Chapter.4

# Chapter.4 - Chapter 4 Relativity and RadiationFrom a Moving...

This preview shows pages 1–3. Sign up to view the full content.

Chapter 4 Relativity and RadiationFrom a Moving Charge The position four vector: We will use the following four-vector notation with = 0,1,2,3: x =( x 0 , x 1 , x 2 , x 3 )=( ct , x , y , z ct , r ) x ct , r ) dx dx cdt ) 2 −| d r | 2 is an invariant! 4.1a 4.1b The metric tensor is given by: [ g  ] = 1 000 0 10 0 00 10 00 0 1 metric tensor 4.2 The vector potential and the current density four-vectors are given by: A ( r . t ( r . t ) , A ( r , t )) J ( r . t c ( r , t ) , J ( r , t )) 4.2a 4.2b The four-vector operator, ,isgivenby ct , ∇) ct , −∇) =−∇ 2 + 1 c 2 2 t 2 4.3a 4.3b 4.3c The wave equation for A The non-homogeneous wave equation for A ( r . t ) and its Green’s function become: : A ( r . t )= 4 c J ( r , t ) G R ( r r , t t ) = 4  ( x x ) 4.4 4.5 where the Green’s function is

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
G ( r r , t t ) = Θ ( ( ct ct )) c | r r | (| r r | c ( t t . The here denotes use of the Retarded ( ) and Advanced ( + ) Green’s functions. The solution for A ( r . t ) using the Green’s function is thus A ( r . t )= A ( r . t ) + ∫∫∫ ∫ 1 c J ( r . t ) Θ ( ( ct ct c | r r | (| r r | c ( t t d 3 r dt . 4.6 Note that this gives the solution for both A ( r , t ) and ( r , t ) .T h e A ( r . t ) represent homogeneous solutions to the wave equation (no sources). They are often treated as incoming or outgoing plane waves. We next consider the following Dirac delta function: (( x x )( x x )) =
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 12/19/2010 for the course PHYS 423 taught by Professor G. during the Spring '10 term at Missouri S&T.

### Page1 / 5

Chapter.4 - Chapter 4 Relativity and RadiationFrom a Moving...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online