Dipole Radiation Analysis With Respect To 2 Sources of Unequal Volume to Obtain Total Potential V

 

             Sketch for analysis of dipole radiation for two separate sources

The key initial electrodynamic equations are:

E =   - Ñ · V   -  ¶A / ¶ t 

B = Ñ X A

1/ m ₀  Ñ X Ñ X A  = - c  Ñ  V' - cA"   + J

And for the Lorentz gauge:  Ñ · A =  -  1/ c ²    ( ¶ V / ¶ t )  

From which we obtain the equations:

i)  Ñ² A  =   -  1/ c ²    ( ¶² A / ¶ t² )  =  - m ₀ J   

2)  Ñ² V   -  1/ c ²    ( ¶²  V / ¶ t² )  =  - r /  ε_o

For the static case:

i)  Ñ² A  = - m ₀ J

ii) Ñ² V  =  - r /  ε_o

From which we obtain solutions:

i) A = m ₀ / 4 p   ò   J (r’) dt / R

ii) V ( r) = 1/ 4 p ε_o   ò   r(r’) dt / R

R  =   ‖r  - r'‖  

We  need to reference an earlier time for t in the potentials so make use of a time delay,

t _r = t -   R /c  

Then in terms of  t _r :

A ( r, t ) = m ₀ / 4 p   ò   J (r’, t _r) dt / R

V ( r, t ) = 1/ 4 p ε_o   ò   r(r’, t _r) dt / R

Since a time delay is involved, both A,V are regarded as retarded potentials.

From the diagram shown we see V1  is a small volume about P.

The total volume is:  V   =   V1   +  V 2

The total  potential is the superposition of potentials from the sources in  

V1   and  V 2:   e.g.  

V  =   V1   +  V2

Where:

V1   = 1/ 4 p ε_o   ò   r(r’, t _r) dt / R 

Similar to static case and corresponds to the solution for:

Ѳ V1  =  - r /  ε_o

Which is exactly true in the limit:  V1  -> 0

For calculation purposes we may change the center of coordinates to a point P using the fact that: R  =   ‖r  - r' ‖  

Whence we can write:

Ñ_R (r / R )  = 1/ R ²   ¶ / ¶ R   {R ¶ / ¶ R {r /  R) }

=   1/ R ²   ¶ / ¶ R   {R²/R ) ¶r / ¶ R -  R² r/ R}

= 1/ R ²  { ¶ r/ ¶ R  + R ¶r² /¶ R²   -  ¶r / ¶ R} =   1/R   ¶r² /¶ R2

Then :

Ѳ V2  ( r, t ) = 1/ 4 p ε_o   ò   (1/ R   ¶r² /¶ R² )   dt

Then the total potential:  V   =   V1   +  V 2

Ѳ V  =  Ñ²  (V1  + V2)  =  Ñ² V1   +  Ñ² V2

=   - r /  ε_o     +  1/ c ²    ( ¶²  V / ¶ t² )    

=>  Ñ² V  =  - 1/ c ²    ( ¶²  V / ¶ t² )    =  - r  /  ε_o