A charge moving through a magnetic field experiences a force
$$ \vec F=q(\vec v\times \vec B) $$
$$ 1 \text{ Tesla}=1\text{ NC}^{-1}\text{m}^{-1}\text{s} $$
If there is a combination of E-field and B-field then:
$$ \vec F=q(\vec E+\vec v\times \vec B) $$
| $q(\vec v\times\vec B)$ | $q\vec E$ |
|---|---|
| Vector | Vector |
| Is only non-zero if the charge is moving | Acts whether the charge moves or not |
| Acts in a direction $\perp$ to $\vec v$ and $\vec B$ | Acts in a direction $\parallel$ to $\vec E$ |
Consider a charge $q$ moving with a velocity $\vec v$ in a field $\vec B$ we get:
$$ \begin{aligned} F&=Bqv_\perp=\frac{mv_\perp^2}{r} \\ &\Rightarrow r=\frac{mv_\perp}{qB} \end{aligned} $$
where $v_\perp$ is part of $\vec v$ that is $\perp$ to $\vec B$
If $\vec I \perp \vec B \Rightarrow BIl$
Current $I$ due to charges moving
$$ \vec I = nAq \vec v $$
$$ \vec F=q(\vec v\times\vec B) $$
$$ \vec F_\text{wire}=nAlq(\vec v\times \vec B)=l(\vec I\times\vec B) $$
$$ v=\frac{I}{nqA} $$
$$ W_\text{m}=\int\vec F_\text{m}\cdot \vec v\,\text dt=\int q(\vec v\times\vec B)\cdot\vec v\,\text dt=0 $$