Magnet - Units and Calculations

Units and Calculations

For most engineering applications, MKS (rationalized) or SI (Système International) units are commonly used. Two other sets of units, Gaussian and CGS-EMU, are the same for magnetic properties and are commonly used in physics.

In all units, it is convenient to employ two types of magnetic field, B and H, as well as the magnetization M, defined as the magnetic moment per unit volume.

1. The magnetic induction field B is given in SI units of teslas (T). B is the magnetic field whose time variation produces, by Faraday's Law, circulating electric fields (which the power companies sell). B also produces a deflection force on moving charged particles (as in TV tubes). The tesla is equivalent to the magnetic flux (in webers) per unit area (in meters squared), thus giving B the unit of a flux density. In CGS, the unit of B is the gauss (G). One tesla equals 104 G.
2. The magnetic field H is given in SI units of ampere-turns per meter (A-turn/m). The turns appears because when H is produced by a current-carrying wire, its value is proportional to the number of turns of that wire. In CGS, the unit of H is the oersted (Oe). One A-turn/m equals 4π×10−3 Oe.
3. The magnetization M is given in SI units of amperes per meter (A/m). In CGS, the unit of M is the oersted (Oe). One A/m equals 10−3 emu/cm3. A good permanent magnet can have a magnetization as large as a million amperes per meter.
4. In SI units, the relation B = μ₀(H + M) holds, where μ₀ is the permeability of space, which equals 4π×10−7 T•m/A. In CGS, it is written as B = H + 4πM. (The pole approach gives μ₀H in SI units. A μ₀M term in SI must then supplement this μ₀H to give the correct field within B, the magnet. It will agree with the field B calculated using Ampèrian currents]

Materials that are not permanent magnets usually satisfy the relation M = χH in SI, where χ is the (dimensionless) magnetic susceptibility. Most non-magnetic materials have a relatively small χ (on the order of a millionth), but soft magnets can have χ on the order of hundreds or thousands. For materials satisfying M = χH, we can also write B = μ₀(1 + χ)H = μ₀μ_rH = μH, where μ_r = 1 + χ is the (dimensionless) relative permeability and μ =μ₀μ_r is the magnetic permeability. Both hard and soft magnets have a more complex, history-dependent, behavior described by what are called hysteresis loops, which give either B vs. H or M vs. H. In CGS, M = χH, but χ_SI = 4πχ_CGS, and μ = μ_r.

Caution: in part because there are not enough Roman and Greek symbols, there is no commonly agreed-upon symbol for magnetic pole strength and magnetic moment. The symbol m has been used for both pole strength (unit A•m, where here the upright m is for meter) and for magnetic moment (unit A•m2). The symbol μ has been used in some texts for magnetic permeability and in other texts for magnetic moment. We will use μ for magnetic permeability and m for magnetic moment. For pole strength, we will employ q_m. For a bar magnet of cross-section A with uniform magnetization M along its axis, the pole strength is given by q_m = MA, so that M can be thought of as a pole strength per unit area.