2D Magnets
Note
- SI Units with the Sommerfeld convetion are used for this discussion: \(\mathbf{B} = \mu_0 \left( \mathbf{H} + \mathbf{M} \right)\)1
- However, the Kennelly Convetion is used for the creation of each magnet object in the library: \(\mathbf{B} = \mu_0\mathbf{H} + \mathbf{J}\)1
- In free space \(\mathbf{B} = \mu_0 \mathbf{H}\)
- In magnetised bodies, the demagnetising field \(\mathbf{H_d} = - N \mathbf{M}\), where \(N\) is the demagnetising factor.
For infinitely long objects, the field problems can be approximated in 2D, where the magnet field consists of:
\[
\mathbf{B} = B_x \mathbf{\hat{x}} + B_y\mathbf{\hat{y}}
\]
Rectangles
The magnetic field due to rectangle magnetised in \(y\) is2:
\[
B_x = \frac{\mu_0 M_r}{4\pi} \left[\ln {\left(
\frac{{\left(x+a\right)}^2 + {\left(y-b\right)}^2}{{\left(x+a\right)}^2
+{\left(y+b\right)}^2}
\right)}
-\ln{\left(
\frac{{\left(x-a\right)}^2+{\left(y-b\right)}^2}{ {\left(x-a\right)}^2 +
{\left(y+b\right)}^2}
\right)}\right]
\]
\[
B_y = \frac{\mu_0 M_r}{2\pi}
\left[{\tan}^{-1}{\left( \frac{2b \left(x+a\right)}{y^2-b^2+{\left(x+a\right)}^2}
\right)}
- {\tan}^{-1}{\left(\frac{2b\left(x-a\right)}{y^2-b^2+{\left(x-a\right)}^2}\right)}\right]
\]
or magnetised in \(x\) is:
TODO:
\[
B_x = \frac{\mu_0 M_r}{2\pi} []
\]
\[
B_y = \frac{\mu_0 M_r}{4\pi}[]
\]
Biaxial Rods (Circle)
A long bipolar rod of radius \(a\) can be approximated as circular source. The magnetic stray field is most conveniently written in polar coordinates, as2:
\[
\mathbf{B} = \frac{\mu_0 M_r}{2} \left( \frac{a^2}{r^2}\right) \left[
\cos(\phi) \mathbf{\hat{r}} + \sin(\phi) \mathbf{\hat{\phi}}
\right]
\]