Editing Dipole (section)

== Field of a static magnetic dipole ==
{{see also|Magnet#Two models for magnets: magnetic poles and atomic currents}}

=== Magnitude ===
The far-field strength, ''B'', of a dipole magnetic field is given by
: <math>B(m, r, \lambda) = \frac{\mu_0}{4\pi} \frac{m}{r^3} \sqrt{1 + 3\sin^2(\lambda)} \, ,</math>
where
: ''B'' is the strength of the field, measured in [[tesla (unit)|tesla]]s
: ''r'' is the distance from the center, measured in metres
: ''λ'' is the magnetic latitude (equal to 90°&nbsp;−&nbsp;''θ'') where ''θ'' is the magnetic colatitude, measured in [[radian]]s or [[degree (angle)|degree]]s from the dipole axis<ref group="note">Magnetic colatitude is 0 along the dipole's axis and 90° in the plane perpendicular to its axis.</ref>
: ''m'' is the dipole moment, measured in [[ampere]]-square metres or [[joule]]s per [[tesla (unit)|tesla]]
: ''μ''<sub>0</sub> is the [[permeability (electromagnetism)|permeability of free space]], measured in [[henry (unit)|henries]] per metre.

Conversion to cylindrical coordinates is achieved using {{nowrap|''r''<sup>2</sup> {{=}} ''z''<sup>2</sup> + ''ρ''<sup>2</sup>}} and
: <math>\lambda = \arcsin\left(\frac{z}{\sqrt{z^2 + \rho^2}}\right)</math>
where ''ρ'' is the perpendicular distance from the ''z''-axis. Then,
: <math>B(\rho, z) = \frac{\mu_0 m}{4 \pi \left(z^2 + \rho^2\right)^\frac32} \sqrt{1 + \frac{3 z^2}{z^2 + \rho^2}}</math>

=== Vector form ===
The field itself is a vector quantity:
: <math>\mathbf{B}(\mathbf{m}, \mathbf{r}) =
  \frac{\mu_0}{4\pi} \ \frac{3(\mathbf{m} \cdot \hat{\mathbf{r}}) \hat{\mathbf{r}} - \mathbf{m}}{r^3}
  </math>
where
: '''B''' is the field
: '''r''' is the vector from the position of the dipole to the position where the field is being measured
: ''r'' is the absolute value of '''r''': the distance from the dipole
: '''r̂''' = {{sfrac|'''r'''|''r''}} is the unit vector parallel to '''r''';
: '''m''' is the (vector) dipole moment
: ''μ''<sub>0</sub> is the permeability of free space

This is ''exactly'' the field of a point dipole, ''exactly'' the dipole term in the multipole expansion of an arbitrary field, and ''approximately'' the field of any dipole-like configuration at large distances.

=== Magnetic vector potential ===
The [[vector potential]] '''A''' of a magnetic dipole is
: <math>\mathbf{A}(\mathbf{r}) = \frac{\mu_0}{4\pi} \frac{\mathbf{m} \times \hat{\mathbf{r}}}{r^2}</math>
with the same definitions as above.