Uses in advanced physics As different aspects of the same phenomenon According to
the special theory of 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 may be perceived by another (in a different frame of reference) as a magnetic force, or a mixture of electric and magnetic forces. The magnetic field existing as electric field in other frames can be shown by consistency of equations obtained from
Lorentz transformation of four force from
Coulomb's Law in particle's rest frame with
Maxwell's laws considering definition of fields from
Lorentz force and for non accelerating condition. The form of magnetic field hence obtained by
Lorentz transformation of
four-force from the form of
Coulomb's law in source's initial frame is given by: \mathbf{B} = \frac q {4 \pi \varepsilon_0 r^3} \frac {1- \beta^2} {(1- \beta^2 \sin^2 \theta)^{3/2}} \frac{\mathbf{v} \times \mathbf{r}}{c^2} = \frac{\mathbf{v} \times \mathbf{E}}{c^2} where q is the charge of the point source, \varepsilon_0 is the
vacuum permittivity, \mathbf{r} is the position vector from the point source to the point in space, \mathbf{v} is the velocity vector of the charged particle, \beta is the ratio of speed of the charged particle divided by the speed of light and \theta is the angle between \mathbf{r} and \mathbf{v}. This form of magnetic field can be shown to satisfy Maxwell's laws within the constraint of particle being non accelerating. The above reduces to
Biot-Savart law for non relativistic stream of current (\beta\ll 1). Formally, special relativity combines the electric and magnetic fields into a rank-2
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
four-momentum. Similarly, the
energy stored in a magnetic field is mixed with the energy stored in an electric field in the
electromagnetic stress–energy tensor.
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
magnetic vector potential , and the
electric scalar potential , are defined using
gauge fixing such that: \begin{align} \mathbf{B} &= \nabla \times \mathbf{A}, \\ \mathbf{E} &= -\nabla \varphi - \frac{ \partial \mathbf{A} }{ \partial t }. \end{align} The vector potential, ''
given by this form may be interpreted as a generalized potential
momentum per unit charge
just as is interpreted as a generalized
potential energy per unit charge''. There are multiple choices one can make for the potential fields that satisfy the above condition. However, the choice of potentials is represented by its respective gauge condition. Maxwell's equations when expressed in terms of the potentials in
Lorenz gauge can be cast into a form that agrees with
special relativity. In relativity, together with forms a
four-potential regardless of the gauge condition, analogous to the
four-momentum that combines the momentum and energy of a particle. Using the four potential instead of the electromagnetic tensor has the advantage of being much simpler—and it can be easily modified to work with quantum mechanics.
Propagation of Electric and Magnetic fields Special theory of relativity imposes the condition for events related by
cause and effect to be time-like separated, that is that causal efficacy propagates no faster than light.
Maxwell's equations for electromagnetism are found to be in favor of this as electric and magnetic disturbances are found to travel at the speed of light in space. Electric and magnetic fields from classical electrodynamics obey the
principle of locality in physics and are expressed in terms of retarded time or the time at which the cause of a measured field originated given that the influence of field travelled at speed of light. The retarded time for a point particle is given as solution of: t_r = \mathbf{t} - \frac{\left|\mathbf{r} - \mathbf{r}_s(t_r)\right|}{c} where t_r is
retarded time or the time at which the source's contribution of the field originated, r_s(t) is the position vector of the particle as function of time, \mathbf{r} is the point in space, \mathbf{t} is the time at which fields are measured and c is the speed of light. The equation subtracts the time taken for light to travel from particle to the point in space from the time of measurement to find time of origin of the fields. The uniqueness of solution for t_r for given \mathbf{t}, \mathbf{r} and r_s(t) is valid for charged particles moving slower than speed of light. which is incorporated into a more complete theory known as the
Standard Model of particle physics. In QED, the magnitude of the electromagnetic interactions between charged particles (and their
antiparticles) is computed using
perturbation theory. These rather complex formulas produce a remarkable pictorial representation as
Feynman diagrams in which
virtual photons are exchanged. 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 mentioned here. However, under most everyday circumstances, the difference between the two theories is negligible.
Uses in geology Earth's magnetic field The Earth's magnetic field is produced by
convection of a liquid iron alloy in the
outer core. In a
dynamo process, the movements drive a feedback process in which electric currents create electric and magnetic fields that in turn act on the currents. The field at the surface of the Earth is approximately the same as if a giant bar magnet were positioned at the center of the Earth and tilted at an angle of about 11° off the rotational axis of the Earth (see the figure). The north pole of a magnetic compass needle points roughly north, toward the
North Magnetic Pole. However, because a magnetic pole is attracted to its opposite, the North Magnetic Pole is actually the south pole of the geomagnetic field. This confusion in terminology arises because the pole of a magnet is defined by the geographical direction it points. Earth's magnetic field is not constant—the strength of the field and the location of its poles vary. Moreover, the poles periodically reverse their orientation in a process called
geomagnetic reversal. The
most recent reversal occurred 780,000 years ago.
Uses in Engineering Rotating magnetic fields The
rotating magnetic field is a common design principle in the operation of
alternating-current motors. A permanent magnet in such a field rotates so as to maintain its alignment with the external field. Magnetic torque is used to drive
electric motors. In one simple motor design, a magnet is fixed to a freely rotating shaft and is subjected to a magnetic field from an array of
electromagnets. By continuously switching the electric current through each of the electromagnets, thereby flipping the polarity of their magnetic fields, like poles are kept next to the rotor; the resultant torque is transferred to the shaft. A rotating magnetic field can be constructed using two coils at right angles with a phase difference of 90 degrees between their AC currents. In practice,
three-phase systems are used where the three currents are equal in magnitude and have a phase difference of 120 degrees. Three similar coils at mutual geometrical angles of 120 degrees create the rotating magnetic field. The ability of the three-phase system to create a rotating field, utilized in electric motors, is one of the main reasons why three-phase systems dominate the world's
electrical power supply systems.
Synchronous motors use DC-voltage-fed rotor windings, which lets the excitation of the machine be controlled—and
induction motors use short-circuited
rotors (instead of a magnet) following the rotating magnetic field of a multicoiled
stator. The short-circuited turns of the rotor develop
eddy currents induced by the rotating field of the stator, and these currents in turn produce a torque on the rotor through the Lorentz force. The Italian physicist
Galileo Ferraris and the Serbian-American
electrical engineer Nikola Tesla independently researched the use of rotating magnetic fields in electric motors. In 1888, Ferraris published his research in a paper to the
Royal Academy of Sciences in
Turin and Tesla gained for his work.
Magnetic circuits An important use of is in
magnetic circuits where inside a linear material. Here, is the
magnetic permeability of the material. This result is similar in form to
Ohm's law , where is the current density, is the conductance and is the electric field. Extending this analogy, the counterpart to the macroscopic Ohm's law () is: \Phi = \frac F R_\mathrm{m}, where \Phi = \int \mathbf{B}\cdot \mathrm{d}\mathbf{A} is the magnetic flux in the circuit, F = \int \mathbf{H}\cdot \mathrm{d}\boldsymbol{\ell} is the
magnetomotive force applied to the circuit, and is the
reluctance of the circuit. Here the reluctance 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.
Uses in material science 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).
Largest magnitude magnetic fields The largest magnitude magnetic field produced over a macroscopic volume outside a lab setting is 2.8 kT (
VNIIEF in
Sarov,
Russia, 1998). The largest magnitude magnetic field produced in a laboratory over a macroscopic volume was 1.2 kT by researchers at the
University of Tokyo in 2018. The largest magnitude microscopic magnetic fields produced in a laboratory occur in particle accelerators, such as
RHIC, inside the collisions of heavy ions, where microscopic fields reach 1014 T.
Magnetars have the strongest known macroscopic magnetic fields of any naturally occurring object, ranging from 0.1 to 100 GT (108 to 1011 T). == Common formulæ==