Analyzing Dipole Radiation - Part 2 - Obtaining the Potential V(r, t)

 

                                      Sketch diagram for analysis of dipole radiation

The  use of a rectangular coordinate system (x,y,z) is again appropriate, as shown in the sketch above showing two test charges: +q and - q referred to an antenna of dimension S,  with position vectors indicated.  We can write:

q(+) =  q ₀  cos w t

q(-) = - q ₀  cos w t  

The dipole moment is then:

P =  q ₀  S cos w t z^ = q S

The electrical potential in the first instance is expressed:  

V_P = 

1/ 4 p ε_o  {q ₀  cos w(t -  R ₊ /c) /R ₊   - q ₀  cos w(t -  R _- /c) /R _-}

The law of cosines yields:

R₊ ²  =   r ²  +  (S/2) ²   +   (r S/2) 2 cos q

=  r ²  +  S²/4  +  r S cos q

or:

R₊ =  [r ²  +  S²/4  +  r S cos q] ^1/2

For: r >> S  (Distance much larger than antenna dimension):

 S² < <   rS (Then terms in  S² can be neglected)

=> R₊ =   [r ²  +  r S cos q] ^1/2

 R₊ =  [r    +  r S cos q] ^1/2

 R₊=  r (1 +  S/ r cos q] ^1/2

 =  r (1 +  S cos q/2 r)

Note the Trigonometric identity used to simplify expressions:

cos (A + B) = cos A cos B  +  sin A sin B

Where we let A =   w(t - r/c)  ,   B  =  s w cos q/2c

Then: cos w(t - r/c) +  cos (s w cos q/2c)  =

cos w(t - r/c) cos (s w cos q/2c)  +     sin w(t - r/c) sin (s w cos q/2c)

We now substitute the above expression into  expression for V (r, t):

V (r, q, t)   = 

1/ 4 p ε_o  [q ₀/ r ( 1 +  S cos q/2 r) cos w(t - r/c) -  S w cos q / 2c  (sin w(t - r/c)) - 

q ₀/r ( 1 -  S cos q/2 r) cos w(t - r/c) +  S w cos q / 2c (sin w(t - r/c))]

Each term contains  q ₀  and  1 / r so these can be factored out to get:

Also cos w(t - r/c)  cancels, as does s w cos q / 2c  (sin w(t - r/c))  So:

V (r, q, t)   =

 q ₀/4 p ε_o r [ s cos q / r (cos w(t - r/c)) - s w cos q / 2c  (sin w(t - r/c))]

But: q ₀S =  P ₀   and factor out cos q :  

V (r, t)   =  

P ₀ cos q /4 p ε_o r [ cos w(t - r/c)/ r  -  w / c sin w(t - r/c) ]

Note:  As   w   -> 0   

V (r, t)   =  P ₀ cos q /4 p ε_o r ²

But in general given the previous approximations, conditions, the potential for the dipole radiation field will be expressed:

V (r, t)   ~     -P ₀ w /4 p c ε_o  (cos q / r) sin w(t - r/c)

See Also:

• The Plasma Physics Of Radio Astronomy (1)