Fundamental Relationships
Read Suggestions
for the Use of SI Units in Magnetism
Fundamental Units
(From Resnick and Halliday, Physics, Part I, 1968, Wiley)
Coulomb's Law
 a force exists between 2 magnetic
poles:

where
is the force
is
the permeability of free space, =
,
are the magnetic pole strength
is
the distance separating the poles
is
the unit radial vector
 unlike gravity, poles come in 2 flavors:

+ (northseeking)
 
(southseeking)
 like poles repel (F is +,
force is outward)
 unlike poles attract (F is , force is inward)
Magnetic Induction, B
 as with gravity, we are interested in force Earth
exerts on a unit pole (like acceleration, with g)
 or, 'magnetic field intensity'
 analogous to gravitational acceleration (but not
acceleration units!)
 force per unit pole strength (force exerted on unit
magnetic pole)
(In our analogy with gravity, m here is the Earth's
"monopole" field, which is a fiction; Stacey incorrectly calls B "magnetic field, which
is H)
Magnetic Field Strength, H
 if we only had to deal with a vacuum (or even air,
since it has negligible magnetic susceptibility), we could always deal with H
(magnetic field strength)..
 however, in presence of "magnetizable"
material, there is a magnetic polarization (or, simply, magnetization)
of material which produces an additional field (J) which adds to H
 combining the field strength, H, and the magnetic
polarization (magnetization), J, is call the magnetic
induction, B
cgs system
 magnetic induction, B, is related to
magnetic field, H, by
B = H + 4pJ
(note that B, H, J, [and M, below] have the same
units)
 J, magnetic polarization (or magnetization)
and M, the magnetic dipole moment per
unit volume are related by
J = M
SI system
B = m_{0}H
+ J
 J = m_{0}M
where m_{0}
= 4p x 10^{7} H/m
(Henry/meter) is
the permeability of free space
Units
cgs system
 from above, all four fundamental terms have the
same units in cgs system, but is has been customary to use:
B 
magnetic induction 
gauss 
G 
H 
magnetic field 
oersted 
Oe 
J 
magnetic polarization,
magnetization 
electromagnetic units 
emu 
M 
magnetic dipole moment per unit volume 
electromagnetic units 
emu 
SI system
 in SI, for force of 1 Newton and 1 unit pole
strength: A/m (H), or Tesla (B)
B 
magnetic induction 
tesla 
T 
H 
magnetic field 
amperes per meter 
A/m 
J 
magnetic polarization,
magnetization 
tesla 
T 
M 
magnetic dipole moment per unit volume 
amperes per meter 
A/m 
 since these units are very large quantities, for
exploration work:
 1 A/m (H), or Tesla (B) = 10^{9} nanotesla
(nT)
 1 Oersted (Oe) = 10^{5} g
(gamma)
 1 g
equivalent to nT
Definition of magnetic permeability
Magnetic Potential
 Since H or B are central force fields (depend only on r),
they are conservative, and can be represented as gradient of a scalar,
potential field. And since the field are proportional to 1/r^{2},
they obey LaPlace's equation. The scalar potential is
and
when there is no
magnetic material is present,
where H is
the magnetic field strength, else,
where
is
relative
magnetic permeability.
Magnetic Dipole
Intensity of magnetization
Magnetic dipole moment is an extensive quantity (like
mass). In analogy with gravity, magnetic dipole moment per unit volume is
an intensive quantity (like density). This is also called the intensity of
magnetization, or
I = M/volume = ml/volume = m/area
where I and M are vector quantities.
Potential due to a Dipole
The potential due to a dipole is found by summing the potentials
of the individual poles:
 so the total potential is (after bringing
outside),
 the basic dipole consists of spinning electron (current
loop), where l and hence l/r are small, so we neglect terms of (l/r)^{2},
and use Taylor's series:
eliminating terms, we get
Field of Magnetic Dipole

Fig. 4.8. Magnetic field of a dipole. Dipole m is
oriented toward top of page. Vectors indicate the direction of B that would
be observed at the center of each vector. Dashed contours indicate constant
values of B, the value decreasing by a factor of 10 at each succeeding
contour from the dipole; that is, if the closest contour to the dipole has
the value B = 1, succeeding contours have the values 0.1, 0.01, and 0.001,
respectively. 
Quotes
 obtain vector field by taking gradient of potential,
and find the components of this vector field:
 in MKS, the magnetic induction components are
 N.B.
 B (and H) depend on 1/r^{3}
 for given r, field is greatest on axis with dipole
 B_{q}
= 0 at magnetic "pole"
 B_{r} = 0 at magnetic "equator"
 sign convention: Positive pole attracted toward
Earth's N pole (northseeking pole)