Matter is charged and when it is as rest, it generates an electric field. Set that charge into motion and you have a new observation: Moving charges generate magnetic field. This field, originally thought to be completely different from the electric field, has its own form of interaction. Like the electric field, the magnetic field also permeates all of space, and it gets weaker as you are further from the source. The magnetic field due to some moving charge has a magnitude and direction at every point in space.

Our introduction to the magnetic field started with a single moving charge. We observe the magnetic field produced by a single moving charge to be,

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

where the vector $\vec{r}$ is still the separation vector between the location of the moving charge at a given time and the observation location at the same time – provided that the charge is moving much more slowly than the speed of light. Again, we cannot derive this equation, it is a model of the point charge magnetic field that fits the data well. Like the electric field of a point charge, this is the starting point for magnetostatics (and, later, electromagnetism, in general).

If another moving charge is brought into the scene, it will interact with the first moving charge through the magnetic field that the first charge generates. (It will also experience the electric force, but we often limit our discussion at first to the isolated magnetic interaction.) This push or pull that the new moving charge experiences is directly related to the cross product of the velocity of that charge and the magnetic field of the source charge. This interaction is simply,

$$\vec{F}_{B} = q_{test}\vec{v}_{test}\times\vec{B}.$$

The direction of the force is determined by the right hand rule and is perpendicular to the plane defined by the vector velocity and magnetic field. This means that a charge will experience no magnetic force if it travels directly along a magnetic field line (or opposite it).

A single moving charge is certainly not the only kind of situation that we encounter. In fact, it's quite often that a collection of charges are moving – forming a current. This collection of moving charges also generate a magnetic field, and, similar to the electric field, the magnetic field obeys superposition. The basic premise is quite similar to the for electric fields,

$$\vec{B}_{net} = \vec{B}_1 + \vec{B}_2 + \vec{B}_3 + \dots = \sum_i \vec{B}_i$$

This result tells us how the total magnetic field due to a distribution of moving charges works out, but what about when there's a true current ($I$)? We can treat each little segment of the wire in a way that asks: how much do you contribute to the magnetic field? When we do this we get a little contribution from each segment (length $dl$) and add them up using an integral,

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