# 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? Commented Jun 22, 2018 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 $\frac{1}{r^3}$. Commented Jun 22, 2018 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. Commented Jun 22, 2018 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. Commented Jun 22, 2018 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". Commented Jun 22, 2018 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? Commented Jun 23, 2018 at 8:35
• Did you take the derivative to see the power terms? Commented Jun 23, 2018 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? Commented Oct 17, 2018 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 Commented Oct 17, 2018 at 12:13