The nature of magnetic interactions
The electric force is mathematically similar to the gravitational force that you studied in Physics 231. For small objects ("point particles") the force depends on an intrinsic property of the objects (q1, q2) and on the distance between them (1/r2). The electric force (Coulomb's Law) is described by the electric field that is created by one particle (k q1 / r2) and felt (F = q2 E) by the other.
Magnetic forces can be produced by electric charges. The situation is complicated because these forces depend not only on the electric charges of the objects and the distance between them, but also on their velocities. The dependence on velocity is total: only moving charges create magnetic fields, and only moving charges feel magnetic forces as a result of those fields. The motion is not always visible, e.g., the force that sticks a permanent magnet to a refrigerator door is caused by the intrinsic "spin" motion of electrons in the atoms of the magnet and the door.
Magnetic forces are the foundation of electric motors and generators. They are also used to store information in tape recordings and computer hard drives, and to focus and steer beams of charged particles such as the electrons in the Cathode Ray Tube (CRT) of a TV or computer monitor.
The dependence of magnetic forces on two velocities makes them complicated to visualize. This is particularly true because even simple examples necessarily involve vectors in all three dimensions of space. From a theoretical standpoint, the dependence on velocities makes magnetic forces vary with the velocity of the observer who is observing them. Understanding how the description of electric and magnetic forces can depend on the reference frame of the observer requires Einstein's special relativity, which will be described later in the course.