Magnetic Force - Yousef's Notes

The magnetic force is a fundamental interaction between moving charged particles. The magnetic force applied on $q_1$ by $q_2$ with velocities $\vec v_1$ and $\vec v_2$ respectively is $$ \vec{F}_{1}=\frac{\mu_{0}}{4 \pi} \frac{q_1 q_{2}}{r^{2}} \vec{v}_1 \times\left(\vec{v}_{2} \times \frac{\vec{r}_{12}}{r}\right) $$ $$ \vec{r}_{12}=\vec{r}_{1}-\vec{r}_{2} $$ $$ r=\|\vec r_{12}\| $$

$\frac{\mu_{0}}{4 \pi}=10^{-7} \mathrm{N} \cdot \mathrm{s}^{2} / \mathrm{C}^{2}$.
$\mu_0$ is the magnetic permeability of vacuum (universal constant). $\mu_{0} =4 \pi \times 10^{-7} \mathrm{N} \cdot \mathrm{s}^{2} / \mathrm{C}^{2}=4 \pi \times 10^{-7} \mathrm{Wb} \mathrm{A} \cdot \mathrm{m}=4 \pi \times 10^{-7} \mathrm{T} \cdot \mathrm{m} / \mathrm{A}$
one can prove that $\vec F_1=-\vec F_2$ (Newton’s third law)

Recall cross product of vectors![[Pasted image 20240306172152.png]]

#Magnetic Force Between Circuits

Force acting on circuit 2 caused by circuit 1

$$ \mathbf{F}_{2}=\frac{\mu_{0}}{4 \pi} I_{1} I_{2} \oint_{1} \oint_{2} \frac{d \vec{l}_{2} \times\left[d \vec{l}_{1} \times\left(\vec{r}_{2}-\vec{r}_{1}\right)\right]}{\left|\vec{r}_{2}-\vec{r}_{1}\right|^{3}} $$ ![[Pasted image 20240306172233.png]] $\oint$ is the integral along the circuit (closed loop); $d \vec{l}$ is an infinitesimal length of wire, with direction given by the current density. The previous expression simplifies when applied to two long parallel wires. The force per length of wire $L$ is simply $$ \frac{F}{L}=\frac{\mu_{0} I I^{\prime}}{2 \pi r} $$