Magnetostatics is the study of magnetic fields
in systems where the currents are steady (not
changing with time). It is the magnetic analogue
of electrostatics, where the charges are stationary.
The magnetization need not be static; the
equations of magnetostatics can be used to
predict fast magnetic switching events that
occur on time scales of nanoseconds or less.
Magnetostatics is even a good approximation
when the currents are not static — as long
as the currents do not alternate rapidly.
Magnetostatics is widely used in applications
of micromagnetics such as models of magnetic
recording devices. Magnetostatic focussing
can be achieved either by a permanent magnet
or by passing current through a coil of wire
whose axis coincides with the beam axis.
== Applications ==
=== Magnetostatics as a special case of Maxwell's
equations ===
Starting from Maxwell's equations and assuming
that charges are either fixed or move as a
steady current
J
{\displaystyle \scriptstyle \mathbf {J} }
, the equations separate into two equations
for the electric field (see electrostatics)
and two for the magnetic field. The fields
are independent of time and each other. The
magnetostatic equations, in both differential
and integral forms, are shown in the table
below.
Where ∇ with the dot denotes divergence,
and B is the magnetic flux density, the first
integral is over a surface
S
{\displaystyle \scriptstyle S}
with oriented surface element
d
S
{\displaystyle \scriptstyle d\mathbf {S} }
. Where ∇ with the cross denotes curl, J
is the current density and H is the magnetic
field intensity, the second integral is a
line integral around a closed loop
C
{\displaystyle \scriptstyle C}
with line element
l
{\displaystyle \scriptstyle \mathbf {l} }
. The 
current going through the loop is
I
enc
{\displaystyle \scriptstyle I_{\text{enc}}}
.
The quality of this approximation may be guessed
by comparing the above equations with the
full version of Maxwell's equations and considering
the importance of the terms that have been
removed. Of particular significance is the
comparison of the
J
{\displaystyle \scriptstyle \mathbf {J} }
term against the
∂
D
/
∂
t
{\displaystyle \scriptstyle \partial \mathbf
{D} /\partial t}
term. If the
J
{\displaystyle \scriptstyle \mathbf {J} }
term is substantially larger, then the smaller
term may be ignored without significant loss
of accuracy.
=== Re-introducing Faraday's law ===
A common technique is to solve a series of
magnetostatic problems at incremental time
steps and then use these solutions to approximate
the term
∂
B
/
∂
t
{\displaystyle \scriptstyle \partial \mathbf
{B} /\partial t}
. Plugging this result into Faraday's Law
finds a value for
E
{\displaystyle \scriptstyle \mathbf {E} }
(which had previously been ignored). This
method is not a true solution of Maxwell's
equations but can provide a good approximation
for slowly changing fields.
== Solving for the magnetic field ==
=== Current sources ===
If all currents in a system are known (i.e.,
if a complete description of the current density
J
(
r
)
{\displaystyle \scriptstyle \mathbf {J} (\mathbf
{r} )}
is available) then the magnetic field can
be determined, at a position r, from the currents
by the Biot–Savart equation:
B
(
r
)
=
μ
0
4
π
∫
J
(
r
′
)
×
(
r
−
r
′
)
|
r
−
r
′
|
3
d
3
r
′
{\displaystyle \mathbf {B} (\mathbf {r} )={\frac
{\mu _{0}}{4\pi }}\int {{\frac {\mathbf {J}
(\mathbf {r} ')\times \left(\mathbf {r} -\mathbf
{r} '\right)}{|\mathbf {r} -\mathbf {r} '|^{3}}}\mathrm
{d} ^{3}\mathbf {r} '}}
This technique works well for problems where
the medium is a vacuum or air or some similar
material with a relative permeability of 1.
This includes air-core inductors and air-core
transformers. One advantage of this technique
is that, if a coil has a complex geometry,
it can be divided into sections and the integral
evaluated for each section. Since this equation
is primarily used to solve linear problems,
the contributions can be added. For a very
difficult geometry, numerical integration
may be used.
For problems where the dominant magnetic material
is a highly permeable magnetic core with relatively
small air gaps, a magnetic circuit approach
is useful. When the air gaps are large in
comparison to the magnetic circuit length,
fringing becomes significant and usually requires
a finite element calculation. The finite element
calculation uses a modified form of the magnetostatic
equations above in order to calculate magnetic
potential. The value of
B
{\displaystyle \scriptstyle \mathbf {B} }
can be found from the magnetic potential.
The magnetic field can be derived from the
vector potential. Since the divergence of
the magnetic flux density is always zero,
B
=
∇
×
A
,
{\displaystyle \mathbf {B} =\nabla \times
\mathbf {A} ,}
and the relation of the vector potential to
current is:
A
(
r
)
=
μ
0
4
π
∫
J
(
r
′
)
|
r
−
r
′
|
d
3
r
′
.
{\displaystyle \mathbf {A} (\mathbf {r} )={\frac
{\mu _{0}}{4\pi }}\int {{\frac {\mathbf {J(\mathbf
{r} ')} }{|\mathbf {r} -\mathbf {r} '|}}\mathrm
{d} ^{3}\mathbf {r} '}.}
=== Magnetization ===
Strongly magnetic materials (i.e., ferromagnetic,
ferrimagnetic or paramagnetic) have a magnetization
that is primarily due to electron spin. In
such materials the magnetization must be explicitly
included using the relation
B
=
μ
0
(
M
+
H
)
.
{\displaystyle \mathbf {B} =\mu _{0}(\mathbf
{M} +\mathbf {H} ).}
Except in metals, electric currents can be
ignored. Then Ampère's law is simply
∇
×
H
=
0.
{\displaystyle \nabla \times \mathbf {H} =0.}
This has the general solution
H
=
−
∇
Φ
M
,
{\displaystyle \mathbf {H} =-\nabla \Phi _{M},}
where
Φ
M
{\displaystyle \Phi _{M}}
is a scalar potential. Substituting this in
Gauss's law gives
∇
2
Φ
M
=
∇
⋅
M
.
{\displaystyle \nabla ^{2}\Phi _{M}=\nabla
\cdot \mathbf {M} .}
Thus, the divergence of the magnetization,
∇
⋅
M
,
{\displaystyle \scriptstyle \nabla \cdot \mathbf
{M} ,}
has a role analogous to the electric charge
in electrostatics and is often referred to
as an effective charge density
ρ
M
{\displaystyle \rho _{M}}
.
The vector potential method can also be employed
with an effective current density
J
M
=
∇
×
M
.
{\displaystyle \mathbf {J_{M}} =\nabla \times
\mathbf {M} .}
== See also ==
Darwin Lagrangian
== Notes
