Magnetic Units

Magnetic poles, moments and magnetic dipoles

The famous inverse square force law between two poles p1 and p2 separated by a distance r was discovered by the English scientist John Michell (1750), and the French scientist Charles Coulomb (1785):

In the cgs (electromagnetic) system, k = 1, and the interaction force between two poles each of unit strength (in emu) separated by 1 cm is equal to 1 dyne. Alternatively, one can visualize this force as an interaction force between the pole p0 and the field H produced by another other pole of strength p:

Therefore, the field produced by p at the location of p0 a distance r away is given by:

The unit of the magnetic field, the Oersted (Oe), is defined as the strength of the field produced by a unit pole at a point 1 cm away (by eq. 3). Alternatively, it is the strength of the field which exerts a force of 1 dyne of a unit pole (by eq. 2).

Faraday’s representation

Michael Faraday represented the field strength in terms of “lines of force” (1 line of force = 1 maxwell (= 1 Mx)). In this representation, the field strength is defined as the number of lines of force passing through a unit area normal to the field. Therefore:

1 Oe = 1 line of force/cm2 = 1 Mx/cm2 (4)

A bar magnet has a magnetic moment given in terms of the pole strength and length of the magnet by:

The unit of the magnetic moment in cgs system is emu = erg/Oe. It is worth mentioning that the magnetic pole does not exist, neither can the distance between the two poles be determined accurately due to non-localization of the pole. However, mathematically speaking, the product of the two quantities is the magnetic moment which can be measured accurately. Further, the number of lines issuing from a pole of strength p is 4πp lines as can be deduced from eqs. 4 and 5.

The energy of a magnetic dipole with moment m in a uniform magnetic field is given by:

⃗⃗ ⃗⃗ The magnetic moment doubles by doubling the volume of the sample according to eq. 5. Therefore, a related quantity, the intensity of magnetization, or the magnetization, is used to characterize the magnetic properties of the sample and determine the degree of magnetization of the sample.

In cgs system the magnetization is expressed in emu/cm3 where V in eq. 7 is the sample volume in cm3. The magnetization is therefore measured in units of emu/cm3. Since the sample volume is temperature dependent, and difficult to measure, the specific magnetization σ is often used in magnetic measurements, which is given by the magnetization divided by the material density ρ:

( )

The magnetization is sometimes expressed per unit cell or per formula unit, in which case the much -21 smaller unit, the Bohr magneton μB = 9.27×10 erg/Oe is used.

The magnetic moment of the sample is the sum of the moments of the fundamental magnetic dipoles μ aligned parallel to the magnetization axis (according to eq. 5).

Currents and magnetic fields As will be shown later in the section of units, a long straight wire produces a circular magnetic field a distance r from the wire given by:

Here the current is measured in amperes and the distance in cm.

Notice that in cgs system, (H = 2i/cR) where c is the speed of light in cm/s and the current is given in esu (1esu = c/10 amperes). This relation leads to eq. 9.

The axial field intensity at the center of a circular current loop of radius R is given by:

The axial magnetic field in a solenoid of N turns and length L is given by:

In this last equation m = NiA/10 erg/Oe is the magnetic moment of the solenoid and V is the volume inside it. Notice that the magnetic moment of the solenoid is simply the sum of the magnetic moments of the N individual current loops of the solenoid. The solenoid can therefore be viewed as a uniformly magnetized bar with a magnetic moment m. The number of lines per cm2 issuing from the magnetization M = m/V is therefore equal to 4πM. Thus the magnetization of bar magnet can be viewed as due to a collection of molecular currents called Amperian currents.

Magnetic materials If a cylinder of a magnetic material is subjected to an external field H, it gets magnetized, and consequently, the number of lines of force per cm2 in the material is no more equal to H. If a thin cavity is made in the material by cutting off a section of the material, magnetic poles develop at the surface of the cavity. If the field is applied from left to right and the material is paramagnetic, ferrimagnetic or ferromagnetic, north poles develop at the left-hand surface and south poles at the right-hand surface. The number of lines per cm2 inside a thin cavity in the material is the sum of H and 4πM. This total line density is the magnetic flux density, or simply, the magnetic induction B. Therefore, in cgs units we have:

H N S N S B N S N S 4πM

Fig. 1: The magnetic field H, the flux density B, and the magnetization directions in a magnetic material.

Notice that the magnetic field, the flux density, and the magnetization in eq. 12 have the same units, namely, Mx/cm2. However, the units for the three quantities were given different names. The unit of B is gauss (G), while that of H is oersteds (Oe). These two units are equivalent physically and numerically according to eq. 12. The unit of M is erg/Oe.cm3 = emu/cm3 as previously mentioned. The quantity 4πM, however, has the dimensions of maxwells/cm2, and is therefore expressed in units of Oe or G. Workers in magnetism usually use the unit G to emphasize the fact that the lines of magnetization contribute to B.

Since M for a paramagnetic material is normally small, an increase of the order of 0.02% in the flux in the material is observed. For a ferromagnetic or a ferrimagnetic material, however, the magnetization is much higher, and the flux density is greatly enhanced in such materials. On the contrary, a diamagnetic material develops poles in opposite sense, and the induced magnetization is opposite to the applied field, resulting in a reduction in the flux density inside the material.

Magnetic susceptibility The magnetic susceptibility is defined as the ratio of the magnetization to the applied field:

This is the volume susceptibility, which has no dimensions since the units of the magnetization and the field are the same (notice eq. 12). The quantity emu/Oe.cm3 therefore has no dimensions (Prove). The mass susceptibility on the other hand is given by:

Fig. 2 shows typical magnetization curves for: a) paramagnetic or antiferromagnetic material (with susceptibility of about 20×10-6), b) diamagnetic material (with susceptibility of about –2×10-6), and c) ferromagnetic or ferrimagnetic material. The figure shows that the susceptibility for the diamagnetic, paramagnetic, or antiferromagnetic materials is constant. However, the susceptibility for a ferromagnetic or an antiferromagnetic material changes with applied field intensity. The curve in c) shows a maximum susceptibility of about 40, which is of the order of a million times greater than that for a paramagnet. In addition, the curve in c) shows saturation at low field, in which case we refer to the sample as magnetically soft. While the saturation magnetization is characteristic of the material, the magnetic hardness is sensitive to the characteristics of the sample and the preparation conditions. Thus, if the same sample with magnetization curve c) were heavily cold-worked, it could need much higher field to saturate, in which case the sample is said to be magnetically hard (see Fig. 3).

3 M(emu/cm3) M(emu/cm )

c

0.2 2000 a Mr

1000 4 H (Oe) 10 – 0.02 b H (Oe) 100 Fig. 2: Magnetization curves for a) paramagnetic or antiferromagnetic material, b) diamagnetic material, and c) ferromagnetic or ferrimagnetic material.

M Fig. 3: a) Magnetization curve of a well annealed Ms magnetically soft sample. The saturation magnetization a b Ms is reached at low applied field (~ 100 Oe). b) Magnetization curve of a cold-worked magnetically hard sample. The field required for saturation H could be tens of thousands of oersteds. In addition to the saturation properties, ferromagnetic and ferrimagnetic materials exhibit irreversibility, or hysteresis in the magnetization process (Fig. 2 c). This property is manifested by the presence of a remanence magnetization Mr as the field is reduced from the saturation value down to zero.

Units The magnetic quantities were traditionally (and commonly nowadays) expressed in cgs (electromagnetic) units. Recently, the SI system of units is adopted, and the demand for converting the magnetic quantities to this system of units is increasing. The SI (rationalized MKSA) system uses the meter, kilogram, second, and ampere as units for length, mass, time, and current, respectively. The cgs system uses centimeter, gram and second as units for the first three quantities. However, the current was given different units in four different fundamental cgs systems. The electromagnetic (emu) and the Gaussian cgs systems are perhaps the most common systems used in literature when dealing with electromagnetic phenomena. In the Gaussian, as well as in the electrostatic (esu) system of units, the charge (and consequently the current) is expressed in electrostatic (esu) units, while in the emu system these quantities are expressed in absolute units (abcoulomb and abampere). The remaining of this section is devoted to deriving the conversion relations for the magnetic quantities between the emu (cgs) and the SI systems of units.

The relation between the esu and the SI units of charge is derived from Coulomb’s law:

́ ́

́

According to eq. 15, the force between two unit charges (1 C each) placed a distance 1 m apart is equal to 9×109 N. The numerical value of 1 C in the esu system is then determined from eq. 16:

The conversion factor in eq. 17 is the same for current.

Current and Magnetic Field H Let us consider the magnetic field produced by a current-carrying long straight wire at a normal distance r from the wire. In electromagnetic units and in the Gaussian system of units, one of Maxwell equations (Amperes law) reads:

⃗⃗

⃗⃗

Here c is the speed of light (≈ 3×1010 cm/s). According to eq. 18, the magnetic field is given by:

However, according to eq. 19, the field is given by:

Now according to eq. 20, a field intensity of 2 Oe (= 2 Mx/cm2) is generated by a current of1 abampere at a distance of 1 cm from the wire. This current (in esu) is given by eq. 21:

Accordingly, with the use of the conversion factor in eq. 17 we obtain:

This last relation provides the numerical conversion relations between the emu, the Gassian, and the SI systems for current.

In the SI system, however, Ampere’s law is given by:

⃗⃗

Accordingly, the magnetic field resulting from the long current-carrying straight wire is given is given by:

Consequently, a long straight wire carrying a current of 1A would produce a field intensity of 1/2π (A/m) at a normal distance of 1 m from the wire. In cgs system, according to eq. 20, this field is:

( )

( )

We therefore have the conversion relation:

Magnetic moment The magnetic moment of a current-carrying loop is the product of the loop area and the current, and is normal to the area. If the area is 1 m2 and the current is 1 A, the magnetic moment of the loop is 1 A.m2 in rationalized MKSA units. In cgs electromagnetic system, the magnetic moment of the loop is

The unit of the magnetic moment in the electromagnetic system is often called emu even though this abbreviation generally refers to the electromagnetic system of units. Therefore:

Magnetization The magnetization M is defined as the magnetic moment per unit volume (eq. 7). In SI system:

This equation indicates that the unit of magnetization is numerically and physically equivalent to that of the magnetic field, which is consistent with the Sommerfeld convention in SI system given by:

-7 In this last equation μ0 = 4π×10 H/m is the permeability of free space B is given in tesla (T), and H and M in A/m. In cgs units, the magnetization is expressed in emu/cm3. The relation between the SI and cgs units of magnetization is therefore:

The specific magnetization σ, on the other hand, is defined as the magnetization per unit mass (eq. 8). In SI system, the specific magnetization is expressed in A.m2/kg. We therefore obtain the conversion relation:

Magnetic Flux and Flux Density B Maxwell equation related to Faraday’s law is given in SI system by:

⃗ ⃗

Accordingly, an electromotive force (emf) develops in a loop as a consequence of the time-varying flux density normal to the area of the loop. This is given by:

Here the electromotive force (in volts) is expressed as the negative of the rate of change of the magnetic flux:

∫ ⃗

The SI unit of magnetic flux (the number of lines of force) is the weber (1 Wb = 1 T.m2 = 1 V.s).

In Gaussian units, Maxwell equation reads:

⃗ ⃗

The emf is given in this system of units by:

Here that the rate of change of flux is expressed in units of Mx/s. The relation between the units of emf in Gaussian (statvolt) and SI (volt) systems is derived from the form of the potential due to a point charge in these two systems.

Accordingly, a point charge of 1 esu produces a potential of 1 statvolt at a point a distance 1 cm away from the point charge. This potential (in volts) is given by eq. 38:

Accordingly, eq. 36 gives the emf in volts as:

We should be reminded here the rate of change of the flux is expressed in Mx/s.

Comparing eq. 33 with eq. 40 we realize that the same emf (of 1 volt for example) develops from a flux rate of 1 Wb/s and a flux rate of 108 Mx/s. This leads to the conversion relation for the magnetic flux in the two systems of units (the SI and the emu systems).

Accordingly, the conversion relation for the magnetic flux density, B can be derived.