Exam Format and Notes: Electrodynamics PHYS30441 Time: 1 hours - PowerPoint PPT Presentation

Exam Format and Notes: Electrodynamics PHYS30441 •Time: 1 ½ hours •Paper consists of 4 questions •Answer question 1 and 2 others •Question 1 is multi-part and covers the whole course •Question 2 is on Statics . •Question 3 is on Radiation from moving charges or Hertzian Dipole •Question 4 is on Relativity •There will be a formula sheet in the exam paper. This will include: Legendre polynomials (if needed); Differential operators (i.e. grad, curl, divergence) for different coordinate systems (if needed); L.W. fields and potentials (if needed); EM field tensor and Lorentz transformation matrix (if needed). Difficult integrals (if needed).

Summary of Essential E.M. Formulae � Gauss' law and Poisson's equation: R ~ R �d 3 r = q; r 2 V = � �="; D � d 2 ~ r = R � ( r 0 ) 1 r 0 j d 3 r 0 and solution: V = 4 �" j ~ r � ~ R J ( r 0 ) ~ � Also, r 2 ~ A = � � 0 ~ J; with a solution: ~ A = � 0 r 0 j d 3 r 0 4 � j ~ r � ~ � Biot-Savart law: R dl 0 � ( ~ R dl 0 � ( ~ d � ! r 0 ) r 0 ) r � ~ r � ~ l 0 = � 0 I B = � 0 ~ (dl' lies along the current source) r 0 j 3 r 0 j 3 4 � 4 � j ~ r � ~ j ~ r � ~ � Maxwell's equations (ME): r � ~ @t ~ H; r � ~ H = ~ @t ~ E = � � @ J + @ D � In media: D = " ~ ~ E = " 0 (1 + � e ) ~ E; ~ B = � 0 (1 + � m ) ~ H � From ME: r � ~ E = �="; r � ~ B = 0 ; r � ~ J = � @ @t � (continuity equation) � Lorentz force: � � ~ ~ v � ~ F = q E + ~ B � Scalar and vector potentials: B = r � ~ ~ A; ~ @t ~ E = �r V � @ A � Coulomb gauge: r : ~ A = 0 � Lorentz gauge: r :A = � 1 c 2 @ @t V � Inhomogeneous wave equation: � � � � r 2 � 1 r 2 � 1 c 2 @ 2 c 2 @ 2 A = � � ~ ~ V = � �="; J @t 2 @t 2 � Vector relations: � � � � � r 2 ~ r � ~ = 0 ; If r � ~ E = 0 ) ~ E = �r V; r � r � ~ r : ~ r : A A = r A A; r = 3 ; r r n = nr n � 2 ~ r = 2 r � ^ r ; r � ~ r; � � r; r 1 r = � ~ r 3 = � ^ r r for example: r r = ^ r 2 � Dirac delta: � ( f ( x )) = P 1 1 f=dx j x = x i � ( x � x i ) ; e.g. � ( ax ) = j a j � ( x ) ; i j d � ^ � = 4 �� 3 ( r ) r 2 1 r 0 j = � 1 r 0 ) ; r � r 4 � � ( ~ r � ~ r j ~ r 2 r � ~ � Multipole expansion: R � cos � 0 � P 1 ( r 0 ) n P n 1 1 r 0 ) d 3 r 0 V ( ~ r ) = � ( ~ h i 4 �" n =0 r n +1 R R R ( r 0 ) 2 � 3 � 2 cos 2 � 0 � 1 r 0 ) d 3 r 0 + 1 r 0 cos � 0 � ( ~ r 0 ) d 3 r 0 + 1 r 0 ) d 3 r 0 + :: 1 1 = � ( ~ � ( ~ 4 �" r r 2 r 3 2 1

Essen tial F orm ulae (Con tin ued) Solution to Lapla e equation in spheri al p olar o ordinates: Larmor form ula: ∞ � A l r l + B l � � V ( r, θ ) = P l (cos θ ) r l +1 l =0 P = µ 0 q 2 a 2 6 πc 1

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