Magnetic Field

Biot-Savart Law

If the current is given, we use this form $d\vec{B}=\frac{{\mu }_0}{4\pi }\frac{Id\vec{l}\times \hat{r}}{r^2}$

We can then integrate $d\vec{B}$ to find $\vec{B}$.

• $\hat{r}$ is a unit vector that always points from A (the charge) to P (where we are measuring the magnetic field)

Approach, pretty straightforward. The calculations are similar for Electric Field Calculations

1. Find the current $I$
2. Find $|d\vec{l}\times \hat{r}|$, see Cross Product
3. Find $r$

Biot-Savart for moving charges

If velocity is given, we replace $I=\frac{dq}{dt}$ we use this form $d\vec{B}=\frac{{\mu }_0}{4\pi }dq\frac{\vec{v}\times \hat{r}}{r^2}$

More details

To master this, you really just need to do lots of exercises.

Notice the placement of $\theta$ in these problems. You tend to place the angle $\theta$ next to $dB$ and then think of ways, but make your life easier by placing theta like in the picture. Then say, for the z axis, it’s using a $\cos$, so you can just do the opposite, therefore $d{B}_z=dB\sin \theta$

Solving Solenoid Problem