# Materials Science and Engineering/Equations/Magnetism

## Force of Charged Particle

When a charged particle moves through a magnetic field B, it feels a force F given by the cross product:

${\displaystyle {\vec {F}}=q{\vec {v}}\times {\vec {B}}}$

## Force on Current-Carrying Wire

The formula for the total force is as follows:

${\displaystyle \mathbf {F} =I\mathbf {L} \times \mathbf {B} \,}$

where

F = Force, measured in newtons
I = current in wire, measured in amperes
B = magnetic field vector, measured in teslas
${\displaystyle \times }$ = vector cross product
L = a vector, whose magnitude is the length of wire (measured in metres), and whose direction is along the wire, aligned with the direction of conventional current flow.

## Magnetic Field from Steady Current

The magnetic field generated by a steady current (a continual flow of electric charge, for example through a wire, which is constant in time and in which charge is neither building up nor depleting at any point), is described by the Biot-Savart law:

${\displaystyle d\mathbf {B} ={\frac {\mu _{0}}{4\pi }}{\frac {Id\mathbf {l} \times \mathbf {\hat {r}} }{r^{2}}}}$

(in SI units), where

${\displaystyle I}$ is the current,
${\displaystyle d\mathbf {l} }$ is a vector, whose magnitude is the length of the differential element of the wire, and whose direction is the direction of conventional current,
${\displaystyle d\mathbf {B} }$ is the differential contribution to the magnetic field resulting from this differential element of wire,
${\displaystyle \mu _{0}}$ is the magnetic constant,
${\displaystyle \mathbf {\hat {r}} }$ is the unit displacement vector from the wire element to the point at which the field is being computed, and
${\displaystyle r}$ is the distance from the wire element to the point at which the field is being computed.

## Magnetic Field Inside Coil - Empty Inductor

 ${\displaystyle B=\mu _{0}nI\;}$


## Energy per Unit Volume of Empty Inductor

 ${\displaystyle {\frac {B^{2}}{2\mu _{0}}}={\frac {\mu _{0}n^{2}I^{2}}{2}}}$


## Total Stored Energy in an Empty Inductor

 ${\displaystyle {\frac {\mu _{0}n^{2}AlI^{2}}{2}}={\frac {LI^{2}}{2}}}$


## Magnetic Field

 ${\displaystyle B=\mu _{0}nI+\mu _{0}M\;}$
${\displaystyle B=\mu _{0}(H+M)\;}$
${\displaystyle B=\mu _{0}\mu _{r}H\;}$


## Relative Permeability of a Material

 ${\displaystyle \mu _{r}={\frac {\mathbf {B} }{\mu _{0}\mathbf {H} }}}$
${\displaystyle \mu _{r}=1+{\frac {\mathbf {M} }{\mathbf {H} }}}$
${\displaystyle \mu _{r}=1+\mathrm {X} _{m}\;}$


## Anisotropy Energy

 ${\displaystyle E_{an}=K\sin ^{2}\phi \;}$