Magnetic fields surround magnetic materials and
electric currents and are detected
by the force they exert on other magnetic materials and moving
electric charges. The magnetic field
at any given point is specified by both a
direction and a
magnitude (or strength); as such it is a
vector field.
For the physics of magnetic materials, see
magnetism and
magnet, more
specifically
ferromagnetism,
paramagnetism, and
diamagnetism. For constant magnetic fields,
such as are generated by magnetic materials and steady currents,
see
magnetostatics. A changing
magnetic field generates an
electric
field and a changing electric field results in a magnetic
field. (See
electromagnetism.)
In view of
special relativity,
the electric and magnetic fields are two interrelated aspects of a
single object, called the
electromagnetic field. A pure electric
field in one
reference frame is
observed as a combination of both an electric field and a magnetic
field in a moving reference frame.
In modern physics, the magnetic (and electric) fields are
understood to be due to a
photon field; in
the language of the
Standard Model
the electromagnetic force is
mediated by photons. Most
often this microscopic description is not needed because the
simpler classical theory covered in this article is sufficient; the
difference is negligible under most circumstances.
B and H

Alternate names for H
name 
used by 
magnetic field intensity 
electrical engineers 
magnetic field strength 
electrical engineers 
auxiliary magnetic field 
physicists 
magnetizing field 
physicists 


The term
magnetic field is used for two different vector
fields, denoted
B and
H. There are many alternative names for
both, though. (See sidebar.) To avoid confusion, this article uses
Bfield and
Hfield for these fields, and uses
magnetic field where either or both fields apply.
The
Bfield can be defined in many equivalent ways based
on the effects it has on its environment. For instance, a particle
having an
electric charge,
q, and moving in a Bfield with a
velocity,
v, experiences a force,
F, called the
Lorentz force
(see
below). In
SI units, the Lorentz force equation is
 \mathbf{F}=q\left(\mathbf{v}\times\mathbf{B}\right)
where
× is the
vector
cross product. The Bfield is measured in
tesla in
SI units and in
gauss in
cgs
units.
Technically,
B is a
pseudovector (also called an
axial
vector). (This is a technical statement about how the magnetic
field behaves when you reflect the world in a mirror; this is known
as
parity) This fact is apparent
from the above definition of
B.
An alternate working definition of the Bfield can be given in
terms of the
torque on a magnetic dipole
placed in a Bfield:
 \boldsymbol{\tau}=\mathbf{m_m}\times\mathbf{B}
for a
magnetic dipole moment
m (in amperesquare meters).
Although views have
shifted over the years,
B is now understood as being the
fundamental quantity, while
H is a
derived field. It is defined as a modification of
B due to
magnetic fields produced by material media, such that (in
SI):
 \mathbf{H}\ \equiv \ \frac{\mathbf{B}}{\mu_0}\mathbf{M},
where
M is the
magnetization
of the material and
μ_{0} is the
permeability of free
space (or
magnetic constant). The Hfield is
measured in
amperes per
meter (A/m) in SI units, and in
oersteds (Oe) in cgs units.
In materials for which
M is proportional to
B the
relationship between
B and
H can be cast into the
simpler form:
H =
B ⁄
μ, where
μ is a material dependent parameter called the
permeability. In
free space, there is no magnetization,
M, so that
H =
B ⁄
μ_{0} (
free space). For
many materials, though, there is no simple relationship between
B and
M. For example,
ferromagnetic materials and
superconductors have a magnetization that is
a
multiplevalued function of
B due to
hysteresis.
See
#History below for further
discussion.
The magnetic field and permanent magnets
Permanent magnets are objects that
produce their own persistent magnetic fields. All permanent magnets
have both a north and a south pole. They are made of
ferromagnetic materials such as
iron and
nickel that have been
magnetized. The strength of a magnet is represented by its
magnetic moment,
m; for simple magnets,
m points in the direction of a line drawn
from the south to the north pole of the magnet. For more details
about magnets see
magnetization below
and the article
ferromagnetism.
Force on a magnet due to a nonuniform B
Like magnetic poles brought near each other repel while opposite
poles attract. This is a specific example of a general rule that
magnets are attracted (or repulsed depending on the orientation of
the magnet) to regions of higher magnetic field. For example,
opposite poles attract because each magnet is pulled into the
larger magnetic field near the pole of the other; the force is
attractive because for each magnet
m is
in the same direction as the magnetic field
B of the other.
Reversing the direction of
m reverses the
resultant force. Magnets with
m opposite
to
B are pushed into regions of
lower magnetic field, provided that the magnet,
and therefore,
m does not flip due to
magnetic torque. This corresponds to the like poles of two magnets
being brought together. The ability of a nonuniform magnetic field
to sort differently oriented dipoles is the basis of the
SternGerlach experiment, which
established the quantum mechanical nature of the magnetic dipoles
associated with atoms and electrons.
Mathematically, the force on a magnet having a magnetic moment
m is:
 \mathbf{F} = \mathbf{\nabla}
\left(\mathbf{m}\cdot\mathbf{B}\right),
where the
gradient ∇ is
the change of the quantity
m·B
per unit distance and the direction is that of maximum increase of
m·B. (The
dot product
m·B =

m
Bcos(
θ),
where   represent the magnitude of the vector and
θ
is the angle between them.) This equation is strictly only valid
for magnets of zero size, but it can often be used as an
approximation for not too large magnets. The magnetic force on
larger magnets is determined by dividing them into smaller regions
having their own
m then summing up the
forces on each of these regions.
The force between two magnets is
quite
complicated and depends on the orientation of both magnets and
the distance of the magnets relative to each other. The force is
particularly sensitive to rotations of the magnets due to magnetic
torque.
In many cases, the force and the torque on a magnet can be modeled
quite well by assuming a 'magnetic charge' at the poles of each
magnet and using a magnetic equivalent to
Coulomb's law. In this model, each magnetic
pole is a source of an
Hfield that is
stronger near the pole. An external
Hfield exerts a force in the direction
of
H on a north pole and opposite to
H on a south pole. In a nonuniform
magnetic field, each pole sees a different field and is subject to
a different force. The difference in the two forces moves the
magnet in the direction of increasing magnetic field and may also
cause a net torque.
Unfortunately, the idea of "poles" does not accurately reflect what
happens inside a magnet (see
ferromagnetism). For instance, a small magnet
placed inside of a larger magnet feels a force in the opposite
direction. The more physically correct description of magnetism
involves atomic sized loops of current distributed throughout the
magnet.
Torque on a magnet due to a Bfield
In the presence of an external
magnetic
field B, a magnet will experience a
torque that tends to align its poles with the
direction of
B. The
torque on a magnet due to an external magnetic field is easy to
observe by placing two magnets near each other while allowing one
to rotate. The torque
τ on a small magnet
is proportional both to the applied
Bfield and to the magnetic moment
m of the magnet:
\boldsymbol{\tau}=\mathbf{m}\times\mathbf{B}, \,
where × represents the vector cross product. The torque
τ will tend to align the magnet's poles
with the
Bfield lines. This phenomenon
explains why the magnetic needle of a
compass points toward the Earth's north pole. By
definition, the direction of the
local magnetic field is the direction that the north pole of a
compass (or of any magnet) tends to point.
Magnetic torque is used to drive simple
electric motors. In one simple motor design,
a magnet is fixed to a freely rotating shaft (forming a
rotor) and subjected to a magnetic field from an array
of
electromagnets—called the
stator. By continuously switching the
electrical current through each of the
electromagnets, thereby flipping the polarity of their magnetic
fields, the stator keeps like poles next to the rotor; The
resultant magnetic torque is transferred to the shaft. The inverse
process, changing mechanical motion to electrical energy, is
accomplished by the inverse of the above mechanism in the
electric generator.
See
Rotating magnetic
fields below for an example using this effect with
electromagnets.
Visualizing the magnetic field using field lines
Mapping out the strength and direction of the magnetic field is
simple in principle. First, measure the strength and direction of
the magnetic field at a large number of locations. Then mark each
location with an arrow (called a
vector) pointing in the direction of the
local magnetic field with a length proportional to the strength of
the magnetic field. An alternative method of visualizing the
magnetic field which greatly simplifies the diagram while
containing the same information is to 'connect' the arrows to form
"magnetic
field lines".
Various physical phenomena have the effect of displaying magnetic
field lines. For example, iron filings placed in a magnetic field
line up in such a way as to visually show the orientation of the
magnetic field (see figure to left). Magnetic fields lines are also
visually displayed in
polar
auroras, in which
plasma
particle dipole interactions create visible streaks of light that
line up with the local direction of Earth's magnetic field.
Field lines provide a simple way to
depict or draw the magnetic field (or any other
vector field). The magnetic field can be
estimated at
any point (whether on a field line or not)
using the direction and density of the field lines nearby. A higher
density of nearby field lines indicates a larger magnetic
field.
Field lines are also a good qualitative tool for visualizing
magnetic forces. In
ferromagnetic
substances like
iron and in
plasmas, magnetic forces can be understood
by imagining that the field lines exert a tension, (like a
rubber band) along their length, and a pressure
perpendicular to their length on neighboring field lines. 'Unlike'
poles of magnets attract because they are linked by many field
lines; 'like' poles repel because their field lines do not meet,
but run parallel, pushing on each other.
The direction of a magnetic field line can be revealed using a
compass. A compass placed near the north pole of a magnet points
away from that pole—like poles repel. The opposite occurs for a
compass placed near a magnet's south pole. The magnetic field
points away from a magnet near its north pole and towards a magnet
near its south pole. Magnetic field lines
outside of a
magnet point from the north pole to the south. Not all
magnetic fields are describable in terms of poles, though. A
straight
currentcarrying
wire, for instance, produces a magnetic field that
points neither towards nor away from the wire, but encircles it
instead.
Bfield lines never end
Field lines are a useful way to represent any
vector field and often reveal sophisticated
properties of fields quite simply. One important property of the
Bfield is that it is a
solenoidal vector field. In field
line terms, this means that magnetic field lines neither start nor
end: They always either form
closed
curves ("loops"), or extend to and from infinity. To date no
exception to this rule has been found. (See
magnetic monopole
below.)
Magnetic field exits a magnet near its north pole and enters near
its south pole but inside the magnet
Bfield lines return from the south pole
back to the north. If a
Bfield line
enters a magnet somewhere it has to leave somewhere else; it is not
allowed to have an end point. For this reason, magnetic poles
always come in N and S pairs. Cutting a magnet in half results in
two separate magnets each with both a north and a south
pole.Magnetic fields are produced by electric currents, which can
be macroscopic currents in wires, or microscopic currents
associated with electrons in atomic orbits. The magnetic field B is
defined in terms of force on moving charge in the Lorentz force
law. The interaction of magnetic field with charge leads to many
practical applications. The SI unit for magnetic field is the
tesla, which can be seen from the
magnetic part of the Lorentz force law F_magnetic = qvB to be
composed of (newton x second)/(coulomb x meter). A smaller magnetic
field unit is the
gauss (1 tesla =
10,000 gauss).
Magnetic monopole (hypothetical)
A magnetic monopole is a hypothetical particle (or class of
particles) that has, as its name suggests, only one magnetic pole
(either a north pole or a south pole). In other words, it would
possess a "magnetic charge" analogous to electric charge.
Modern interest in this concept stems from
particle theories, notably
Grand Unified Theories and
superstring theories, that predict either
the existence or the possibility of magnetic monopoles. These
theories and others have inspired extensive efforts to search for
monopoles. Despite these efforts, no magnetic monopole has been
observed to date.
In recent research materials known as
spin
ices can simulate monopoles, but do not contain actual
monopoles.
Hfield lines begin and end near magnetic poles
Outside a magnet
Hfield lines are
identical to
Bfield lines, but inside
they point in opposite directions. Whether inside or out of a
magnet,
Hfield lines start near the S
pole and end near the N. The
Hfield,
therefore, is analogous to the electric field
E which starts as a positive charge and
ends at a negative charge. It is tempting, therefore, to model
magnets in terms of magnetic charges localized near the poles.
Unfortunately, this model is incorrect; it often fails when
determining the magnetic field inside of magnets for instance. (See
#Force on a
magnet due to a nonuniform B above.)
The magnetic field and electrical currents
Currents of electrical charges both generate a magnetic field and
feel a force due to magnetic Bfields.
Magnetic field due to moving charges and electrical
currents
All moving charges produce magnetic fields. Moving
point charges produces a complicated but well
known magnetic field that depends on the charge, velocity, and
acceleration of the particle. It forms closed loops around a line
pointing in the direction the charge is moving.
Current carrying wires generate magnetic field lines that form
concentric circles around them The
direction of the magnetic field in these loops is determined by the
right hand grip rule. When
moving along the current, to the left the magnetic field points up
while to the right it points down. (See figure to the right.) The
strength of the magnetic field decreases with distance from the
wire.
Bending a current carrying wire into a loop concentrates the
magnetic field inside the loop and weakens it outside. Stacking
many such loops to form a
solenoid (or long
coil) enhances this effect. Such devices, called
electromagnets, are important because they
generate strong well controlled magnetic fields. An infinitely long
electromagnet has a uniform magnetic field inside and no magnetic
field outside. A finite length electromagnet produces essentially
the same magnetic field as a uniform permanent magnet of the same
shape and size with a strength (and polarity) that is controlled by
the input current.
The magnetic field generated by a steady
current I (a constant flow of
charges in which charge is neither
accumulating nor depleting at any point) is described by the
BiotSavart law:
 \mathbf{B} = \frac{\mu_0I}{4\pi}\oint\frac{d\boldsymbol{\ell}
\times \mathbf{\hat r}}{r^2},
where the integral sums over the entire loop of a wire with
d'l
a particular infinitesimal piece of that
loop, μ_{0} is the magnetic constant, r is the
distance between the location of d'l and the
location at which the magnetic field is being calculated, and
\scriptstyle\mathbf{\hat r} is a unit vector in the direction of
r.
A slightly more general way of relating the current
I to
the
Bfield is through
Ampère's law:
 \oint \mathbf{B} \cdot d\boldsymbol{\ell} = \mu_0
I_{\mathrm{enc}},
where the integral is over any arbitrary loop and
I_{enc} is the current enclosed by that loop.
Ampère's law is always valid for steady currents and can be used to
calculate the
Bfield for certain highly
symmetric situations such as an infinite wire or an infinite
solenoid.
In a modified form that accounts for time varying electric fields,
Ampère's law is one of four
Maxwell's equations that describe
electricity and magnetism.
Force due to a Bfield on a moving charge
Force on a charged particle
Beam of electrons moving in a
circle.
Lighting is caused by excitation of atoms of gas in a
bulb.
A
charged particle moving in a
Bfield experiences a
sideways
force that is proportional to the strength of the magnetic field,
the component of the velocity that is perpendicular to the magnetic
field and the charge of the particle. This force is known as the
Lorentz force, and is given by
 \mathbf{F} = q (\mathbf{v} \times \mathbf{B}),
where
F is the
force,
q is the
electric charge of
the particle,
v is the instantaneous
velocity of the particle, and
B is
the magnetic field (in
teslas).
The Lorentz force is always perpendicular to both the velocity of
the particle and the magnetic field that created it. Neither a
stationary particle nor one moving in the direction of the magnetic
field lines experiences a force. For that reason, charged particles
move in a circle (or more generally, in a
helix) around magnetic field lines; this is called
cyclotron motion. Because the
magnetic force is always perpendicular to the motion, the magnetic
fields can do no
work on an isolated
charge. It can and does, however, change the particle's direction,
even to the extent that a force applied in one direction can cause
the particle to drift in a perpendicular direction. It is often
claimed that the magnetic force can do work to a nonelementary
magnetic dipole, or to charged
particles whose motion is constrained by other forces, but this is
not the case because the work in those cases is performed by the
electric forces of the charges deflected by the magnetic
field.
Force on currentcarrying wire
The force on a current carrying wire is similar to that of a moving
charge as expected since a charge carrying wire is a collection of
moving charges. A current carrying wire feels a sideways force in
the presence of a magnetic field. The Lorentz force on a
macroscopic current is often referred to as the
Laplace force.
Direction of force
The direction of force on a positive charge or a current is
determined by the
righthand rule.
See the figure on the right. Using the right hand and pointing the
thumb in the direction of the moving positive charge or positive
current and the fingers in the direction of the magnetic field the
resulting force on the charge points outwards from the palm. The
force on a negative charged particle is in the opposite direction.
If both the speed and the charge are reversed then the direction of
the force remains the same. For that reason a magnetic field
measurement (by itself) cannot distinguish whether there is a
positive charge moving to the right or a negative charge moving to
the left. (Both of these cases produce the same current.) On the
other hand, a magnetic field combined with an electric field
can distinguish between these, see
Hall effect below.
An alternative, similar trick to the right hand rule is
Fleming's left hand rule.
H and B inside and outside of magnetic
materials
The formulas derived for the magnetic field above are correct when
dealing with the entire current. A magnetic material placed inside
a magnetic field, though, generates its own
bound current which can be a challenge to
calculate. (This bound current is due to the sum of atomic sized
current loops and the
spin of the
subatomic particles such as electrons that make up the material.)
The
Hfield as defined above helps factor
out this bound current; but in order to see how it helps to
introduce the concept of
magnetization first.
Magnetization
The magnetization field
M represents how
strongly a region of material is magnetized and is defined as the
volume density of the net
magnetic dipole moment in that
region. The unit of magnetization
M in
SI is
Ampereturn/
meter which is
identical to that of the
Hfield since
the unit of magnetic moment is Ampereturn m
^{2}. The
direction of the magnetization
M is that
of the average magnetic dipole moment in the region and is the same
as the local
Bfield it produces.
Magnetization can be thought of as the magnetic equivalent of the
polarization density
P used for electrical charges. In other
words,
M begins and ends at
bound magnetic charges. (Unlike
B, magnetization must begin and end near
the poles; there is no magnetization outside of the material.) In
this model, the source of the
M field are
bound magnetic charges such that
−
∇ · μ
_{0}M =
ρ_{b},
where
ρ_{b} is the bound magnetic charge density.
For uniform
M this bound charge is zero
everywhere except near the poles.
An equivalent, and more physically correct, way to represent
magnetization is to add all of the currents of the dipole moments
that produce the magnetization. See
#Magnetic dipoles below and
magnetic poles vs. atomic currents for more information. The
resultant current is called bound current and is the source of the
magnetic field due to the magnet. Mathematically, the
curl of
M equals the bound
current.
Magnetism
Most materials produce their own magnetization
M and therefore their own
Bfield in response to an applied
Bfield. Typically, the response is very
weak and exists only when the magnetic field is applied. The term
magnetism is used to describe how
these materials respond on the microscopic level and is used to
categorize the magnetic phase of a material. Materials are divided
into groups based upon their magnetic behavior:
Hfield and magnetic materials
In the case of paramagnetism, and diamagnetism the magnetization
M is often proportional to the applied
magnetic field such that:
 \mathbf{B} = \mu \mathbf{H},
where \mu is a material dependent parameter called the permeability
(see
constitutive equations).
In some cases the permeability may be a second rank
tensor so that
H may not
point in the same direction as
B. These
relations between
B and
H are examples of
constitutive equations. However,
superconductors and ferromagnets have a more complex
B to
H
relation, see
hysteresis.
In all cases, the definition of
H given
above:
 \mathbf{H}\ \equiv \
\frac{\mathbf{B}}{\mu_0}\mathbf{M},(definition of
H in SI
units)
(along with its Gaussian counterpart) is still valid.
The advantage of the
Hfield is that its
bound sources are treated so differently that they can often be
isolated from the free sources. For example, a
line integral of the
Hfield in a closed loop yields the total
free current in the loop (not including the bound current).
Similarly, a
surface integral of
H over any
closed surface picks out the 'magnetic
charges' within that closed surface. Examining the definition of
H helps flesh out this statement.
Taking the divergence of this definition results in

\mathbf{\nabla}\cdot\mathbf{B}=\mu_0\mathbf{\nabla}\cdot\mathbf{H}+\mathbf{\nabla}\cdot\mu_0\mathbf{M}=
0,
where the equation has been rearranged so that its parallel to the
displacement field is more
obvious. Noting that
−
∇ · μ
_{0}M = ρ
_{b}
the bound magnetic charge density from the definition of
M above and that
∇ · B = 0 represents
the absence of free magnetic charges this definition
of H
requires that
μ_{0}∇ · H = ρ
_{tot}.
In other words, as described above, the definition of
H requires that its field lines begin at
positive magnetic charge (near south pole) and end at a negative
magnetic charge (north pole).
Taking the curl of the definition of
H
yields that:

\mathbf{\nabla}\times\mathbf{H}=\frac{1}{\mu_0}\mathbf{\nabla}\times\mathbf{B}\mathbf{\nabla}\times\mathbf{M}=\mathbf{J}_{tot}\mathbf{J}_{b}
= \mathbf{J}_f
where
J_{f} represents the free
current.
Energy stored in magnetic fields
In asking how much energy is needed to create a specific magnetic
field using a particular current it is important to distinguish
between free and bound currents. It is the free current that we
directly 'push' on to create the magnetic field. The bound currents
create a magnetic field that the free current has to work against
without doing any of the work.
It is not surprising, therefore, that the
Hfield is important in magnetic energy
calculations since it treats the two sources differently. In
general the incremental amount of work per unit volume
δW
needed to cause a small change of magnetic field
δ'B
is:
 \delta W = \mathbf{H}\cdot\delta\mathbf{B}.
If there are no magnetic materials around then we can replace
H with
B ⁄
μ_{0},
 u = \frac{\mathbf{B}\cdot\mathbf{B}}{2\mu_o}.
For linear materials (such that
B =
μ
H ), the
energy density can be expressed as:
 u = \frac{\mathbf{B}\cdot\mathbf{B}}{2\mu} =
\frac{\mu\mathbf{H}\cdot\mathbf{H}}{2}. (Valid
only for linear materials)
Nonlinear materials cannot use the above equation but must return
to the first equation which is always valid. In particular, the
energy density stored in the fields of hysteretic materials such as
ferromagnets and superconductors depends on how the magnetic field
was created.
Electromagnetism: the relationship between magnetic and
electric fields
The magnetic field due to a changing electric field
A changing electric field generates a magnetic field proportional
to the time rate of the change of the electric field. This fact is
known as
Maxwell's correction to Ampere's Law. Therefore the full
Ampere's Law is:
 \nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0
\varepsilon_0 \frac{\partial \mathbf{E}} {\partial t},\
where
J is the current density, and
partial derivatives indicate
spatial location is fixed when the
time
derivative is taken. The last term is Maxwell's correction.
This equation is valid even when magnetic materials are involved,
but in practice it is often easier to use an alternate
equation.
Electric force due to a changing Bfield
Above is a discussion of how a changing
Efield creates a
Bfield. The inverse process also occurs:
a changing magnetic field, such as a magnet moving through a
stationary coil, generates an
electric
field (and therefore tends to drive a current in the coil).
(These two effects bootstrap together to form
electromagnetic waves, such as light.)
This is known as
Faraday's
Law and forms the basis of many electric
generators and
electrical motors.
Faraday's law is commonly represented as:
 \mathcal{E} =  \frac{d\Phi_m}{dt},
where \scriptstyle\mathcal{E} is the
electromotive force or
EMF (the
voltage generated around a closed loop) and
Φ_{m} is the
magnetic flux—the product of the area times
the magnetic field normal to that area. (This definition of
magnetic flux is why engineers often refer to
B as "magnetic flux density".) This law
includes both flux changes because of the magnetic field generated
by a time varying
Efield
(
transformer EMF) and flux changes because of movement
through a magnetic field (
motional EMF).
A form of Faraday's law of induction that does not include
motional EMF is the MaxwellFaraday equation:
 \nabla \times \mathbf{E} = \frac{\partial \mathbf{B}}
{\partial t},\
one of
Maxwell's equations. This
equation is valid even in the presence of magnetic material.
A complete expression for Faraday's law of induction in terms of
the electric
E and magnetic fields can be
written as:
\mathcal{E} =  \frac{d\Phi_m}{dt} = \oint_{\partial \Sigma
(t)}\left( \mathbf{E}( \mathbf{r},\ t) +\mathbf{ v \times
B}(\mathbf{r},\ t)\right) \cdot d\boldsymbol{\ell}\ \ =\frac {d}
{dt} \iint_{\Sigma (t)} d \boldsymbol {A} \cdot \mathbf
{B}(\mathbf{r},\ t) \ ,
where
∂Σ(
t) is the moving closed path
bounding the moving surface
Σ(
t), and
d
A is an element of surface area of
Σ(
t). The first integral calculates the
work done moving a charge a distance d
ℓ
based upon the
Lorentz force law. In
the case where the bounding surface is stationary, the
KelvinStokes theorem
can be used to show this equation is equivalent to the
MaxwellFaraday equation.
Maxwell's equations
Like all vector fields the
Bfield has
two important mathematical properties that relates it to its
sources. These two properties, along with the two
corresponding properties of the electric field, make up
Maxwell's Equations. Maxwell's Equations
together with the Lorentz force law form a complete description of
classical electrodynamics
including both electricity and magnetism.
The first property is the
divergence of a
vector field
A,
∇ · A which represents
how A
'flows' outward from a given
point. As discussed above
a Bfield line never starts nor ends at a
point but instead forms a complete
loop. This is
mathematically equivalent to saying that the divergence of B is
zero. (Such
vector fields are called solenoidal vector fields.) This
property is called Gauss's law
for magnetism and is equivalent to the statement that there are
no magnetic charges or magnetic
monopoles.
The electric field on the other hand begins
and ends at electrical charges so that its divergence is nonzero
and proportional to the charge
density (See Gauss's
law).
The second mathematical property is called the
curl,
∇ × such that
∇ × A represents how A
curls or 'circulates' around a given
point. The result of the curl
is called a 'circulation source' The curl of B
and of E are given above and
are called the AmpèreMaxwell equation and
Faraday's law
respectively.
The complete set of Maxwell's equations then are:
 \nabla \cdot \mathbf{B} = 0,
 \nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0},
 \nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0
\varepsilon_0 \frac{\partial \mathbf{E}} {\partial t},
 \nabla \times \mathbf{E} =  \frac{\partial \mathbf{B}}
{\partial t},
where
J = complete microscopic
current density and
ρ is the charge
density.
As discussed above, materials respond to an applied electric
E field and an applied magnetic
B field by producing their own internal
'bound' charge and current distributions that contribute to
E and
B but are
difficult to calculate. To circumvent this problem the auxiliary
H and
D fields
are defined so that Maxwell's equations can be refactored in terms
of the
free current density
J_{f} and
free charge
density ρ_{f}:
 \nabla \cdot \mathbf{B} = 0,
 \nabla \cdot \mathbf{D} = \rho_f,
 \nabla \times \mathbf{H} = \mathbf{J}_f + \frac{\partial
\mathbf{D}} {\partial t},
 \nabla \times \mathbf{E} =  \frac{\partial \mathbf{B}}
{\partial t}.
These equations are not any more general then the original
equations (if the 'bound' charges and currents in the material are
known'). They also need to be supplemented by the relationship
between
B and
H
as well as that between
E and
D. On the other hand, for simple
relationships between these quantities this form of Maxwell's
equations can circumvent the need to calculate the bound charges
and currents.
Electric and magnetic fields: different aspects of the same
phenomenon
According to
special relativity,
the partition of the
electromagnetic force into separate
electric and magnetic components is not fundamental, but varies
with the
observational
frame of reference; an electric force perceived by one observer
is perceived by another (in a different frame of reference) as a
mixture of electric and magnetic forces. (Too, a magnetic force in
one reference frame is perceived as a mixture of electric and
magnetic forces in another.)
More specifically, special relativity combines the electric and
magnetic fields into a rank2
tensor, called
the
electromagnetic tensor.
Changing reference frames
mixes these components. This is
analogous to the way that special relativity
mixes space
and time into
spacetime, and mass,
momentum and energy into
fourmomentum.
Magnetic vector potential
In advanced topics such as
quantum
mechanics and
relativity it is often
easier to work with a potential formulation of electrodynamics
rather than in terms of the electric and magnetic fields. In this
representation, the
vector
potential,
A, and the
scalar potential,
φ, are defined
such that:
 \mathbf{B} = \nabla \times \mathbf{A},
 \mathbf{E} =  \nabla \phi  \frac { \partial \mathbf{A} } {
\partial t }.
The vector potential
A may be interpreted
as a
generalized potential momentum per unit charge just
as
φ is interpreted as a
generalized potential energy
per unit charge.
Maxwell's equations when expressed in terms of the potentials can
be cast into a form that agrees with special relativity with little
effort. In relativity
A together with
φ forms the
fourpotential
analogous to the fourmomentum which combines the momentum and
energy of a particle. Using the four potential instead of the
electromagnetic tensor has the advantage of being much simpler;
further it can be easily modified to work with quantum
mechanics.
Quantum electrodynamics
In modern physics, the electromagnetic field is understood to be
not a
classical field, but rather a
quantum field; it is represented not as a
vector of three
numbers at each point,
but as a vector of three
quantum
operators at each point. These theories explain that the
electromagnetic field is derived from the
photon
field; indeed, all electromagnetic interactions are mediated by
this field. This model when extended to include all of the
elementary particles and the four
fundamental interactions
(electromagnetism,
gravity,
weak force, and
strong
force) is known as the
Standard
Model.
Quantum electrodynamics,
QED, describes the electromagnetic interaction between charged
particles (and their
antiparticles) as
due to the exchange of
virtual
photons. The magnitude of these interactions is computed using
perturbation
theory; these rather complex formulas have a remarkable
pictorial representation as
Feynman
diagrams.
QED does not predict what will happen in an experiment; it predicts
the
probability of what will happen in an experiment. This
is how (statistically) it is experimentally verified. Predictions
of QED agree with experiments to an extremely high degree of
accuracy: currently about 10
^{−12} (and limited by
experimental errors); for details see
precision tests of QED. This makes
QED one of the most accurate physical theories constructed thus
far.
All equations in this article are in the
classical approximation, which is
less accurate than the
quantum
description as mentioned above. However, under most everyday
circumstances, the difference between the two theories is
negligible.
Measuring the Bfield
Devices used to measure the local magnetic field are called
magnetometers. Important classes of
magnetometers include using a rotating
coil,
Hall effect magnetometers,
NMR magnetometer,
SQUID magnetometer, and a
fluxgate magnetometer. The magnetic fields
of distant
astronomical objects
can be determined by noting their effects on local charged
particles. For instance, electrons spiraling around a field line
produce
synchotron radiation
which is detectable in
radio
waves.
The
smallest magnetic field measured is on the order of attoteslas
(10^{−18} tesla); the largest magnetic field produced in a
laboratory is 2,800 T (VNIIEF in
Sarov, Russia, 1998) The
magnetic field of some astronomical objects such as magnetars are much higher; magnetars range from
0.1 to 100 GT (10^{8} to
10^{11} T). See
orders of magnitude
.
History
Perhaps the earliest description of a magnetic field was performed
by
Petrus Peregrinus and published
in his “Epistola Petri Peregrini de Maricourt ad Sygerum
deFoucaucourt Militem de Magnete” and is dated 1269 A.D. Petrus
Peregrinus mapped out the magnetic field on the surface of a
spherical magnet. Noting that the resulting field lines crossed at
two points he named those points 'poles' in analogy to Earth's
poles. Almost three centuries later, near the end of the sixteenth
century,
William Gilbert of
Colchester replicated Petrus Peregrinus' work and was the first to
state explicitly that Earth itself was a magnet. William Gilbert's
great work
De Magnete was
published in
1600 A.D. and helped to establish
the study of magnetism as a science.
The modern understanding that the
Bfield
is the more fundamental field with the
Hfield being an
auxiliary field
was not easy to arrive at. Indeed, largely because of mathematical
similarities to the
electric field,
the
Hfield was developed first and was
thought at first to be the more fundamental of the two.
The modern distinction between the
B and
H fields was not needed until
SiméonDenis Poisson (
1781–
1840) developed one of the
first mathematical theories of magnetism. Poisson's model,
developed in 1824, assumed that magnetism was due to magnetic
charges. In analogy to electric charges, magnetic charges produce
an
Hfield. In modern notation, Poisson's
model is exactly analogous to electrostatics with the
Hfield replacing the
electric field
Efield and the
Bfield replacing the auxiliary
Dfield.
Poisson's model was, unfortunately, incorrect. Magnetism is not due
to magnetic charges. Nor is magnetism created by the
Hfield
polarizing magnetic
charge in a material. The model, however, was remarkably successful
for being fundamentally wrong. It predicts the correct relationship
between the
Hfield and the
Bfield, even though it wrongly places
H as the fundamental field with
B as the
auxiliary field. It
predicts the correct forces between magnets.
It even predicts the correct energy stored in the magnetic fields.
By the definition of magnetization, in this model, and in analogy
to the physics of springs, the work done per unit volume, in
stretching and twisting the
bonds between
magnetic
charge to increment the magnetization by
μ_{0}δ'M
is
W = 'H ·
μ_{0}δM.
In this model,
B =
μ_{0} (
H
+
M ) is an effective magnetization which
includes the
Hfield term to account for
the energy of setting up the magnetic field in a vacuum. Therefore
the total energy density increment needed to increment the magnetic
field is
W = 'H ·
δ'B.
This is the correct result, but it is derived from an incorrect
model.
In retrospect the success of this model is due largely to the
remarkable coincidence that from the 'outside' the field of an
electric dipole has the exact same
form as that of a
magnetic dipole.
It is therefore only for the physics of magnetism 'inside' of
magnetic material where the simpler model of
magnetic
charges fails. It is also important to note that this model is
still useful in many situations dealing with magnetic material. One
example of its utility is the concept of
magnetic circuits.
The formation of the correct theory of magnetism begins with a
series of revolutionary discoveries in 1820, four years before
Poisson's model was developed. (The first clue that something was
amiss, though, was that unlike electrical charges magnetic poles
cannot be separated from each other or form magnetic currents.) The
revolution began when
Hans
Christian Oersted discovered that an electrical current
generates a magnetic field that encircles the wire. In a quick
succession that discovery was followed by
Andre Marie Ampere showing that parallel
wires having currents in the same direction attract, and by
JeanBaptiste Biot and
Felix Savart developing the correct equation,
the
BiotSavart Law, for the
magnetic field of a current carrying wire. In 1825, Ampere extended
this revolution by publishing his
Ampere's
Law which provided a more mathematically subtle and correct
description of the magnetic field generated by a current than the
BiotSavart Law.
Subsequent development in the nineteenth century interlinked
magnetic and electric phenomena even tighter, until the concept of
magnetic charge was not needed. Magnetism became an electric
phenomenon with even the magnetism of permanent magnets being due
to small loops of current in their interior. This development was
aided greatly by
Michael Faraday,
who in 1831 showed that a changing magnetic field generates an
encircling electric field.
In 1861, James ClerkMaxwell wrote a paper entitled 'On Physical
Lines of Force'
[6229] in which he attempted to explain
Faraday's magnetic lines of force in terms of a sea of tiny
molecular vortices. These molecular vortices occupied all space and
they were aligned in a solenoidal fashion such that their rotation
axes traced out the magnetic lines of force. When two like magnetic
poles repel each other, the magnetic lines of force spread outwards
from each other in the space between the two poles. Maxwell
considered that magnetic repulsion was the consequence of a lateral
pressure between adjacent lines of force, due to
centrifugal force in the equatorial plane
of the molecular vortices. When deriving the equation for magnetic
force in part I of his 1861 paper, Maxwell used a quantity which
was closely related to the circumferential speed of the vortices.
This quantity was therefore a measure of the vorticity in the
magnetic lines of force, and Maxwell referred to it as the
intensity of the magnetic force. In the 1861 paper, the magnetic
intensity which we denote as v, was always multiplied by the term μ
as a weighting for the cross sectional density of the lines of
force. The quantity v corresponds reasonably closely to the modern
magnetic field vector
H, and the product μv
corresponds very closely to the modern magnetic flux density
B, where μ is referred to as the magnetic
permeability.
Although the
classical theory of
electrodynamics was essentially complete with Maxwell's equations,
the twentieth century saw a number of improvements and extensions
to the theory.
Albert Einstein, in
his great paper of 1905 that established relativity, showed that
both the electric and magnetic fields were part of the same
phenomena viewed from different reference frames. Finally, the
emergent field of
quantum
mechanics was merged with electrodynamics to form
quantum electrodynamics or
QED.
In the late nineteenth century the
moving magnet and conductor
problem developed as an important
thought experiment that eventually helped
Albert Einstein to develop
special relativity. This thought
experiment revolves around the interpretation of
Faraday's law, as explained next:
Imagine a conducting loop moving relative to a magnet
as seen by two different observers: one on the magnet the other on
the loop.
Both observers see the identical EMF generated in the
coil using the flux form of Faraday's law, but explain the result
using two different reasons.
The observer on the magnet sees the magnet as
stationary with an unchanging magnetic field, while the conducting
loop moves.
All of the charges within the loop move with the loop,
and due to the Bfield experience a
sideways Lorentz force, which generates the EMF.
On the other hand, an observer on the loop sees a
changing magnetic field due to a moving magnet (relative
to the loop's reference frame) and no Lorentz force (charges in the
loop are not moving).
This changing magnetic field means
∂B / ∂t ≠ 0, which creates an
electric field that generates the current.
Prior to special relativity, it was customary to draw a sharp
distinction between these two cases; a stationary magnet and a
moving loop only produces
motional EMF due to the Lorentz
force from the
Bfield, while a moving
magnet through a stationary loop produces only
transformer
EMF due to the electric field
E
generated by a changing
B. See
Faraday's law as two different phenomena. Einstein, on the
other hand, proposed the equivalence of these two scenarios in the
first postulate of relativity that the physics depends on only
relative motion. Motional EMF and transformer EMF,
therefore are the same phenomenon as seen in different reference
frames. Likewise, the same is true of
E
and
B, which are not separate, but are
aspects of the same
electromagnetic tensor.
Important uses and examples of magnetic field
Magnetic circuits
An important use of
H is in magnetic
circuits where inside a linear material
B
= μ
H. Here, μ is the magnetic
permeability of the material. This result is similar in form to
Ohm's Law J = σ
E, where
J is
the current density, σ is the conductance and
E is the electric field. Extending this
analogy we derive the counterpart to the macroscopic Ohm's law (
I = V ⁄ R ) as:
 \Phi = \frac F R_m,
where \Phi = \int \mathbf{B}\cdot d\mathbf{A} is the magnetic flux
in the circuit, F = \int \mathbf{H}\cdot d\mathbf{l} is the
magnetomotive force applied to
the circuit, and R_m is the
reluctance of
the circuit. Here the reluctance R_m is a quantity similar in
nature to
resistance for the
flux.
Using this analogy it is straightforward to calculate the magnetic
flux of complicated magnetic field geometries, by using all the
available techniques of
circuit
theory.
Hall effect
The
charge carriers of a current
carrying conductor placed in a
transverse magnetic field experience a
sideways
Lorentz force; this results
in a charge separation in a direction perpendicular to the current
and to the magnetic field. The resultant voltage in that direction
is proportional to the applied magnetic field. This is known as the
Hall effect.
The
Hall effect is often used to measure the magnitude of
a magnetic field. It is used as well to find the sign of the
dominant charge carriers in materials such as semiconductors
(negative electrons or positive holes).
Magnetic field shape descriptions
 An azimuthal magnetic field is one that runs
eastwest.
 A meridional magnetic field is one that runs
northsouth. In the solar dynamo model
of the Sun, differential
rotation of the solar plasma causes the meridional magnetic
field to stretch into an azimuthal magnetic field, a process called
the omegaeffect. The reverse process is called the
alphaeffect.
 A quadrupole
magnetic field is one seen, for example, between the poles of
four bar magnets. The field strength grows linearly with the radial
distance from its longitudinal axis.
 A solenoidal magnetic field is similar to a
dipole magnetic field, except that a solid bar magnet is replaced
by a hollow electromagnetic coil magnet.
 A toroidal magnetic field occurs in a
doughnutshaped coil, the electric current spiraling around the
tubelike surface, and is found, for example, in a tokamak.
 A poloidal magnetic field is generated by a
current flowing in a ring, and is found, for example, in a tokamak.
 A radial magnetic field is one in which the
field lines are directed from the center outwards, similar to the
spokes in a bicycle wheel. An example can be found in a loudspeaker transducers (driver).
 A helical magnetic field is corkscrewshaped,
and sometimes seen in space plasmas such as the Orion Molecular Cloud.
Magnetic dipoles
The magnetic field of a magnetic dipole is depicted on the right.
From outside, the ideal magnetic dipole is identical to that of an
ideal electric dipole of the same strength. Unlike the electric
dipole, a magnetic dipole is properly modeled as a current loop
having a current
I and an area
a. Such a current
loop has a magnetic moment of:
 m=Ia, \,
where the direction of
m is perpendicular
to the area of the loop and depends on the direction of the current
using the
right hand rule. An ideal
magnetic dipole is modeled as a real magnetic dipole whose area
a has been reduced to zero and its current
I
increased to infinity such that the product
m =
Ia is finite. In this model it is
easy to see the connection between angular momentum and magnetic
moment which is the basis of the
Einsteinde Haas effect "rotation by
magnetization" and its inverse, the
Barnett effect or "magnetization by
rotation". Rotating the loop faster (in the same direction)
increases the current and therefore the magnetic moment, for
example.
It is sometimes useful to model the magnetic dipole similar to the
electric dipole with two equal but opposite magnetic charges (one
south the other north) separated by distance
d. This model
produces an
Hfield not a
Bfield. Such a model is deficient,
though, both in that there are no magnetic charges and in that it
obscures the link between electricity and magnetism. Further, as
discussed above it fails to explain the inherent connection between
angular momentum and
magnetism.
Earth's magnetic field
Because of
Earth's magnetic field, a
compass placed anywhere on Earth turns so
that the "north pole" of the magnet inside
the compass points roughly north, toward
Earth's north magnetic pole in
northern Canada. This
is the traditional definition of the "north pole" of a magnet,
although other equivalent definitions are also possible. One
confusion that arises from this definition is that if Earth itself
is considered as a magnet, the
south pole of that magnet
would be the one nearer the north magnetic pole, and viceversa.
(Opposite poles attract, so the north pole of the compass magnet is
attracted to the south pole of Earth's interior magnet.) The north
magnetic pole is so named not because of the polarity of the field
there but because of its geographical location.
The figure to the right is a sketch of Earth's magnetic field
represented by field lines. For most locations, the magnetic field
has a significant up/down component in addition to the North/South
component. (There is also an East/West component; Earth's magnetic
poles do not coincide exactly with Earth's geological pole.) The
magnetic field is as if there were a
magnet
deep in Earth's interior.
Earth's magnetic field is
probably due to a
dynamo that produces
electric currents in the outer
liquid part of its core. Earth's magnetic field is not constant:
Its strength and the location of its poles vary. The poles even
periodically reverse direction, in a process called
geomagnetic reversal.
Rotating magnetic fields
The rotating magnetic field is a key principle in the operation of
alternatingcurrent motors.
A permanent magnet in such a field rotates so as to maintain its
alignment with the external field. This effect was conceptualized
by
Nikola Tesla, and later utilized in
his, and others', early AC (
alternatingcurrent) electric motors. A
rotating magnetic field can be constructed using two orthogonal
coils with 90 degrees phase difference in their AC currents.
However, in practice such a system would be supplied through a
threewire arrangement with unequal currents. This inequality would
cause serious problems in standardization of the conductor size and
so, in order to overcome it,
threephase
systems are used where the three currents are equal in magnitude
and have 120 degrees phase difference. Three similar coils having
mutual geometrical angles of 120 degrees create the rotating
magnetic field in this case. The ability of the threephase system
to create a rotating field, utilized in electric motors, is one of
the main reasons why threephase systems dominate the world's
electrical power supply
systems.
Because magnets degrade with time,
synchronous motors and
induction motors use shortcircuited
rotors (instead of a magnet)
following the rotating magnetic field of a multicoiled
stator. The shortcircuited turns of the rotor
develop
eddy currents in the rotating
field of the stator, and these currents in turn move the rotor by
the
Lorentz force.
In 1882, Nikola Tesla identified the concept of the rotating
magnetic field. In 1885,
Galileo
Ferraris independently researched the concept. In 1888, Tesla
gained for his work.
Also in 1888, Ferraris published his research
in a paper to the Royal Academy of Sciences in Turin.
See also
General
Mathematics
 Ampère's law — law describing
how currents act as circulation sources for magnetic fields.
 BiotSavart law — the magnetic
field set up by a steadily flowing line current.
 Magnetic helicity — extent to
which a magnetic field "wraps around itself".
 Maxwell's equations — four
equations describing the behavior of electric and magnetic fields
and their interaction with matter.
Applications
Notes and references
 Technically, magnetic field is a pseudo vector; pseudovectors, which also
include torque and rotational velocity, are similar to vectors
except that they remain unchanged when the coordinates are
inverted.
 The standard graduate textbook by J. D. Jackson "Classical Electrodynamics"
specifically follows the historical tradition, specifically, "In
the presence of magnetic materials the dipole tends to align itself
in a certain direction. That direction is by definition the
direction of the magnetic flux density, denoted by
B, provided the dipole is sufficiently
small and weak that it does not perturb the existing field".
Similarly, in Section 5 of Jackson, H is
referred to as the magnetic field. Hence, Edward
Purcell, in Electricity and Magnetism, McGrawHill, 1963,
writes, Even some modern writers who treat 'B as
the primary field feel obliged to call it the magnetic induction
because the name magnetic field was historically preempted
by H. This seems clumsy and pedantic. If you go
into the laboratory and ask a physicist what causes the pion
trajectories in his bubble chamber to curve, he'll probably answer
"magnetic field", not "magnetic induction." You will seldom hear a
geophysicist refer to the Earth's magnetic induction, or an
astrophysicist talk about the magnetic induction of the galaxy. We
propose to keep on calling B the magnetic field.
As for H, although other names have been invented
for it, we shall call it "the field H" or even
"the magnetic field H." In a similar vein,
says: “So we may think of both B and
H as magnetic fields, but drop the word
'magnetic' from H so as to maintain the
distinction … As Purcell points out, 'it is only the names that
give trouble, not the symbols'.”
 Nave, C. R., "Magnetic Field Strength H",
HyperPhysics, Dept. of Physics and Astronomy, Georgia
State Univ., 2005.
 Magnetic Field Strength Converter,
UnitConversion.org.
 See Eq. 11.42 in
 The use of iron filings to display a field presents something
of an exception to this picture; the filing alter the magnetic
field so that it is much larger along the "lines" of iron, due to
the large permeability of iron relative to
air.
 To see that this must be true imagine placing a compass inside
a magnet. There, the north pole of the compass will point toward
the north pole of the magnet since magnets stacked on each other
point in the same direction.
 Two experiments produced candidate events that were initially
interpreted as monopoles, but these are now regarded to be
inconclusive. For details and references, see magnetic
monopole.
 In special relativity this means that electric and magnetic
fields are two parts of the same phenomenon. For, a moving charge
produces both an electric and a magnetic field. But, in a reference
frame where the particle is not moving there is only an electric
field. Yet, the physics is the same in all reference systems. In
this reference system the electric field changes as well to
produces the same force as the original reference frame. It is
probably a mistake, though, to say that the electric field causes
the magnetic field when relativity is accounted for, since relativity
favors no particular reference frame. (One could just as easily say
that the magnetic field caused an electric field). More importantly
it is not always possible to move into a coordinate system
in which all of the charges are stationary. See classical
electromagnetism and special relativity.
 In practice the BiotSavart law and other laws of
magnetostatics can often be used even when the charge is changing
in time as long as it is not changing too quickly. This situation
is known as being quasistatic.
 The BiotSavart law contains the additional restriction
(boundary condition) that the Bfield
must go to zero fast enough at infinity. It also depends on the
divergence of B being zero, which is
always valid. (There are no magnetic charges.)
 For a good qualitative introduction see:
 [1] Gravity Probe B
 Kouveliotou, C.; Duncan, R. C.; Thompson, C. (February 2003). "
Magnetars". Scientific American; Page
36.
 The Solar Dynamo, retrieved Sep 15, 2007.
 I. S. Falconer and M. I. Large (edited by I. M. Sefton), "
Magnetism: Fields and Forces" Lecture E6, The
University of Sydney, retrieved 3 Oct 2008
 Robert Sanders, " Astronomers find magnetic Slinky in Orion", 12
January 2006 at UC Berkeley. Retrieved 3 Oct 2008
 (See magnetic moment for further
information.)
Further reading
Web
Books
External links
Information
Field density
 Jiles, David (1994). Introduction to Electronic Properties of
Materials (1st ed.). Springer. ISBN 0412495805.
Rotating magnetic fields
Diagrams
Journal Articles