Magnetic field is defined using the magnetic force a charged particle experiences while moving in that field:

$$ \vec{F} = q \vec{v} \times \vec{B} = qvB\sin\theta $$

where $\vec{v}$ is the velocity of the particle, $\vec{B}$ is the magnetic field, and $\theta$ the angle between the two.

SI unit is tesla (T), and another unit is gauss (G). 1 G = $10^{-4}$ T.

$$ F = qvB \Rightarrow \frac{mv^2}{r}=qvB \Rightarrow r = \frac{mv}{qB} $$

which is the radius of the curvature of the path of a negative charge. The period

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

If the velocity is not perpendicular to the field, then we have a helical motion.

The $v_{perp}=v\sin\theta$ provides the circular motion and $v_{para}=v\cos\theta$ provides the forward motion resulting in a pitch $p=v_{para}T$.

Electric current $I=neAv_d$.

The magnetic force on a single charge is $e\vec{v}_d\times \vec{B}$. Total force on charges within the volume $nAdl$ would then be

$$ d\vec{F} = nAdle \vec{v}_d\times \vec{B}. $$

If $d\vec{l}$ is a vector pointing in the direction of $\vec{v}_d$, then

$$ d\vec{F} = neAv_d d\vec{l}\times \vec{B} = I d\vec{l} \times \vec{B}. $$

If the wire is straight and the field is uniform,

$$ \vec{F} = I \vec{l} \times \vec{B}. $$

Apply $\vec{F} = I\vec{l}\times \vec{B}$ on 4 sides.