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Having established this equation, let us take a moment to analyze its implications. First, it is clear that a charge moving parallel to the magnetic field experiences no force, as the cross product is zero. Second, the magnitude of the force on the charge varies directly not only with the magnitude of the charge, but of the velocity as well. The faster a charged particle travels, the more force it will feel in the presence of a given magnetic field.

This equation forms a basis for our study of electromagnetism. From it we will be able to derive the fields created by various wires and magnets, and derive some properties of the magnetic field.

### Relating Magnetic and Electric Forces

Using the definition of the magnetic field we have just developed, we become able to generate a complete expression for the force exerted on a charged particle, q, in the presence of both electric and magnetic fields. Recall that in the presence of an electric field alone the force felt by a point charge q is simply proportional to the field at that point, or F = qE. Thus, if this point charge is in the presence of both an electric field and a magnetic field, we can find the total force on the charge by simple vector addition: = q + This equation only applies to vector quantities--usually the force due to the electric field and the magnetic field are not in the same direction, and cannot be added algebraically.