Magnetism

${\boldsymbol {F}}_{B}=q{\boldsymbol {v}}\times {\boldsymbol {B}}$

The magnetic force exerted on a moving particle in a magnetic field is the cross product of the magnetic field' and the velocity of the particle, multiplied by the charge of the particle.

Because the magnetic force is perpendicular to the particle's velocity, this causes uniform circular motion. That motion can be explained by the following

$r={\frac {mv}{qB}}$

The radius of this circle is directly proportional to the mass and the velocity of the particle and inversely proportional to the charge of the particle and the field strength of the magnetic field.

$T={\frac {2\pi m}{qB}}$

$f={\frac {qB}{2\pi m}}={\frac {v}{2\pi r}}$

The period and frequency of this motion (referred to as the cyclotron period and frequency) can be derived as well.

$B={\frac {\mu _{0}I}{2\pi y}}$

The magnetic field created by charge flowing through a straight wire is equal to a constant, ${\frac {\mu _{0}}{2\pi }}$ , multiplied by the current flowing through the wire and divide by the distance from the wire.

$B={\frac {\mu _{0}\mu }{4\pi x^{3}}}$

The magnetic field created by a magnetic dipole (at distances much greater than the size of the dipole) is approximately equal to a constant, ${\frac {\mu _{0}}{4\pi }}$ , multiplied by the dipole moment divided by the cube of the distance from the dipole. EDIT: This forumla is incomplete. The field from a dipole is a vector that depends not only on the distance from the dipole, but also the angle relative to the orientation of the magnetic moment. This is because of the vector nature of the magnetic moment and its associated magnetic field. The field component pointing in the same directions as the magnetic moment is the above formula multiplied by $(3*(\cos \theta )^{2}-1)$ .

$B=\mu _{0}nI$

The magnetic field created by an ideal solenoid is equal to a constant, $\mu _{0}$ , times the number of turns of the solenoid times the current flowing through the solenoid.

$B={\frac {\mu _{0}nI}{2\pi r}}$

The magnetic field created by an ideal toroid is equal to a constant, $\mu _{0}$ , times the number of turns of the toroid times the current flowing through the toroid divided by the circumference of the toroid.

$F_{B}={\frac {\mu _{0}I_{1}I_{2}\ell }{2\pi d}}$

The magnetic force between two wires is equal to a constant, ${\frac {\mu _{0}}{2\pi }}$ , times the current in one wire times the current in the other wire times the length of the wires divided by the distance between the wires.

$\tau =I{\vec {A}}\times {\vec {B}}$

The torque on a current loop in a magnetic field is equal to the cross product of the magnetic field and the area enclosed by the current loop (the area vector is perpendicular to the current loop).

${\vec {\mu }}=I{\vec {A}}n$

The dipole moment of a current loop is equal to the current in the loop times the area of the loop times the number of turns of the loop.

$U_{B}=-{\vec {\mu }}\cdot {\vec {B}}$

The magnetic potential energy is the opposite of the dot product of the magnetic field and the dipole moment.

Variables

F: Force (N)
q: Charge (C)
v: Velocity (m/s)
B: Magnetic field (teslas (T))
r: Radius (m)
m: Mass (kg)
T: Period (s)
f: Frequency (Hz)
$\mu _{0}$ : A constant, 4π×10^-7 N/A
y: Distance (m)
$\mu$ : Dipole moment
x: Distance (m)
n: Number of turns
$\ell$ : Length (m)
d: Distance (m)
$\tau$ : Torque (N·m)
A: Area (m²)
U: Potential energy (J)