Problem.Set.3.F08 - Problem Set 3 1(Jackson 6.20 An example...

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Problem Set 3 : 1. ( Jackson 6.20 ). An example of the preservation of causality and finite speed of propagation in spite of the use of the Coulomb gauge is afforded by a unit strength dipole source that is flashed on and off at t = 0. The charge and current densities are ( r , t ) = ( x ) ( y ) ( z ) ( t ) J ( r , t ) = − e 3 ( x ) ( y ) ( z ) ( t ) where a prime means differentiation with respect to the argument. This dipole is of unit strength and it points in the negative z direction. ( a ) Show that the instantaneous Coulomb potential (6.23 in text) is ( r , t ) = − 1 4  o z r 3 ( t ) . ( b ) Show that the transverse current, J t , is J t ( r , t ) = − e 3 2 3 ( 3 ) ( r ) + 1 4 3 n ( n e 3 ) − e 3 r 3 ( t ) where n = r ̂ , a unit vector along the r direction and the. 2 3 factor multiplying the delta function comes from treating the gradient of z r 3 according to (4.20 in text.) ( c ) Show that the electric and magnetic fields are causal and that the electric field components are given by: E ( r , t ) = e 3 c r ′′ ( ct r ) + 1 r 2
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