Question

Consider the general form for an electric field propagating in free space, i.e. E = E + Ey + E,2. (a) Prove that V x (x E) = V( VE) - VE by considering the definition of the V operator in cartesian coordinates. [7 points) (b) The divergence of the electric field in free space is zero (i.e. no charges) such that V. E=0. Derive an expression for V X (V x E) in terms of E and its derivatives. [3 points)

Answer 1

TE - î Ex +ġ Ey +K EZ 言 - ta + + K 9 08 41 t .܀ ExÊ 一 뜰 品 M 38 Ey ge . Ex a Ez sy -3) lage be sex BY - + t( - 7 ; *(a+1)*e: 二 els (ela ge 236 ze Exas 42 9 -- 对 ass EQ әа | QY No 以 taking a component mly 包后。 57) - 最3E ə Ez 3 C 品 - Fx 18 Q1 arex EVEZ xane - Tone ate xeze 2YY ลฯ

for CTPE)}, + Vez - Det Fie tone the xfue arey วYE2 + Zad2 arex aze2 hexe zzie arex ay2 222 ze xe 22 24 2 Ex + 24 +387 - Ex - tu ar aya to on 2 av (5.) - DEX U Similarly for 7 and a component (5xFrē)), (7.7) – DIVER z (7.7) – TVEZI Bxläre). ho (= KE XC)} = (5xFRB)+44*60, A (5XF xĒ) TK Z

+ AXEXE) = ( 2 2 2 2 + 1 2 3 4 k 3x)=2) - oli Ex+Ey+ K E2) 1) 14-(29) proved. 6 Now DĚ TO FxBX) = -- TVE x component ar Ey tarey arey aren ZVEZ azr ax2 Dray axaz ayz xze arex azr the Ey-Ex) tage Haexe + arey axay arez xay + ( E2-Ex). ť an aya 1.2 (Ez-Ex) Ey- Ex) ( (3:4) LE 3 Similarly for Y Z comportet are ( Ex-Fy) + (Ex Ey). 227 S dar ਆਓ ਵੇ) (5.2) dar (Ex-) + 242 (Ey-E