Having gone through the history of the development of magnetism, we may now begin our quantitative analysis of magnetic fields and forces. The study of magnetic fields is quite complicated, much more complicated than that of electric fields, and requires more effort to fully describe. The largest challenge regarding magnetic fields is the absence of magnetic charge. With electric fields, we were able to describe the field created in terms of a single charge, then generalize for a number of charges. With magnetic field, there are multiple sources of fields, not a single indivisible quantity, such as charge.
As such, we must begin by defining the forces felt by moving particles in magnetic fields without defining the source of the field. Don't worry in this section if you find the situations a little hard to grasp conceptually. We are working with an incomplete picture of magnetic fields, something that will be remedied in the next SparkNote. After defining forces felt by moving particles, we generalize our concepts for many moving charges (currents).
This SparkNote puts in place the first piece of the puzzle of magnetic fields. From the definitions we here derive we will be able to move on to describe not only the forces from magnetic fields, buy the sources of the fields.