advertisement

March 9, 2000 PHY 712 { Problem Set # 16 Read Chapter 6 of Jackson. Consider the electric eld produced by a point charge q moving on a trajectory described 2 2 by 0( ) with (r; t) qÆ 3 ( 0 (t)). Assume that 0 (t) @ 0 (t)[email protected] and @ 0 (t)[email protected] = 0. Show that the electric eld can be written in the form: r t r r v r r (1) E(r; t) = 4q (1 (vR =c v)(0R R=cv)0R=c) ; where R jR(t )j, R(t ) r r0 (t ), and where all quantities which depend on time on 2 0 2 3 0 r r r the right hand side of the equation are evaluated at the retarded time tr t R(tr )=c. This is the result which we will obtain from the Lienard-Wiechert potentials in Chapter 14. (See equation 14.14.) You may wish to consult Problem #6.2 and section 6.5 of your text. If you can prove them, you may wish to use some of the following results: Z 3 d r dt Æ 0 0 3 r r0(t )Æ ( 0 0 t 0 jr t+ r0j ! = Z dt Æ(t 0 c @tr @t @R(tr ) @t = = 0 t + R(t0 )=c) R v0 R=c R R v0 R : v0 R=c = 1 1 v0 R=cR : (2) (3) (4)