Chapter.3.F08

# Chapter.3.F08 - Chapter 3 Simple Radiating Systems A simple...

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

Chapter 3 Simple Radiating Systems A simple radiating system consists of a localized charge density ρ ( r ,t ) and a localized current density J ( r ) .Th e system is considered to be localized if its dimensions are small compared to the wavelength of the radiation. We will consider radiating sources in a vacuum and begin with the equations for the vector and scalar potentials in the Lorentz gauge and Gaussian units: 2 φ ( r ) 1 c 2 2 ∂t 2 φ ( r ,t )= 4 πρ ( r ); 2 A ( r ) 1 c 2 2 2 A ( r ,t 4 π c J ( r ) . (3.334) To analyze a radiating system we use the retarded Green’s function for the basic wave equation (Eq. 68) : G ( r r 0 t 0 Θ ( t t 0 ) c | r r 0 | δ ( | r r 0 | c ( t t 0 )) . (3.335) The vector and scalar potentials are given by: A ( r A h ( r )+ ZZZ Z J μ r 0 | r r 0 | c Θ ( t t 0 ) c | r r 0 | δ ( | r r 0 | c ( t t 0 )) d 3 r 0 dt 0 & φ ( r φ h ( r ρ μ r 0 | r r 0 | c Θ ( t t 0 ) c | r r 0 | δ ( | r r 0 | c ( t t 0 )) d 3 r 0 dt 0 . Using δ ( | r r 0 | c ( t t 0 )) = 1 c δ ( t 0 t + | r r 0 | /c )) to do the t 0 integration one f nds A ( r A h ( r 1 c ZZZ J μ r 0 | r r 0 | c 1 | r r 0 | d 3 r 0 and (3.336) φ ( r φ h ( r 1 c ρ μ r 0 | r r 0 | c 1 | r r 0 | d 3 r 0 , (3.337) where A h and φ h are solutions to the homogenous wave equations (no sources)& they will be taken to be zero. 3.1 Localized harmonically oscillating source We consider f rst sources which oscillate at a f xed frequency, ω : ρ ( r ρ 0 ( r )exp( iωt ) (3.338) J ( r J 0 ( r iωt ) . (3.339) The real part of these source functions and of the potentials will be the physical parameters. The vector potential generated by this current source is A ( r exp ( iωt ) c J 0 ( r 0 ) exp ( | r r 0 | /c ) | r r 0 | d 3 r 0 . (3.340) The spatial variation of the vector potential is contained in A 0 ( r 1 c J 0 ( r 0 ) exp ( | r r 0 | /c ) | r r 0 | d 3 r 0 . (3.341) 73

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

View Full Document
The spatial dependence of the magnetic induction is given by B 0 ( r )= × A 0 ( r ) (3.342) and (since at the observation point J ( r , t )=0 ) the electric f eld is given by × B ( r ,t 1 c ∂t E ( r , t i ω c E ( r , t ) , (3.343) E ( r ic ω × B ( r ) . (3.344) The spatial dependence of the electric f eld is E 0 ( r ic ω × B 0 ( r ) . We have already restricted our sources to be localized with a characteristic dimension, d , satisfying d ¿ c/ω = λ/ 2 π. The exponential term in the integrand for the vector potential suggests that the potential will have a different spatial dependence depending on the range of r : d ¿ r ¿ c/ω (the near f eld or static zone) r À c/ω (the far f eld or radiation zone) To evaluate A 0 ( r ) one needs an appropriate expansion for the following Green’s function: G 0 ( r r 0 exp ( | r r 0 | /c ) | r r 0 | where [ 2 + k 2 ] G 0 ( r r 0 4 πδ ( r r 0 ) and k = ω/c . This Green’s function for the Helmholtz equation (the wave equation with 2 G c 2 2 = ω 2 c 2 2 G = k 2 G ) can be written in spherical coordinates as follows: G ( r r 0 X l =0 l X m = l g l ( r, r 0 ) Y lm ( θ,φ ) Y lm ¡ θ 0 0 ¢ where (3.345) where g l ( r, r 0 ) is a solution of d dr μ r 2 d dr g l ( r, r 0 ) + ¡ kr 2 l ( l +1) ¢ g l ( 0 4 ( r r 0 ) ,k = ω c . (3.346) The g l ( 0 ) can be expanded in solutions to the spherical Bessel equation: d dr μ r 2 d dr f C ( ) + ¡ k 2 r 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 10

Chapter.3.F08 - Chapter 3 Simple Radiating Systems A simple...

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

View Full Document
Ask a homework question - tutors are online