Reviion: Current and Magnetotatic 3. Dielectric and Magnetic Material 4. Time Varying Sytem; Maxwell Equation 5. Plane Electromagnetic Wave 6. Wave in Media 7. Polarization 8. Relativity and Electrodynamic 3

4 Source D.J. Griffith, Introduction to Electrodynamic, 3rd ed., 1999 (or 4th edition, 2012) J.D. Jackon, Claical Electrodynamic, 3rd ed., 1999 (earlier edition non-si) G.L. Pollack & D.R. Stump, Electromagnetim, 2002 P. Lorrain & D.R. Coron, Electromagnetic Field and Wave, 2nd ed., 1970 (out of print) R.K. Wangne, Electromagnetic Field, 2nd ed.,

5 Firt Year EM: (a) Electrotatic 1. Coulomb law: force between charge F = 1 q q (magnitude) 4πε 0 r 2 2. Define electric field E a force per unit charge 3. Gau Law E. da = Φ = Q enc ε 0 (derived from electric field of a point charge) 4. Define potential V a PE per unit charge b p.d. V a V b = E. dl a, tatic E conervative 5. Potential gradient: E x = dv dx E = V + Capacitor etc 5

6 (b) Magnetotatic 6. Define magnetic field B by force on moving charge: F = qv B. 7. Magnetic flux: Φ = B da 8. B produced by moving charge: B = kq v r r 2 Define μ 0 ε 0 = 1 c 2 Idl r 9. Biot-Savart Law: db = μ 0 4π r2 due to current element 10. B of long traight wire = circle around wire. => Net flux through cloed urface B. da = Ampere Law: B. dl = μ 0 I (Biot-Savart and Ampere ued to find B for given I.) 6

7 (c) Induced Electric Field 12. emf E = E n dl ( non-electrotatic E ) 13. Faraday Law (with Lenz Law ( ign)) relating induced emf to flux: E = dφ dt 14. Hence induced E: c E dl = d dt B da Thi contitute pre-maxwell electromagnetic theory, i.e. ca

8 Electric and Magnetic Field The Lorentz Force The electric field E i defined a the force per unit charge i.e. F = qe The magnetic field i defined in term of the force on a moving charge: F = qv B In general electric and magnetic field, a (moving) charge experience force due to both field, and F = q E + v B The Lorentz Force In fact we can how that B arie from the application of the relativitic (Lorentz) tranformation to the force between two charge in a moving frame. (Later!) 8

9 Gau Law in Integral Form Conider charge a ource of E. The field i repreented by field line whoe denity repreent the intenity of the field. Electric flux through a urface Φ E = E da i the total number of field line paing through the urface. Gau Law relate the electric flux through a cloed urface to the total charge encloed: E da = Q ε 0 = 1 ε 0 ρ dv V 9

10 Faraday Law in Integral Form The familiar form of Faraday Law i written E = dφ m (negative ign from Lenz Law) dt Now the emf can be conidered a the line integral of the induced electric field, i.e. E = E dl (or c E i the potential gradient ) Φ m i of coure the magnetic flux or the integral of B over the urface S bounded by C: B da Thu in integral form c E dl = d B da dt Faraday Law 10

11 Static Field: Scalar Potential c E dl = d dt B da Faraday Law For tatic field ( d dt = 0), Faraday Law give E dl = 0 c The electrotatic field i conervative Thi implie that the electrotatic field i decribed by a calar potential V which depend only on poition. In 1-D E = dv or in 3-D E = V dx ince the curl of the gradient of any calar i zero. 11

12 Solenoidal Magnetic Field Fundamental Law: Unlike electric field, magnetic field do not have a beginning or an end there are no magnetic monopole. For any cloed urface the number of magnetic field line entering mut equal the number leaving. i.e. the net magnetic flux through any cloed urface i zero: B da = 0 Sometime called Gau Law for magnetic field ( ). It implie that B i given by a vector potential (ee later). 12

13 Ampere Law + Diplacement Current Familiar form of Ampere Law: B. dl = μ 0 I I = J da (current denity J = nqv), i the current encloed by the cloed loop around which the line integral of B i taken Thi clearly how current I (i.e. moving charge) a a ource of B. Maxwell: thi eqn. i incomplete. B alo generated by changing E (diplacement current): I D = ε 0 d dt E da 13

14 Modified Ampere Law The total current through any urface i then the conduction current + diplacement current i.e. I total = I + I D Ampere Law hould read B. dl = μ 0 I total = μ 0 I + I D B. dl = μ 0 I + μ 0 ε 0 d dt E da d B. dl = μ 0 J da + μ 0 ε 0 E da dt In e.g. a wire (a good conductor), the diplacement current i negligible and o the total current i jut I, In e.g. a capacitor pace, I i zero and the total current i jut the diplacement current due to the changing E field. 14 or

15 Maxwell Equation in Integral Form The 4 fundamental equation of electromagnetim: 1. Gau Law: E da = 1 ε 0 ρ dv V 2. Faraday Law: c E dl = d dt 3. No Monopole : B da = 0 B da 4. Ampere Law with diplacement current: d B. dl = μ 0 J da + μ 0 ε 0 E da The field which are the olution to thee equation are coupled through the d dt term. 15 dt

16 Maxwell Equation continued In the tatic cae (charge not moving, contant current) Maxwell equation are decoupled. Thi allow u to tudy electrotatic and magnetotatic eparately, a in firt year. The equation are then 1. Gau Law: E da = Q ε 0 2. Faraday Law: E dl c = 0 3. No Monopole : B da = 0 4. Ampere Law: B. dl = μ 0 I = J da 16