In physics (particularly in electromagnetism)
the Lorentz force (or electromagnetic force)
is the combination of electric and magnetic
force on a point charge due to electromagnetic
fields. A particle of charge q moving with
a velocity v in an electric field E and a
magnetic field B experiences a force
F
=
q
E
+
q
v
×
B
{\displaystyle \mathbf {F} =q\mathbf {E} +q\mathbf
{v} \times \mathbf {B} }
(in SI units). Variations on this basic formula
describe the magnetic force on a current-carrying
wire (sometimes called Laplace force), the
electromotive force in a wire loop moving
through a magnetic field (an aspect of Faraday's
law of induction), and the force on a charged
particle which might be traveling near the
speed of light (relativistic form of the Lorentz
force).
The first derivation of the Lorentz force
is commonly attributed to Oliver Heaviside
in 1889, although other historians suggest
an earlier origin in an 1865 paper by James
Clerk Maxwell. Hendrik Lorentz derived it
in 1895, a few years after Heaviside.
== Equation ==
=== Charged particle ===
The force F acting on a particle of electric
charge q with instantaneous velocity v, due
to an external electric field E and magnetic
field B, is given by (in SI units):
where × is the vector cross product (all
boldface quantities are vectors). In terms
of cartesian components, we have:
F
x
=
q
(
E
x
+
v
y
B
z
−
v
z
B
y
)
,
{\displaystyle F_{x}=q(E_{x}+v_{y}B_{z}-v_{z}B_{y}),}
F
y
=
q
(
E
y
+
v
z
B
x
−
v
x
B
z
)
,
{\displaystyle F_{y}=q(E_{y}+v_{z}B_{x}-v_{x}B_{z}),}
F
z
=
q
(
E
z
+
v
x
B
y
−
v
y
B
x
)
.
{\displaystyle F_{z}=q(E_{z}+v_{x}B_{y}-v_{y}B_{x}).}
In general, the electric and magnetic fields
are functions of the position and time. Therefore,
explicitly, the Lorentz force can be written
as:
F
(
r
,
r
˙
,
t
,
q
)
=
q
[
E
(
r
,
t
)
+
r
˙
×
B
(
r
,
t
)
]
{\displaystyle \mathbf {F} (\mathbf {r} ,\mathbf
{\dot {r}} ,t,q)=q[\mathbf {E} (\mathbf {r}
,t)+\mathbf {\dot {r}} \times \mathbf {B}
(\mathbf {r} ,t)]}
in which r is the position vector of the charged
particle, t is time, and the overdot is a
time derivative.
A positively charged particle will be accelerated
in the same linear orientation as the E field,
but will curve perpendicularly to both the
instantaneous velocity vector v and the B
field according to the right-hand rule (in
detail, if the fingers of the right hand are
extended to point in the direction of v and
are then curled to point in the direction
of B, then the extended thumb will point in
the direction of F).
The term qE is called the electric force,
while the term qv × B is called the magnetic
force. According to some definitions, the
term "Lorentz force" refers specifically to
the formula for the magnetic force, with the
total electromagnetic force (including the
electric force) given some other (nonstandard)
name. This article will not follow this nomenclature:
In what follows, the term "Lorentz force"
will refer to the expression for the total
force.
The magnetic force component of the Lorentz
force manifests itself as the force that acts
on a current-carrying wire in a magnetic field.
In that context, it is also called the Laplace
force.
The Lorentz force is a force exerted by the
electromagnetic field on the charged particle,
that is, it is the rate at which linear momentum
is transferred from the electromagnetic field
to the particle. Associated with it is the
power which is the rate at which energy is
transferred from the electromagnetic field
to the particle. That power is
v
⋅
F
=
q
v
⋅
E
{\displaystyle \mathbf {v} \cdot \mathbf {F}
=q\,\mathbf {v} \cdot \mathbf {E} }
.Notice that the magnetic field does not contribute
to the power because the magnetic force is
always perpendicular to the velocity of the
particle.
=== Continuous charge distribution ===
For a continuous charge distribution in motion,
the Lorentz force equation becomes:
d
F
=
d
q
(
E
+
v
×
B
)
{\displaystyle \mathrm {d} \mathbf {F} =\mathrm
{d} q\left(\mathbf {E} +\mathbf {v} \times
\mathbf {B} \right)\,\!}
where dF is the force on a small piece of
the charge distribution with charge dq. If
both sides of this equation are divided by
the volume of this small piece of the charge
distribution dV, the result is:
f
=
ρ
(
E
+
v
×
B
)
{\displaystyle \mathbf {f} =\rho \left(\mathbf
{E} +\mathbf {v} \times \mathbf {B} \right)\,\!}
where f is the force density (force per unit
volume) and ρ is the charge density (charge
per unit volume). Next, the current density
corresponding to the motion of the charge
continuum is
J
=
ρ
v
{\displaystyle \mathbf {J} =\rho \mathbf {v}
\,\!}
so the continuous analogue to the equation
is
The total force is the volume integral over
the charge distribution:
F
=
∭
(
ρ
E
+
J
×
B
)
d
V
.
{\displaystyle \mathbf {F} =\iiint \!(\rho
\mathbf {E} +\mathbf {J} \times \mathbf {B}
)\,\mathrm {d} V.\,\!}
By eliminating ρ and J, using Maxwell's equations,
and manipulating using the theorems of vector
calculus, this form of the equation can be
used to derive the Maxwell stress tensor σ,
in turn this can be combined with the Poynting
vector S to obtain the electromagnetic stress–energy
tensor T used in general relativity.In terms
of σ and S, another way to write the Lorentz
force (per unit volume) is
f
=
∇
⋅
σ
−
1
c
2
∂
S
∂
t
{\displaystyle \mathbf {f} =\nabla \cdot {\boldsymbol
{\sigma }}-{\dfrac {1}{c^{2}}}{\dfrac {\partial
\mathbf {S} }{\partial t}}\,\!}
where c is the speed of light and ∇· denotes
the divergence of a tensor field. Rather than
the amount of charge and its velocity in electric
and magnetic fields, this equation relates
the energy flux (flow of energy per unit time
per unit distance) in the fields to the force
exerted on a charge distribution. See Covariant
formulation of classical electromagnetism
for more details.
The density of power associated with the Lorentz
force in a material medium is
J
⋅
E
{\displaystyle \mathbf {J} \cdot \mathbf {E}
}
.If we separate the total charge and total
current into their free and bound parts, we
get that the density of the Lorentz force
is
f
=
(
ρ
f
−
∇
⋅
P
)
E
+
(
J
f
+
∇
×
M
+
∂
P
∂
t
)
×
B
{\displaystyle \mathbf {f} =(\rho _{f}-\nabla
\cdot \mathbf {P} )\mathbf {E} +(\mathbf {J}
_{f}+\nabla \times \mathbf {M} +{\frac {\partial
\mathbf {P} }{\partial t}})\times \mathbf
{B} }
.where: ρf is the density of free charge;
P is the polarization density; Jf is the density
of free current; and M is the magnetization
density. In this way, the Lorentz force can
explain the torque applied to a permanent
magnet by the magnetic field. The density
of the associated power is
(
J
f
+
∇
×
M
+
∂
P
∂
t
)
⋅
E
{\displaystyle \left(\mathbf {J} _{f}+\nabla
\times \mathbf {M} +{\frac {\partial \mathbf
{P} }{\partial t}}\right)\cdot \mathbf {E}
}
.
=== Equation in cgs units ===
The above-mentioned formulae use SI units
which are the most common among experimentalists,
technicians, and engineers. In cgs-Gaussian
units, which are somewhat more common among
theoretical physicists as well as condensed
matter experimentalists, one has instead
F
=
q
c
g
s
(
E
c
g
s
+
v
c
×
B
c
g
s
)
.
{\displaystyle \mathbf {F} =q_{\mathrm {cgs}
}\left(\mathbf {E} _{\mathrm {cgs} }+{\frac
{\mathbf {v} }{c}}\times \mathbf {B} _{\mathrm
{cgs} }\right).}
where c is the speed of light.
Although this equation looks slightly different,
it is completely equivalent, since
one has the following relations:
q
c
g
s
=
q
S
I
4
π
ϵ
0
,
E
c
g
s
=
4
π
ϵ
0
E
S
I
,
B
c
g
s
=
4
π
/
μ
0
B
S
I
,
c
=
1
ϵ
0
μ
0
.
{\displaystyle q_{\mathrm {cgs} }={\frac {q_{\mathrm
{SI} }}{\sqrt {4\pi \epsilon _{0}}}},\quad
\mathbf {E} _{\mathrm {cgs} }={\sqrt {4\pi
\epsilon _{0}}}\,\mathbf {E} _{\mathrm {SI}
},\quad \mathbf {B} _{\mathrm {cgs} }={\sqrt
{4\pi /\mu _{0}}}\,{\mathbf {B} _{\mathrm
{SI} }},\quad c={\frac {1}{\sqrt {\epsilon
_{0}\mu _{0}}}}.}
where ε0 is the vacuum permittivity and μ0
the vacuum permeability. In practice, the
subscripts "cgs" and "SI" are always omitted,
and the unit system has to be assessed from
context.
== History ==
Early attempts to quantitatively describe
the electromagnetic force were made in the
mid-18th century. It was proposed that the
force on magnetic poles, by Johann Tobias
Mayer and others in 1760, and electrically
charged objects, by Henry Cavendish in 1762,
obeyed an inverse-square law. However, in
both cases the experimental proof was neither
complete nor conclusive. It was not until
1784 when Charles-Augustin de Coulomb, using
a torsion balance, was able to definitively
show through experiment that this was true.
Soon after the discovery in 1820 by H. C.
Ørsted that a magnetic needle is acted on
by a voltaic current, André-Marie Ampère
that same year was able to devise through
experimentation the formula for the angular
dependence of the force between two current
elements. In all these descriptions, the force
was always given in terms of the properties
of the objects involved and the distances
between them rather than in terms of electric
and magnetic fields.The modern concept of
electric and magnetic fields first arose in
the theories of Michael Faraday, particularly
his idea of lines of force, later to be given
full mathematical description by Lord Kelvin
and James Clerk Maxwell. From a modern perspective
it is possible to identify in Maxwell's 1865
formulation of his field equations a form
of the Lorentz force equation in relation
to electric currents, however, in the time
of Maxwell it was not evident how his equations
related to the forces on moving charged objects.
J. J. Thomson was the first to attempt to
derive from Maxwell's field equations the
electromagnetic forces on a moving charged
object in terms of the object's properties
and external fields. Interested in determining
the electromagnetic behavior of the charged
particles in cathode rays, Thomson published
a paper in 1881 wherein he gave the force
on the particles due to an external magnetic
field as
F
=
q
2
v
×
B
.
{\displaystyle \mathbf {F} ={\frac {q}{2}}\mathbf
{v} \times \mathbf {B} .}
Thomson derived the correct basic form of
the formula, but, because of some miscalculations
and an incomplete description of the displacement
current, included an incorrect scale-factor
of a half in front of the formula. Oliver
Heaviside invented the modern vector notation
and applied it to Maxwell's field equations;
he also (in 1885 and 1889) had fixed the mistakes
of Thomson's derivation and arrived at the
correct form of the magnetic force on a moving
charged object. Finally, in 1895, Hendrik
Lorentz derived the modern form of the formula
for the electromagnetic force which includes
the contributions to the total force from
both the electric and the magnetic fields.
Lorentz began by abandoning the Maxwellian
descriptions of the ether and conduction.
Instead, Lorentz made a distinction between
matter and the luminiferous aether and sought
to apply the Maxwell equations at a microscopic
scale. Using Heaviside's version of the Maxwell
equations for a stationary ether and applying
Lagrangian mechanics (see below), Lorentz
arrived at the correct and complete form of
the force law that now bears his name.
== Trajectories of particles due to the Lorentz
force ==
In many cases of practical interest, the motion
in a magnetic field of an electrically charged
particle (such as an electron or ion in a
plasma) can be treated as the superposition
of a relatively fast circular motion around
a point called the guiding center and a relatively
slow drift of this point. The drift speeds
may differ for various species depending on
their charge states, masses, or temperatures,
possibly resulting in electric currents or
chemical separation.
== Significance of the Lorentz force ==
While the modern Maxwell's equations describe
how electrically charged particles and currents
or moving charged particles give rise to electric
and magnetic fields, the Lorentz force law
completes that picture by describing the force
acting on a moving point charge q in the presence
of electromagnetic fields. The Lorentz force
law describes the effect of E and B upon a
point charge, but such electromagnetic forces
are not the entire picture. Charged particles
are possibly coupled to other forces, notably
gravity and nuclear forces. Thus, Maxwell's
equations do not stand separate from other
physical laws, but are coupled to them via
the charge and current densities. The response
of a point charge to the Lorentz law is one
aspect; the generation of E and B by currents
and charges is another.
In real materials the Lorentz force is inadequate
to describe the collective behavior of charged
particles, both in principle and as a matter
of computation. The charged particles in a
material medium not only respond to the E
and B fields but also generate these fields.
Complex transport equations must be solved
to determine the time and spatial response
of charges, for example, the Boltzmann equation
or the Fokker–Planck equation or the Navier–Stokes
equations. For example, see magnetohydrodynamics,
fluid dynamics, electrohydrodynamics, superconductivity,
stellar evolution. An entire physical apparatus
for dealing with these matters has developed.
See for example, Green–Kubo relations and
Green's function (many-body theory).
== Lorentz force law as the definition of
E and B ==
In many textbook treatments of classical electromagnetism,
the Lorentz force Law is used as the definition
of the electric and magnetic fields E and
B. To be specific, the Lorentz force is understood
to be the following empirical statement:
The electromagnetic force F on a test charge
at a given point and time is a certain function
of its charge q and velocity v, which can
be parameterized by exactly two vectors E
and B, in the functional form:
F
=
q
(
E
+
v
×
B
)
{\displaystyle \mathbf {F} =q(\mathbf {E}
+\mathbf {v} \times \mathbf {B} )}
This is valid, even for particles approaching
the speed of light (that is, magnitude of
v = |v| ≈ c). So the two vector fields E
and B are thereby defined throughout space
and time, and these are called the "electric
field" and "magnetic field". The fields are
defined everywhere in space and time with
respect to what force a test charge would
receive regardless of whether a charge is
present to experience the force.
As a definition of E and B, the Lorentz force
is only a definition in principle because
a real particle (as opposed to the hypothetical
"test charge" of infinitesimally-small mass
and charge) would generate its own finite
E and B fields, which would alter the electromagnetic
force that it experiences. In addition, if
the charge experiences acceleration, as if
forced into a curved trajectory by some external
agency, it emits radiation that causes braking
of its motion. See for example Bremsstrahlung
and synchrotron light. These effects occur
through both a direct effect (called the radiation
reaction force) and indirectly (by affecting
the motion of nearby charges and currents).
Moreover, net force must include gravity,
electroweak, and any other forces aside from
electromagnetic force.
== Force on a current-carrying wire ==
When a wire carrying an electric current is
placed in a magnetic field, each of the moving
charges, which comprise the current, experiences
the Lorentz force, and together they can create
a macroscopic force on the wire (sometimes
called the Laplace force). By combining the
Lorentz force law above with the definition
of electric current, the following equation
results, in the case of a straight, stationary
wire:
F
=
I
ℓ
×
B
{\displaystyle \mathbf {F} =I{\boldsymbol
{\ell }}\times \mathbf {B} }
where ℓ is a vector whose magnitude is the
length of wire, and whose direction is along
the wire, aligned with the direction of conventional
current flow I.
If the wire is not straight but curved, the
force on it can be computed by applying this
formula to each infinitesimal segment of wire
dℓ, then adding up all these forces by integration.
Formally, the net force on a stationary, rigid
wire carrying a steady current I is
F
=
I
∫
d
ℓ
×
B
{\displaystyle \mathbf {F} =I\int \mathrm
{d} {\boldsymbol {\ell }}\times \mathbf {B}
}
This is the net force. In addition, there
will usually be torque, plus other effects
if the wire is not perfectly rigid.
One application of this is Ampère's force
law, which describes how two current-carrying
wires can attract or repel each other, since
each experiences a Lorentz force from the
other's magnetic field. For more information,
see the article: Ampère's force law.
== EMF ==
The magnetic force (qv × B) component of
the Lorentz force is responsible for motional
electromotive force (or motional EMF), the
phenomenon underlying many electrical generators.
When a conductor is moved through a magnetic
field, the magnetic field exerts opposite
forces on electrons and nuclei in the wire,
and this creates the EMF. The term "motional
EMF" is applied to this phenomenon, since
the EMF is due to the motion of the wire.
In other electrical generators, the magnets
move, while the conductors do not. In this
case, the EMF is due to the electric force
(qE) term in the Lorentz Force equation. The
electric field in question is created by the
changing magnetic field, resulting in an induced
EMF, as described by the Maxwell–Faraday
equation (one of the four modern Maxwell's
equations).Both of these EMFs, despite their
apparently distinct origins, are described
by the same equation, namely, the EMF is the
rate of change of magnetic flux through the
wire. (This is Faraday's law of induction,
see below.) Einstein's special theory of relativity
was partially motivated by the desire to better
understand this link between the two effects.
In fact, the electric and magnetic fields
are different facets of the same electromagnetic
field, and in moving from one inertial frame
to another, the solenoidal vector field portion
of the E-field can change in whole or in part
to a B-field or vice versa.
== Lorentz force and Faraday's law of induction
==
Given a loop of wire in a magnetic field,
Faraday's law of induction states the induced
electromotive 
force (EMF) in the wire is:
E
=
−
d
Φ
B
d
t
{\displaystyle {\mathcal {E}}=-{\frac {\mathrm
{d} \Phi _{B}}{\mathrm {d} t}}}
where
Φ
B
=
∬
Σ
(
t
)
d
A
⋅
B
(
r
,
t
)
{\displaystyle \Phi _{B}=\iint _{\Sigma (t)}\mathrm
{d} \mathbf {A} \cdot \mathbf {B} (\mathbf
{r} ,t)}
is the magnetic flux through the loop, B is
the magnetic field, Σ(t) is a surface bounded
by the 
closed contour ∂Σ(t), at all at time t,
dA is an infinitesimal vector area element
of Σ(t) (magnitude is the area of an infinitesimal
patch of surface, direction is orthogonal
to that surface patch).
The sign of the EMF is determined by Lenz's
law. Note that this is valid for not only
a stationary wire – but also for a moving
wire.
From Faraday's law of induction (that is valid
for a moving wire, for instance in a motor)
and the Maxwell Equations, the Lorentz Force
can be deduced. The reverse is also true,
the Lorentz force and the Maxwell Equations
can be used to derive the Faraday Law.
Let Σ(t) be the moving wire, moving together
without rotation and with constant velocity
v and Σ(t) be the internal surface of the
wire. The EMF around the closed path ∂Σ(t)
is given by:
E
=
∮
∂
Σ
(
t
)
⁡
d
ℓ
⋅
F
/
q
{\displaystyle {\mathcal {E}}=\oint _{\partial
\Sigma (t)}\mathrm {d} {\boldsymbol {\ell
}}\cdot \mathbf {F} /q}
where
E
=
F
/
q
{\displaystyle \mathbf {E} =\mathbf {F} /q}
is the electric field and dℓ is an infinitesimal
vector element of the contour ∂Σ(t).
NB: Both dℓ and dA have a sign ambiguity;
to get the correct sign, the right-hand rule
is used, as explained in the article Kelvin–Stokes
theorem.
The above result can be compared with the
version of Faraday's law of induction that
appears in the modern Maxwell's equations,
called here the Maxwell–Faraday equation:
∇
×
E
=
−
∂
B
∂
t
.
{\displaystyle \nabla \times \mathbf {E} =-{\frac
{\partial \mathbf {B} }{\partial t}}\ .}
The Maxwell–Faraday equation also can be
written in an integral form using the Kelvin–Stokes
theorem.So we have, the Maxwell Faraday equation:
∮
∂
Σ
(
t
)
⁡
d
ℓ
⋅
E
(
r
,
t
)
=
−
∬
Σ
(
t
)
d
A
⋅
d
B
(
r
,
t
)
d
t
{\displaystyle \oint _{\partial \Sigma (t)}\mathrm
{d} {\boldsymbol {\ell }}\cdot \mathbf {E}
(\mathbf {r} ,\ t)=-\ \iint _{\Sigma (t)}\mathrm
{d} \mathbf {A} \cdot {{\mathrm {d} \,\mathbf
{B} (\mathbf {r} ,\ t)} \over \mathrm {d}
t}}
and the Faraday Law,
∮
∂
Σ
(
t
)
⁡
d
ℓ
⋅
F
/
q
(
r
,
t
)
=
−
d
d
t
∬
Σ
(
t
)
d
A
⋅
B
(
r
,
t
)
.
{\displaystyle \oint _{\partial \Sigma (t)}\mathrm
{d} {\boldsymbol {\ell }}\cdot \mathbf {F}
/q(\mathbf {r} ,\ t)=-{\frac {\mathrm {d}
}{\mathrm {d} t}}\iint _{\Sigma (t)}\mathrm
{d} \mathbf {A} \cdot \mathbf {B} (\mathbf
{r} ,\ t).}
The two are equivalent if the wire is not
moving. Using the Leibniz integral rule and
that div B = 0, results in,
∮
∂
Σ
(
t
)
⁡
d
ℓ
⋅
F
/
q
(
r
,
t
)
=
−
∬
Σ
(
t
)
d
A
⋅
∂
∂
t
B
(
r
,
t
)
+
∮
∂
Σ
(
t
)
v
×
B
d
ℓ
{\displaystyle \oint _{\partial \Sigma (t)}\mathrm
{d} {\boldsymbol {\ell }}\cdot \mathbf {F}
/q(\mathbf {r} ,t)=-\iint _{\Sigma (t)}\mathrm
{d} \mathbf {A} \cdot {\frac {\partial }{\partial
t}}\mathbf {B} (\mathbf {r} ,t)+\oint _{\partial
\Sigma (t)}\!\!\!\!\mathbf {v} \times \mathbf
{B} \,\mathrm {d} {\boldsymbol {\ell }}}
and using the Maxwell Faraday equation,
∮
∂
Σ
(
t
)
⁡
d
ℓ
⋅
F
/
q
(
r
,
t
)
=
∮
∂
Σ
(
t
)
⁡
d
ℓ
⋅
E
(
r
,
t
)
+
∮
∂
Σ
(
t
)
v
×
B
(
r
,
t
)
d
ℓ
{\displaystyle \oint _{\partial \Sigma (t)}\mathrm
{d} {\boldsymbol {\ell }}\cdot \mathbf {F}
/q(\mathbf {r} ,\ t)=\oint _{\partial \Sigma
(t)}\mathrm {d} {\boldsymbol {\ell }}\cdot
\mathbf {E} (\mathbf {r} ,\ t)+\oint _{\partial
\Sigma (t)}\!\!\!\!\mathbf {v} \times \mathbf
{B} (\mathbf {r} ,\ t)\,\mathrm {d} {\boldsymbol
{\ell }}}
since this is valid for any wire position
it implies that,
F
=
q
E
(
r
,
t
)
+
q
v
×
B
(
r
,
t
)
.
{\displaystyle \mathbf {F} =q\,\mathbf {E}
(\mathbf {r} ,\ t)+q\,\mathbf {v} \times \mathbf
{B} (\mathbf {r} ,\ t).}
Faraday's law of induction holds whether the
loop of wire is rigid and stationary, or in
motion or in process of deformation, and it
holds whether the magnetic field is constant
in time or changing. However, there are cases
where Faraday's law is either inadequate or
difficult to use, and application of the underlying
Lorentz force law is necessary. See inapplicability
of Faraday's law.
If the magnetic field is fixed in time and
the conducting loop moves through the field,
the magnetic flux ΦB linking the loop can
change in several ways. For example, if the
B-field varies with position, and the loop
moves to a location with different B-field,
ΦB will change. Alternatively, if the loop
changes orientation with respect to the B-field,
the B ⋅ dA differential element will change
because of the different angle between B and
dA, also changing ΦB. As a third example,
if a portion of the circuit is swept through
a uniform, time-independent B-field, and another
portion of the circuit is held stationary,
the flux linking the entire closed circuit
can change due to the shift in relative position
of the circuit's component parts with time
(surface ∂Σ(t) time-dependent). In all
three cases, Faraday's law of induction then
predicts the EMF generated by the change in
ΦB.
Note that the Maxwell Faraday's equation implies
that the Electric Field E is non conservative
when the Magnetic Field B varies in time,
and is not expressible as the gradient of
a scalar field, and not subject to the gradient
theorem since its rotational is not zero.
== Lorentz force in terms of potentials ==
The E and B fields can be replaced by the
magnetic vector potential A and (scalar) electrostatic
potential ϕ by
E
=
−
∇
ϕ
−
∂
A
∂
t
{\displaystyle \mathbf {E} =-\nabla \phi -{\frac
{\partial \mathbf {A} }{\partial t}}}
B
=
∇
×
A
{\displaystyle \mathbf {B} =\nabla \times
\mathbf {A} }
where ∇ is the gradient, ∇⋅ is the divergence,
∇× is the curl.
The force becomes
F
=
q
[
−
∇
ϕ
−
∂
A
∂
t
+
v
×
(
∇
×
A
)
]
{\displaystyle \mathbf {F} =q\left[-\nabla
\phi -{\frac {\partial \mathbf {A} }{\partial
t}}+\mathbf {v} \times (\nabla \times \mathbf
{A} )\right]}
and using an identity for the triple product
simplifies to
(v has no dependence on position, so there's
no need to use Feynman's subscript notation).
Using the chain rule, the total derivative
of A is:
d
A
d
t
=
∂
A
∂
t
+
(
v
⋅
∇
)
A
{\displaystyle {\frac {\mathrm {d} \mathbf
{A} }{\mathrm {d} t}}={\frac {\partial \mathbf
{A} }{\partial t}}+(\mathbf {v} \cdot \nabla
)\mathbf {A} }
so the above expression can be rewritten as:
F
=
q
[
−
∇
(
ϕ
−
v
⋅
A
)
−
d
A
d
t
]
{\displaystyle \mathbf {F} =q\left[-\nabla
(\phi -\mathbf {v} \cdot \mathbf {A} )-{\frac
{\mathrm {d} \mathbf {A} }{\mathrm {d} t}}\right]}
.With v = ẋ, we can put the equation into
the convenient Euler–Lagrange form
where
∇
x
=
x
^
∂
∂
x
+
y
^
∂
∂
y
+
z
^
∂
∂
z
{\displaystyle \nabla _{\mathbf {x} }={\hat
{x}}{\dfrac {\partial }{\partial x}}+{\hat
{y}}{\dfrac {\partial }{\partial y}}+{\hat
{z}}{\dfrac {\partial }{\partial z}}}
and
∇
x
˙
=
x
^
∂
∂
x
˙
+
y
^
∂
∂
y
˙
+
z
^
∂
∂
z
˙
{\displaystyle \nabla _{\dot {\mathbf {x}
}}={\hat {x}}{\dfrac {\partial }{\partial
{\dot {x}}}}+{\hat {y}}{\dfrac {\partial }{\partial
{\dot {y}}}}+{\hat {z}}{\dfrac {\partial }{\partial
{\dot {z}}}}}
.
== Lorentz force and analytical mechanics
==
The Lagrangian for a charged particle of mass
m and charge q in an electromagnetic field
equivalently describes the dynamics of the
particle in terms of its energy, rather than
the force exerted on it. The classical expression
is given by:
L
=
m
2
r
˙
⋅
r
˙
+
q
A
⋅
r
˙
−
q
ϕ
{\displaystyle L={\frac {m}{2}}\mathbf {\dot
{r}} \cdot \mathbf {\dot {r}} +q\mathbf {A}
\cdot \mathbf {\dot {r}} -q\phi }
where A and ϕ are the potential fields as
above. Using Lagrange's equations, the equation
for the Lorentz force can be obtained.
The potential energy depends on the velocity
of the particle, so the force is velocity
dependent, so it is not conservative.
The relativistic Lagrangian is
L
=
−
m
c
2
1
−
(
r
˙
c
)
2
+
q
A
(
r
)
⋅
r
˙
−
q
ϕ
(
r
)
{\displaystyle L=-mc^{2}{\sqrt {1-\left({\frac
{\dot {\mathbf {r} }}{c}}\right)^{2}}}+q\mathbf
{A} (\mathbf {r} )\cdot {\dot {\mathbf {r}
}}-q\phi (\mathbf {r} )\,\!}
The action is the relativistic arclength of
the path of the particle in space time, minus
the potential energy contribution, plus an
extra contribution which quantum mechanically
is an extra phase a charged particle gets
when it is moving along a vector potential.
== Relativistic form of the Lorentz force
==
=== Covariant form of the Lorentz force ===
==== Field tensor ====
Using the metric signature (1, −1, −1,
−1), the Lorentz force for a charge q can
be written in covariant form:
where pα is the four-momentum, defined as
p
α
=
(
p
0
,
p
1
,
p
2
,
p
3
)
=
(
γ
m
c
,
p
x
,
p
y
,
p
z
)
,
{\displaystyle p^{\alpha }=\left(p_{0},p_{1},p_{2},p_{3}\right)=\left(\gamma
mc,p_{x},p_{y},p_{z}\right)\,,}
τ the proper time of the particle, Fαβ
the contravariant electromagnetic tensor
F
α
β
=
(
0
−
E
x
/
c
−
E
y
/
c
−
E
z
/
c
E
x
/
c
0
−
B
z
B
y
E
y
/
c
B
z
0
−
B
x
E
z
/
c
−
B
y
B
x
0
)
{\displaystyle F^{\alpha \beta }={\begin{pmatrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\E_{x}/c&0&-B_{z}&B_{y}\\E_{y}/c&B_{z}&0&-B_{x}\\E_{z}/c&-B_{y}&B_{x}&0\end{pmatrix}}}
and U is the covariant 4-velocity of the particle,
defined as:
U
β
=
(
U
0
,
U
1
,
U
2
,
U
3
)
=
γ
(
c
,
−
v
x
,
−
v
y
,
−
v
z
)
,
{\displaystyle U_{\beta }=\left(U_{0},U_{1},U_{2},U_{3}\right)=\gamma
\left(c,-v_{x},-v_{y},-v_{z}\right)\,,}
in which
γ
(
v
)
=
1
1
−
v
2
c
2
=
1
1
−
v
x
2
+
v
y
2
+
v
z
2
c
2
{\displaystyle \gamma (v)={\frac {1}{\sqrt
{1-{\frac {v^{2}}{c^{2}}}}}}={\frac {1}{\sqrt
{1-{\frac {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}{c^{2}}}}}}}
is the Lorentz factor.
The fields are transformed to a frame moving
with constant relative velocity by:
F
′
μ
ν
=
Λ
μ
α
Λ
ν
β
F
α
β
,
{\displaystyle F'^{\mu \nu }={\Lambda ^{\mu
}}_{\alpha }{\Lambda ^{\nu }}_{\beta }F^{\alpha
\beta }\,,}
where Λμα is the Lorentz transformation
tensor.
==== Translation to vector notation ====
The α = 1 component (x-component) of the
force is
d
p
1
d
τ
=
q
U
β
F
1
β
=
q
(
U
0
F
10
+
U
1
F
11
+
U
2
F
12
+
U
3
F
13
)
.
{\displaystyle {\frac {\mathrm {d} p^{1}}{\mathrm
{d} \tau }}=qU_{\beta }F^{1\beta }=q\left(U_{0}F^{10}+U_{1}F^{11}+U_{2}F^{12}+U_{3}F^{13}\right).}
Substituting the components of the covariant
electromagnetic tensor F yields
d
p
1
d
τ
=
q
[
U
0
(
E
x
c
)
+
U
2
(
−
B
z
)
+
U
3
(
B
y
)
]
.
{\displaystyle {\frac {\mathrm {d} p^{1}}{\mathrm
{d} \tau }}=q\left[U_{0}\left({\frac {E_{x}}{c}}\right)+U_{2}(-B_{z})+U_{3}(B_{y})\right].}
Using the components of covariant four-velocity
yields
d
p
1
d
τ
=
q
γ
[
c
(
E
x
c
)
+
(
−
v
y
)
(
−
B
z
)
+
(
−
v
z
)
(
B
y
)
]
=
q
γ
(
E
x
+
v
y
B
z
−
v
z
B
y
)
=
q
γ
[
E
x
+
(
v
×
B
)
x
]
.
{\displaystyle {\begin{aligned}{\frac {\mathrm
{d} p^{1}}{\mathrm {d} \tau }}&=q\gamma \left[c\left({\frac
{E_{x}}{c}}\right)+(-v_{y})(-B_{z})+(-v_{z})(B_{y})\right]\\&=q\gamma
\left(E_{x}+v_{y}B_{z}-v_{z}B_{y}\right)\\&=q\gamma
\left[E_{x}+\left(\mathbf {v} \times \mathbf
{B} \right)_{x}\right]\,.\end{aligned}}}
The calculation for α = 2, 3 (force components
in the y and z directions) yields similar
results, so collecting the 3 equations into
one:
d
p
d
τ
=
q
γ
(
E
+
v
×
B
)
,
{\displaystyle {\frac {\mathrm {d} \mathbf
{p} }{\mathrm {d} \tau }}=q\gamma \left(\mathbf
{E} +\mathbf {v} \times \mathbf {B} \right)\,,}
and since differentials in coordinate time
dt and proper time dτ are related by the
Lorentz factor,
d
t
=
γ
(
v
)
d
τ
,
{\displaystyle dt=\gamma (v)d\tau \,,}
so we arrive at
d
p
d
t
=
q
(
E
+
v
×
B
)
.
{\displaystyle {\frac {\mathrm {d} \mathbf
{p} }{\mathrm {d} t}}=q\left(\mathbf {E} +\mathbf
{v} \times \mathbf {B} \right)\,.}
This is precisely the Lorentz force law, however,
it is important to note that p is the relativistic
expression,
p
=
γ
(
v
)
m
0
v
.
{\displaystyle \mathbf {p} =\gamma (v)m_{0}\mathbf
{v} \,.}
=== Lorentz force in spacetime algebra (STA)
===
The electric and magnetic fields are dependent
on the velocity of an observer, so the relativistic
form of the Lorentz force law can best be
exhibited starting from a coordinate-independent
expression for the electromagnetic and magnetic
fields
F
{\displaystyle {\mathcal {F}}}
, and an arbitrary time-direction,
γ
0
{\displaystyle \gamma _{0}}
. This can be settled through Space-Time Algebra
(or the geometric algebra of space-time),
a type of Clifford's Algebra defined on a
pseudo-Euclidean space, as
E
=
(
F
⋅
γ
0
)
γ
0
{\displaystyle \mathbf {E} =({\mathcal {F}}\cdot
\gamma _{0})\gamma _{0}}
and
i
B
=
(
F
∧
γ
0
)
γ
0
{\displaystyle i\mathbf {B} =({\mathcal {F}}\wedge
\gamma _{0})\gamma _{0}}
F
{\displaystyle {\mathcal {F}}}
is a space-time bivector (an oriented plane
segment, just like a vector is an oriented
line segment), which has six degrees of freedom
corresponding to boosts (rotations in space-time
planes) and rotations (rotations in space-space
planes). The dot product with the vector
γ
0
{\displaystyle \gamma _{0}}
pulls a vector (in the space algebra) from
the translational part, while the wedge-product
creates a trivector (in the space algebra)
who is dual to a vector which is the usual
magnetic field vector.
The relativistic velocity is given by the
(time-like) changes in a time-position vector
v
=
x
˙
{\displaystyle v={\dot {x}}}
, where
v
2
=
1
,
{\displaystyle v^{2}=1,}
(which shows our choice for the metric) and
the velocity is
v
=
c
v
∧
γ
0
/
(
v
⋅
γ
0
)
.
{\displaystyle \mathbf {v} =cv\wedge \gamma
_{0}/(v\cdot \gamma _{0}).}
The proper (invariant is an inadequate term
because no transformation has been defined)
form of the Lorentz force law is simply
Note that the order is important because between
a bivector and a vector the dot product is
anti-symmetric. Upon a space time split like
one can obtain the velocity, and fields as
above yielding the usual expression.
=== Lorentz force in general relativity ===
In the general theory of relativity the equation
of motion for a particle with mass
m
{\displaystyle m}
and charge
e
{\displaystyle e}
, moving in a space with metric tensor
g
a
b
{\displaystyle g_{ab}}
and electromagnetic field
F
a
b
{\displaystyle F_{ab}}
, is given as
m
d
u
c
d
s
−
m
1
2
g
a
b
,
c
u
a
u
b
=
e
F
c
b
u
b
,
{\displaystyle m{\frac {du_{c}}{ds}}-m{\frac
{1}{2}}g_{ab,c}u^{a}u^{b}=eF_{cb}u^{b}\;,}
where
u
a
=
d
x
a
/
d
s
{\displaystyle u^{a}=dx^{a}/ds}
(
d
x
a
{\displaystyle dx^{a}}
is taken along the trajectory),
g
a
b
,
c
=
∂
g
a
b
/
∂
x
c
{\displaystyle g_{ab,c}=\partial g_{ab}/\partial
x^{c}}
, and
d
s
2
=
g
a
b
d
x
a
d
x
b
{\displaystyle ds^{2}=g_{ab}dx^{a}dx^{b}}
.
The equation can also be written as
m
d
u
c
d
s
−
m
Γ
a
b
c
u
a
u
b
=
e
F
c
b
u
b
,
{\displaystyle m{\frac {du_{c}}{ds}}-m\Gamma
_{abc}u^{a}u^{b}=eF_{cb}u^{b}\;,}
where
Γ
a
b
c
{\displaystyle \Gamma _{abc}}
is the Christoffel symbol (of the torsion-free
metric connection in general relativity),
or as
m
D
u
c
d
s
=
e
F
c
b
u
b
,
{\displaystyle m{\frac {Du_{c}}{ds}}=eF_{cb}u^{b}\;,}
where
D
{\displaystyle D}
is the covariant differential in general relativity
(metric, torsion-free).
== Applications ==
The Lorentz force occurs in many devices,
including:
Cyclotrons and other circular path particle
accelerators
Mass spectrometers
Velocity Filters
Magnetrons
Lorentz force velocimetryIn its manifestation
as the Laplace force on an electric current
in a conductor, this force occurs in many
devices including:
== See also ==
== Footnotes
