### HIGH FREQUENCY ELECTROMAGNETS

The force between two magnets equals
B^{2}A/(2KU_{0})
- where:
- B is the magnetic field (measured in teslas),
- A is the cross-sectional area (in square meters),
- K is the relative permeability of the magnet (non-dimensional),
- U
_{0} is the permeability of vacuum (4(PI)E-7 henry per meter).

Electromagnets operating at a low frequency are efficient, because
they can use soft magnetic materials having a high saturation
induction. For example, a transformer operating at 60 Hz can use
alloys having a saturation induction of 2 teslas and almost no core
loss.

Electromagnets operating at a frequency of 1 MHz must use
high-frequency ferrites (Mn-Zn or Ni-Zn), which have a saturation
induction of only 0.33 tesla. The high-frequency ferrites are (2 /
0.33)^{2} times heavier than the low-frequency alloys.
Furthermore, electric insulation of the high-frequency electromagnets
is thick and heavy.

Superconductive electromagnets,
transformers, motors, and generators exist, but they melt down when
the magnetic field change rate exceeds several teslas per second. A
linear motor accelerating cargo to the orbital velocity
experiences a magnetic field change rate of millions T/s.

H. Nakamura, H. Matsumoto, H. Hasegawa, K. Nakanishi, K. Inoue,
and M. Sakai, "Static Excitation Test Results of the Partial Rotor
Model for 70 MW Class Superconducting Generator with Quick Response
Excitation," *IEEE Transactions on Magnetics*, Vol. 32, No. 4,
July 1996, pp. 2357-2360.