Hw12nb 2 b in the coulomb gauge the scalar potential

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Unformatted text preview: mode expansion satisfies the Coulomb gauge condition. HW12.nb 2 (b) In the Coulomb gauge, the scalar potential vanishes in the absence of electric charges, and the electric field is simply ÿ Ø Ø 1 E = ÅÅÅÅ A. On the other hand, we can use integration by parts and the Coulomb gauge condition to simplify the term c ‡ d x B = ‡ d x J“ ä AN ÿ J“ ä AN = - ‡ d x A ÿ D A 2 ØØ ØØ Therefore, ·d Ø ÿ2 i Øj 1 Ø j j ÅÅÅÅÅÅ A x j c2 j j k For the first term, 1 ÅÅÅÅÅpÅ Å 8 H= 1 1 ÅÅÅÅÅpÅ · d x ÅÅÅÅÅÅ A Å 8 c2 Ø ÿ Ø 2 Ø Ø ØØ Ø Ø Øy z z - A ÿ D Az z z z { ØØ ØØ Ø1 ØØ Ø Ø *Ø Ø 1 2p— 2 = ÅÅÅÅÅpÅ ‡ d x ÅÅÅÅÅÅ J"#########c###Å# ‚Ø "####p# ‚ I- i e ± I pM a± I pM ei pÿ x ë— + i e ± I pM a± † I pM ei pÿ x ë— MN Å ÅÅÅÅÅÅÅÅ3ÅÅÅÅÅÅÅ Å w ## 8 c2 L p ± = ÅÅÅÅ—ÅÅÅÅÅ ‚Ø Ø "############# ‚ Å w p wq 4 L3 Ÿdx Ø l,l' p, q 2 J- i e l I pM al I pM ei pÿ x ë— + i e l I pM al I pM ei pÿ x ë — N J- i e l' I q M al' I q M ei q ÿ x ë— + i e l' I q M al' Iq M ei q ÿ x ë— N Ø Ø Ø Ø* Ø ØØ = ÅÅÅÅ—ÅÅÅÅÅ ‚Ø Ø "############# ⁄l,l' L3 Å w p wq 4 L3 Ø Ø Ø Ø* I e l I pM ÿ e l' I q M p, q ؆ Ø ØØ Ø Ø al I pM al' † I q M dØ,Ø + e l I pM ÿ e l' I q M al † I pM al' I q M pq Ø Ø Ø Ø* Ø Ø Ø Ø* Ø ØØ Ø† ØØ Ø - e l I pM ÿ e l' I q M al I pM al' I q M dØ,-Ø - e l I pM ÿ e l' Iq M al † I pM al' † Iq M d...
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This note was uploaded on 01/17/2014 for the course PHYSICS 221b taught by Professor Staff during the Winter '08 term at University of California, Berkeley.

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