e.m.u., emu or electromagnetic unit of magnetic moment - a physical quantity of magnetic dipole moment m commonly used in the CGS system of units¹⁾²⁾ such that m _CGS = 1 emu _CGS = 1 erg/Oe, which is equivalent in the SI system of units to m _SI = 4·π·10^-10 Wb·m. “Emu” is not a basic electromagnetic unit within the CGS system, but it is widely used as it shortens the notation.

However, there are other meanings associated with the acronym “EMU” which creates confusion in the subject with relation to the CGS (or Gaussian) system of units, even among those who operate with the CGS units. In the SI system such confusion does not arise.

Uniform magnetic field B (or H) acts with torque τ on magnetic dipole m, tending to align m parallel to B, with the potential energy proportional to the cosine of the angle between the two vectors

^{S. Zurek, E-Magnetica.pl, CC-BY-4.0}

In many publications which employ the CGS system of units the symbol “emu” can have many different meanings, which are not necessarily obvious even from the context, so great attention must be paid to the numerical values and the units. Also different sources might even define different meanings for “emu”.

For example, volume magnetisation M_vol (in CGS) has the units of moment/volume so strictly erg/(Oe·cm³). However, erg/Oe = emu, and therefore for brevity it can be written to have the units of emu/cm³.

Also, the mass magnetisation M_m (in CGS) can be calculated by dividing the volume magnetisation M_vol by mass density d thus leading to the unit of emu/g. ⁶⁾

However, the full expression of such CGS unit is sometimes omitted to the point of simply stating that the result is given in the “CGS unit” or “emu unit”, or similar, so that even the quantity itself is not necessarily immediately obvious (let alone its unit). To give some examples, the following notation (and many more) can be encountered in the literature (exact quotes):

• magnetic moment m (in emu units) ⁷⁾
• Magnetic susceptibility of pure copper […] χ × 10⁶, cgs ⁸⁾
• magnetic susceptibility […] in cgs-emu units ⁹⁾
• specific magnetization […] emu/g ¹⁰⁾
• mass susceptibility […] c.g.s. e.m.u. per gram ¹¹⁾
• magnetization […] emu/cm^{3 12)}
• magnetic saturation […] emu g^{-1 13)}
• mean NRM of the samples is (2.69 ± 0.28) × 10^-3 EMU cm^-3 and the mean susceptibility is (2.39 ± 0.003) × 10^-4 EMU oersted^-1 cm^{-3 14)}
• in the same book: ¹⁵⁾
  • saturation magnetization of 1.23 × 10^-4 Wb m kg^-1 (98 emu g^-1)
  • mass mag. moment (emu/g)
• Magnetic moments as small as 5 × 10⁴ A m² (5 × 10^-5 emu) ¹⁶⁾
• The unit of magnetic moment is Am² although old unit emu is also used (1 Am²=10³ emu). ¹⁷⁾
• saturation magnetization (magnetic moment per unit particle volume) […] M_sat = 86.1 emu/g Fe (= 315 emu/cm³ for maghemite) and a mean magnetic moment μ = 4.38 × 10^-17 emu/particle, which corresponds to a μ of 7.98 × 10^-21 emu per Fe ion. In units of the Bohr magneton (BM), 9.27 × 10^-21 erg/G, μ corresponds to 0.86 BM per iron ion. […] giving X_c = 9.07 × 10^-4 (which is dimensionless in emu units) ¹⁸⁾

The three basic magnetic quantities are related by an equation in the CGS system, however, they all have different units: ¹⁹⁾²⁰⁾

• magnetic flux density B in (G)
• magnetic field strength H in (Oe)
• magnetisation M in (erg/(Oe·cm³)) = (emu/cm³)
• but the component “4·π·M” should have the units of (G) or (Oe) even though “4·π” is unitless

Moreover, in the CGS system, magnetic susceptibility χ is the ratio of M/H, which can have the units expressed in several different ways:

When converted from CGS to SI, the magnetic moment is scaled by the absolute magnetic permeability of vacuum and thus has the unit of (Wb·m)≡(A·m²·H/m), whereas the magnetic moment expressed directly in the SI system has just the unit of (A·m²).

In the CGS system the apparent scaling is achieved simply by multiplying just by the factor “4·π” (which itself is unitless). The numerical values can be converted between CGS and SI systems for various quantities, but not for the physical units:

• B in (G) can be converted to (T) which can be expressed in the basic SI units as (T)≡(kg/(A·s²))
• H in (Oe) can be converted to (A/m) which is already expressed in the basic SI units
• in CGS for vacuum B = H but they have different SI units and thus cannot be added directly, even after converting numerical values - the additional required operation is scaling by the vacuum permeability μ₀ (H/m), because (A/m)·(H/m)≡(A/m)·(kg·m²/(A²·s²)/m)≡(kg/(A·s²))

Further confusion in the subject is added by the prior existence of CGS EMU (electromagnetic) and CGS ESU (electrostatic) systems of units.^21)22)23)

In this context the acronym “EMU” or “emu” (also “em” ²⁴⁾) simply means “electromagnetic (type of) unit” rather than a specific physical unit.

Similarly “ESU” or “esu” (also “es” ²⁵⁾) means “electrostatic (type of) unit”.

Special attention must be paid when operating with such quantities, because of the additional numerical scaling factors arising due to incompatibilities of each system.²⁶⁾ For example 1 abampere in CGS emu = 1/c statamperes in CGS esu = 10 A in SI (where c is the speed of light). And for 1 abvolt emu = c · statvolt esu = 10⁸ volts in SI.

• Magnetic dipole moment
• Magnetic susceptibility
• CGS system of units
• SI system of units