Question

12.15. Verify that the temporal component (M = 0) of (12.56) is the same as V.B=0, and the spatial components (n =i = 1,2,3) are the same as V E=-ƏB/at.

eq12.56

dual tensor

Answer 1

GIVEN: . Gur= so I-B. -By L-B z By By By : o - Eric An Ezlco Eylc Exic Bz Eulc - Exlc Əyv 7. Ř =0 Putting r=0 in o) OG Әҳv H 2400 axo 602 2603 = 0 Now, OS • Gur, 00 = 0 ol = Br O2 = By 603 = Bz [ Goo Gol 902 903 "a G" G" G12 G13 LG 20 G21 G22 G 23 - LG 30 G 31 G 32 G²3 2 and x = t , x'=x, x² = 4 x3=z. from (1) Thus 120 +2 (8x) + 2(88) + 2(83=0 TT

Dale І і "ПІ - го - COD) - авч зву + 2 Эрч, әВ OZ = га Jana . B = tai ta, I. Св, іt B, it в К). - әВ, ә 49 әВ ә2. әЧ Thus, VoB to from ello for ex E = — гр. Әв от Putting M = 1 in cio, nibe ӘxУ act2 be+ 13 ә, 13 = 0 . ago 24° (0 = Тут әуі əz- 2x2 3 ) — — В, - Ezic 12 13 = = - Brelic Es (с from cell), әЯ 2 (-4 ) + (0) + + 2 (Е, /с) (- Ey/с) – от

2, in C12, гла Ә41" во х" + оч п ә420 2 G2 + 2 2x Эх з whenel 420 = - By 421 = Evlc 22 2 бг G+* = — Es (с 2. о (- B4) + 2 (0) + ә at + ә СЕ lc) 2kz 2 (- Eyle) =o Э? ГЭЕz - ӘEx Clax 27 . әВя » (2 - 2 1 - (v1. 1 — (м» at feting Ә9° = 0 = 3, % (1), — axu с 3° = the -Вz 9 - Eric

(-0- 2 (-Elt) te (ENo +7 (0). — О. — ӘEx - ӘEя - сач ӘХ ( 2 - 2 1 - этом. — ӘВz ЭЕ Multiplying ean (u) with i . ean (V) with Î, an ean cully with Ŕ on both sides and adding them together, we get - Ex əz J тәEя - Et с C 23 24 ТГЭEx - - С 24 2x J a Ez Ləx b = aBy i et 2E₂ + - I (ак, -ар, ) i+ (а - в) — Т colok әЕ, — ГЭЕ- PE2 I lay Э ? Эх. 2E4 – ЭЕ Е Т . орох эЯ ә ( Вit By it B2B). at 1-4 ке је в ак ЭЕ — әд( ўхЁ ) . - в OR ət , In natural units, ca . Түх E = – әВ ! — lence shown

$\hat i,\ \hat j,\ \hat k$ are the unit vectors along x, y, and z axes respectively.

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