Editing Dipole (section)

== Dipole radiation ==
[[File:Electric dipole radiation.gif|thumb|Modulus of the Poynting vector for an oscillating electric dipole (exact solution). The two charges are shown as two small black dots.]]
<!-- use of the far-field approximation for the near field yielded incorrect results, see archived discussion
[[File:dipole.gif|thumb|right|250px|Evolution of the magnetic field of an oscillating electric dipole. The field lines, which are horizontal rings around the axis of the vertically oriented dipole, are perpendicularly crossing the ''xy''-plane of the image. Shown as a colored [[Contour line|contour plot]] is the ''z''-component of the field. Cyan is zero magnitude, green–yellow–red and blue–pink–red are increasing strengths in opposing directions.]] -->
{{see also|Dipole antenna}}
In addition to dipoles in electrostatics, it is also common to consider an electric or magnetic dipole that is oscillating in time. It is an extension, or a more physical next-step, to [[spherical wave]] radiation.

In particular, consider a harmonically oscillating electric dipole, with [[angular frequency]] ''ω'' and a dipole moment ''p''<sub>0</sub> along the '''ẑ''' direction of the form
: <math>\mathbf{p}(\mathbf{r}, t) = \mathbf{p}(\mathbf{r})e^{-i\omega t} = p_0\hat{\mathbf{z}}e^{-i\omega t} .</math>

In vacuum, the exact field produced by this oscillating dipole can be derived using the [[retarded potential]] formulation as:
: <math>\begin{align}
  \mathbf{E} &= \frac{1}{4\pi\varepsilon_0} \left\{
      \frac{\omega^2}{c^2 r} \left( \hat{\mathbf{r}} \times \mathbf{p} \right) \times \hat{\mathbf{r}} +
      \left( \frac{1}{r^3} - \frac{i\omega}{cr^2} \right)
        \left( 3\hat{\mathbf{r}} \left[\hat{\mathbf{r}} \cdot \mathbf{p}\right] - \mathbf{p} \right)
    \right\} e^\frac{i\omega r}{c} e^{-i\omega t} \\
  \mathbf{B} &= \frac{\omega^2}{4\pi\varepsilon_0 c^3} (\hat{\mathbf{r}} \times \mathbf{p}) \left( 1 - \frac{c}{i\omega r} \right) \frac{e^{i\omega r/c}}{r} e^{-i\omega t}.
\end{align}</math>

For {{sfrac|''rω''|''c''}}&nbsp;≫&nbsp;1, the far-field takes the simpler form of a radiating "spherical" wave, but with angular dependence embedded in the cross-product:<ref>[[David J. Griffiths]], Introduction to Electrodynamics, Prentice Hall, 1999, page 447</ref>
: <math>\begin{align}
  \mathbf{B}
    &=  \frac{\omega^2}{4\pi\varepsilon_0 c^3} (\hat{\mathbf{r}} \times \mathbf{p}) \frac{e^{i\omega (r/c - t)}}{r}
     =  \frac{\omega^2 \mu_0 p_0 }{4\pi c} (\hat{\mathbf{r}} \times \hat{\mathbf{z}}) \frac{e^{i\omega (r/c - t)}}{r}
     = -\frac{\omega^2 \mu_0 p_0 }{4\pi c} \sin(\theta) \frac{e^{i\omega (r/c - t)}}{r} \mathbf{\hat{\phi}} \\
  \mathbf{E}
    &=  c \mathbf{B} \times \hat{\mathbf{r}}
     = -\frac{\omega^2 \mu_0 p_0}{4\pi} \sin(\theta) \left(\hat{\phi} \times \mathbf{\hat{r}}\right) \frac{e^{i\omega (r/c - t)}}{r}
     = -\frac{\omega^2 \mu_0 p_0}{4\pi} \sin(\theta) \frac{e^{i\omega (r/c - t)}}{r} \hat{\theta}.
\end{align}</math>

The time-averaged [[Poynting vector]]
: <math>\langle \mathbf{S} \rangle = \left(\frac{\mu_0 p_0^2\omega^4}{32\pi^2 c}\right) \frac{\sin^2(\theta)}{r^2} \mathbf{\hat{r}}</math>
is not distributed isotropically, but concentrated around the directions lying perpendicular to the dipole moment, as a result of the non-spherical electric and magnetic waves. In fact, the [[spherical harmonic]] function (sin ''θ'') responsible for such [[torus|toroidal]] angular distribution is precisely the ''l''&nbsp;=&nbsp;1 "p" wave.

The total time-average power radiated by the field can then be derived from the Poynting vector as
: <math>P = \frac{\mu_0 \omega^4 p_0^2}{12\pi c}.</math>

Notice that the dependence of the power on the fourth power of the frequency of the radiation is in accordance with the [[Rayleigh scattering]], and the underlying effects why the sky consists of mainly blue colour.

A circular polarized dipole is described as a superposition of two linear dipoles.