Editing Magnetic field (section)

==== As different aspects of the same phenomenon ====
According to [[special relativity|the special theory of relativity]], the partition of the [[electromagnetic force]] into separate electric and magnetic components is not fundamental, but varies with the [[Frame of reference#Observational frames of reference|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|Coulomb's Law]] in particle's rest frame with [[Maxwell's equations|Maxwell's laws]] considering definition of fields from [[Lorentz force#Lorentz force law as the definition of E and B|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:<ref name="Rosser1968">{{Cite book |last1=Rosser |first1=W. G. V. |url=https://link.springer.com/book/10.1007/978-1-4899-6559-2 |title=Classical Electromagnetism via Relativity |year=1968 |isbn=978-1-4899-6258-4 |publisher=Springer |location=Boston, MA |language=en |doi=10.1007/978-1-4899-6559-2}}</ref>{{rp|pages=29–42}}

<math display="block">\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}</math>

where <math>q</math> is the charge of the point source, <math>\varepsilon_0</math> is the [[vacuum permittivity]], <math>\mathbf{r}</math> is the position vector from the point source to the point in space, <math>\mathbf{v}</math> is the velocity vector of the charged particle, <math>\beta</math> is the ratio of speed of the charged particle divided by the speed of light and <math>\theta</math> is the angle between <math>\mathbf{r}</math> and <math>\mathbf{v}</math>. This form of magnetic field can be shown to satisfy Maxwell's laws within the constraint of particle being non accelerating.<ref>{{Cite book |last=Purcell |first=Edward |url=http://dx.doi.org/10.1017/cbo9781139005043 |title=Electricity and Magnetism |date=2011-09-22 |publisher=Cambridge University Press |isbn=978-1-107-01360-5 |doi=10.1017/cbo9781139005043}}</ref> The above reduces to [[Biot–Savart law|Biot-Savart law]] for non relativistic stream of current (<math>\beta\ll 1</math>).

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]].<ref>C. Doran and A. Lasenby (2003) ''Geometric Algebra for Physicists'', Cambridge University Press, p. 233. {{ISBN|0521715954}}.</ref> Similarly, the [[Magnetic energy|energy stored in a magnetic field]] is mixed with the energy stored in an electric field in the [[electromagnetic stress–energy tensor]].