1) What is the decay of the magnetic field of a magnetic dipole?

It can be modelled as a ciruclar current with infinitesimal small radius. The decay of a current carrying wire can be easily calculated with Amperes law to be $$\vec{H}_\varphi=\frac{I}{2\pi R}\vec{e}_\varphi$$ with the radius \$R\$ and the current \$I\$ (cylinder coordinates). For a magnetic dipole, the field distribution can be calculated via Biot-Savarts law, and in my calculations it results in a \$\propto\frac{1}{r^2}\$ decay of the field. Measurements show a decay of \$\propto\frac{1}{r^2}\$ and \$\propto\frac{1}{r^3}\$ further away from the source. I should also mention the theoretical formulas for a magnetic dipole $$\vec{H}(\vec{r})=m\frac{1}{4\pi\mu_0}\bigg\{\vec{e}_r2\bigg(\frac{jk}{r}+\frac{1}{r^2}\bigg)\frac{e^{.jkr}}{r}\cos\vartheta+\vec{e}_\vartheta\bigg(-k^2+\frac{jk}{r}+\frac{1}{r^2}\bigg)\frac{e^{-jkr}}{r}\sin\vartheta\bigg\}.$$ From this formula (spheric coordinates: \$r\$, \$\vartheta\$, \$\varphi\$ ), I conclude that there are three different regions such as farfield etc. However, I do not know the boundaries of the regions and neither how this formula can coexist with Biot-Savarts law which yields (in my calculations) only one region. So:

2) What are the boundaries for farfield, nearfield?

3) How can I merge the analytic results from Biot-Savart and the magnetic dipole Formula above?

  • \$\begingroup\$ What did your measurement experiment look like? \$\endgroup\$
    – Andy aka
    Commented Jun 22, 2018 at 16:52
  • \$\begingroup\$ Used SQUID Sensors in BMSR-2 in Berlin and varied the distance in vertical direction. Then plottet the measured values and additionally \$\frac{1}{r^2}\$ and \$\frac{1}{r^3}\$ curves. For closer distances it fits the \$\frac{1}{r^2}\$, for distances further away it fits \$\frac{1}{r^3}\$. \$\endgroup\$
    – Irenaius
    Commented Jun 22, 2018 at 16:55
  • \$\begingroup\$ But you cannot make a magnetic dipole so what did you use? Draw a picture because I'm not going to try and guess what you did. \$\endgroup\$
    – Andy aka
    Commented Jun 22, 2018 at 17:02
  • \$\begingroup\$ The measurements are irrelevant for my question since it is purely theoretical. However, in first approximation the field of a heart is similar to a magnetic dipole. \$\endgroup\$
    – Irenaius
    Commented Jun 22, 2018 at 18:45
  • \$\begingroup\$ Biot savart will show a reciprocal square turning into a reciprocal cube for a small loop. And a small loop made smaller is going to end up as a "ciruclar current with infinitesimal small radius". \$\endgroup\$
    – Andy aka
    Commented Jun 22, 2018 at 18:54

1 Answer 1


The right angled normal decay of a magnetic field due to current is solely dependent upon the source geometry.

For a planar surface it is 1/r.

For a wire it is quadratic.

For a point source it is cubic like the far-field dipole on monopole.

In between depends on the \$r/\lambda\$ interpolation of quadratic and cubic.

  • \$\begingroup\$ If the decay of a point source was cubic, then why does the formula above states otherwise - or is it incorrect? The \$r/\lambda\$ dependence is exactly what I am looking for in question 2), but at what ratios are the boundaries exactly? \$\endgroup\$
    – Irenaius
    Commented Jun 23, 2018 at 8:35
  • \$\begingroup\$ Did you take the derivative to see the power terms? \$\endgroup\$ Commented Jun 23, 2018 at 12:56
  • \$\begingroup\$ No, but why would I? I want to have the H field decay, not the power decay? Or do I missunderstand the comment? \$\endgroup\$
    – Irenaius
    Commented Oct 17, 2018 at 11:30
  • \$\begingroup\$ The answer lies in the geometry of the source relative to distance for a point source, line source, plane source , the same for RF and light. Partial derivatives of the geometry at some distance when equal contribute due to partial sum and order of exponent \$\endgroup\$ Commented Oct 17, 2018 at 12:13

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