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Magnetic Field 1D Methods

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.

Cylinder

The magnetic field directly above the centre of a cylinder is2:

Magnetic cylinder schematic
\[ B_z = \frac{\mu_0 M_r}{2} \left[ \frac{z+L}{\sqrt{(z+L)^2 + R^2} } - \frac{z}{\sqrt{z^2 + R^2}} \right] \]

Cuboid

While for a cuboid, this equation is:

Magnetic cuboid
\[ B_z = \frac{\mu_0 M_r}{2} {\left[ \tan^{-1}{\left( \frac{(z+L)\sqrt{a^2 + b^2 + (z+L)^2} }{ab} \right)} - \tan^{-1}{\left( \frac{z\sqrt{a^2 + b^2 + z^2} }{ab} \right)} \right]} \]

  1. J. M. D. Coey, Magnetism and Magnetic Materials (Cambridge University Press, 2010). ↩↩

  2. E. P. Furlani, Permanent Magnet and Electromechanical Devices (Academic Press, San Diego, 2001). ↩