It is an experimental fact that moving electric charges generate magnetic fields in all of space. When we observe a magnetic field, we know that is often due to some charge or collection of charges that are moving relative to our location in space (unless it’s due to a changing electric field as we will see soon).

The model of the magnetic field generate by a moving point charge is a bit more complicated than that for the electric field of a point charge because we observe that the magnetic field points perpendicular to the velocity vector and relation position vector,

$$\vec{B} = \dfrac{\mu_0}{4\pi} \dfrac{q \vec{v} \times \hat{r}}{r^2}$$

From this model, we can generate the magnetic field for a number of distributions of moving charges: line currents, rings of current, solenoids, and permanent magnets — for which the story is a bit more complicated than the others. The magnetic field, much like the electric field, superposes, so we can just add up the contributions to little chunks of moving charge (chunks of current) to find the field:

$$\vec{B} = \dfrac{\mu_0}{4 \pi} \int \dfrac{I d\vec{l} \times \hat{r}}{r^2}$$

This is called the Biot-Savart Law, but is really just an expression for superposition of the magnetic field. Later we will find that the pattern of the magnetic field in some cases suggests a short cut to finding the magnetic field that doesn’t involve superposition integrals.

Magnetic fields can exert forces on moving charges, but these forces are always perpendicular to the motion of said charges. The model for a single point charge in the presence of a magnetic field is:

$$\vec{F} = q \vec{v} \times \vec{B}$$

Which shows that the force is both perpendicular to the motion of the charge ($\vec{v}$) and the local magnetic field in which the charge is moving ($\vec{B}$). By superposing the forces on little chunks of moving charge in a wire, we can determine how a magnetic field will generate a force on a wire. We derived this model:

$$\vec{dF} = I d\vec{l} \times \vec{B} \longrightarrow \vec{F} = \int I d\vec{l} \times \vec{B}$$

Typically, the most common experience people have with magnetic force comes from permanent magnets, where the story is more complicated. We haven’t gotten into the mathematical model for how forces in that situation work because while superposition continues to work, it means computing every magnetic field generated by every domain in one magnet and determining the force exerted on each domain of the other magnet and adding that all up!

The definition of magnetic force shows us that magnetic forces cannot change the kinetic energy of particles - the force is always perpendicular to the motion, so magnetic fields can not do work! Magnetic forces are used to change the trajectory of moving charges without adding energy to them. This is very useful in applications like mass spectroscopy.

However, this leads to a little bit of a difference between electric and magnetic fields, which is that there’s no such thing as a scalar magnetic potential (unlike electric potential). This doesn’t mean there’s no such thing as magnetic energy - there most definitely is, but it is a more complex and abstract idea than we had for electric fields. In fact, because there are no magnetic monopoles, we can define a vector potential for magnetic field, which carries with it some information related to energy, but that is beyond the scope of this course.