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:

$\overrightarrow{F}=q\overrightarrow{v}\times \overrightarrow{B}$

Force on Current-Carrying Wire

The formula for the total force is as follows:

$𝐅=I𝐋\times 𝐁$

where

F = Force, measured in newtons

I = current in wire, measured in amperes

B = magnetic field vector, measured in teslas

$\times$ = 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:

$d𝐁=\frac{{\mu }_0}{4\pi }\frac{Id𝐥\times \widehat{𝐫}}{r^2}$

(in SI units), where

$I$ is the current,

$d𝐥$ is a vector, whose magnitude is the length of the differential element of the wire, and whose direction is the direction of conventional current,

$d𝐁$ is the differential contribution to the magnetic field resulting from this differential element of wire,

${\mu }_0$ is the magnetic constant,

$\widehat{𝐫}$ is the unit displacement vector from the wire element to the point at which the field is being computed, and

$r$ is the distance from the wire element to the point at which the field is being computed.

Magnetic Field Inside Coil - Empty Inductor

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

Energy per Unit Volume of Empty Inductor

 $\frac{B^2}{2{\mu }_0}=\frac{{\mu }_0{n}^2{I}^2}{2}$

Total Stored Energy in an Empty Inductor

 $\frac{{\mu }_0{n}^{2}Al{I}^2}{2}=\frac{L{I}^2}{2}$

Magnetic Field

 $B={\mu }_{0}nI+{\mu }_{0}M$
 $B={\mu }_0(H+M)$
 $B={\mu }_0{\mu }_{r}H$

Relative Permeability of a Material

 ${\mu }_r=\frac{𝐁}{{\mu }_0𝐇}$
 ${\mu }_r=1+\frac{𝐌}{𝐇}$
 ${\mu }_r=1+{\mathrm{X}}_m$

Anisotropy Energy

 $E_{an}=K{\sin }^2\varphi$