# Field decay of magnetic dipole

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?

• What did your measurement experiment look like? – Andy aka Jun 22 '18 at 16:52
• 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 $\frac{1}{r^3}$. – Irenaius Jun 22 '18 at 16:55
• 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. – Andy aka Jun 22 '18 at 17:02
• 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. – Irenaius Jun 22 '18 at 18:45
• 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". – Andy aka Jun 22 '18 at 18:54

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.

• 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? – Irenaius Jun 23 '18 at 8:35
• Did you take the derivative to see the power terms? – Tony Stewart Sunnyskyguy EE75 Jun 23 '18 at 12:56
• No, but why would I? I want to have the H field decay, not the power decay? Or do I missunderstand the comment? – Irenaius Oct 17 '18 at 11:30
• 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 – Tony Stewart Sunnyskyguy EE75 Oct 17 '18 at 12:13