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# Relativity PHYS 428

Mason

GPA 3.66

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This 16 page Class Notes was uploaded by Sonny Breitenberg on Monday September 28, 2015. The Class Notes belongs to PHYS 428 at George Mason University taught by Joseph Weingartner in Fall. Since its upload, it has received 21 views. For similar materials see /class/215199/phys-428-george-mason-university in Physics 2 at George Mason University.

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Date Created: 09/28/15

Electrodynamics e 2 electric field b magnetic field ugtltb C Recall Lorentz force law f q 9 f depends linearly on q and u but is not parallel to u Simplest relativistic expression with these properties F p EEILV U V Epu is called the electromagnetic field tensor q is assumed invariant We expect F u to be rest mass preserving which requires FUFMU 0 gt EMVU Uquot0 forallU gt El EH ie El is antisymmetric The Lorentz force law works out exactly if we take 0 b3 b2 61 0 b3 b2 61 b3 0 b1 62 b3 0 b1 62 E Nquot I b2 b1 0 63 E b2 b1 0 63 61 62 63 0 el 62 63 0 From sample problem e transforms as a 3 Vector under spatial rotations b does too as we39ll see later F EEW UV El ucquot ce 11 X be uh 1 1E Recall F 7u 7 gt F 7u f 1E c M Edt ugtltb gt f q 9 re the Lorentz force law Explicit demonstration that El 11 c ce 11 X b e uh This paiticular multiplication and sum is equivalent to a matrix multiplication 0 b3 b2 61 U1 u2b3 U3b2 61 b3 0 b1 62 uz u1b3 U3b1 62 b2 b1 0 63 U3 7 u1b2 u2b1 CB3 61 62 e3 0 c U161 u262 U363 ce ugtltbeu Maxwell39s eqn39s V e 47rp V X e Ei Relativistic sources p0 proper charge density ie in frame c0m0Ving with the charge p 2 p0 yu total p Z povi Note sum is over each moving 2 charge dist i is not an index 3 current density ji Piu39i j Zji Z 439Current J2 00 Ui WWW 112W inc1239 density J 2J2 LC3 L J LC3 Recall the continuity eqn In 4 tensor notation 4 tensor form of Maxwells39 eqns 8p V J at J 0 J u 1171 1272 1373 I474 n n am 8x1 8x2 8x3 8ct 8p 7 V J at EH 7 4JJquot c Elli7U Ewan Eamu 0 47r Elwyn Jquot 4 equations C 4 V 21 E1171 E212 E313 E414 If 0 ab3 ab 1 411 852 81173 C c 13 4 1St componentof be 9 7T 6 at i c J smularly for v 2 3 47r v 4 E1471 E2472 E37473 E4474 7 i J4 399 a 3 46 3 I 3151 9112 853 c 6 ie V e 47W Envy EVU Emu 64 equations u1 v2 032 E123 E231 E312 0 3b 3b 3b 73 71 J V b 0 u v a 2 3 4 E234 E342 E423 0 18121 963 362 cat 8x2 8x3 1St component of laib V X e 0 c at u v a 3 4 1 4 1 2 yield 2 and 3 1 components Other combinations of u v a yield repeats of old eqns or 0 0 4 Note EWW IJquot gt J 0 continuity eqn 6 since EWW is symmetric in its subscripts equality of mixed partial derivs but antisymmetric in its superscripts The field tensor can be expressed in terms of a 4 potential PM Em I m I M This definition immediately yields the 2nd tensor field eqn Elm EM Emu I ww I MU I mu DI4w I IMV 1 0tu 0 X y39 X y39 Notes The field does not uniquely determine the potential PM is a 4 Vector if we pick it in one IF then its transforms in all other lFs will yield the field in those lFs s HA 4 A 1 tensor field eqn E 7 7 J c 47r guAEuAM guAJ 4 El w E J E 1w 9 E514 9 I ww I mu 4 2 gm7 117L7M Dagp 6 J11 It is possible to choose PM such that WW see Rindler 74 called the Lorenz gauge condition With this choice the 2 term in the field eqn vanishes QW I UJM VJM MM 404 07V 0 l 39 I 47f I 47r Leaves us With 9 lawn JV or g 1qu J 11 l l l 4w 9 1s a d1fferent1al operator IW 7 7 7 7 7 7W9 62 8152 8x2 8y2 822 2 47r ia7v2g mg the d39AleInbertian 1 1 In vacuum J 0 gt El PM 0 the wave equation gt wave eqn for the field too DEM g PVM EMMA WWN QUA I WW mourn D4 0 07 0 HA 71 4t Transformation of e and h under a standard LT 61 61 7 62 762 7 b1 b1 b2 7012 363 sample problem 63 7033 W2 b3 V073 62 Note symmetry transformation is unchanged if b a e and e a b b1 b1 7 412 7b2 363 473 7473 362 61 61 62 762 Bbg 63 763 Bbz If we write the field transformation as e b39 T e b then b e39 T b e Form the dual of EW by interchanging components e H b bee 0 63 62 b1 0 63 62 b1 63 0 61 b2 63 0 61 b2 B N V 62 61 0 b3 B 62 61 0 b3 b1 b2 b3 0 b1 b2 b3 0 BW is a tensor since its components transform in the same way as those of EW gt b transforms as a 3 Vector under spatial rotations Maxwell39s equations can be written in the form 47r EM 7 Jquot 3W 0 There are two invariants associated with EW 1 1 X E Equot102 eZ 73 31 1 Y1EHBWeb gt if electric and magnetic fields have equal magnitude in one IF then they have equal magnitudes in all lFs if they39re orthogonal in one IF then they39re orthogonal in all lFs Example 1 the field of a uniformly moving point charge 14 Suppose charge q is moving at speed BC along the x axis and is at the origin at t 0 In q s rest frame 9 13 at y z b 0 22y2z12 TI v reversed transformation law for e 61 61 62 762 Bb3l 63 763 Bbzl gt 61 61 62 762 63 763 LTwitht02 xx v yy zz39 r e 113 7392 39yzxz y2 z2 77quot a2 2 12X2 zz 7 r 7 3 y z 72721 82 sin2 0 6 2 angle btwn x axjs and r r gt 9 V2730 32 Sinz 032 transformation eqns for b by b1 7 b2 7072 363 7 b3 753 362 withb 0 1210 122 4363 173362 vgtlte 1 gt b 0711163711162 C C Example 2 the field of an infinite straight current Consider a line with rest frame charge density A0 moving in positive direction in frame S along x axis with speed v i l I 20 AI In line39s rest frame Gauss39s Law yields 9 77quot In S length contraction yields A 3 A0 current i A v Consider the point x 0 y r z 0 In S x 0 y r z 0 e 0 2Or 0 b 0 61 61 62 762 Bbg 63 763 Bbz with a by b1 b2 7b2 863 b3 7b3 ez v reversal gt e 07 27A07397 0 b 07072718A0739 2AA 2M 2i 77 bii 739 CT CT 01 9 3X

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