Coulomb's Law and the Definition of Electric Field

Coulomb's Law Revisited

Our study of electrostatics began with Coulomb's law:

where r is the distance between charges q1 and q. We considered F to be the force between the charges, and realized that, by Newton's third law, the force on charge q1 was equal and opposite to the force on charge q. F here is a scalar; it is the magnitude of the force.

All forces are vectors, of course, and the electric force is no exception. Our scalar equation does not seem to assign a direction to the force, but we can do that by remembering that opposite charges attract, and like charges repel.

When would you calculate the force to be negative? Physically, what does this mean?

Definition of Electric Field

Now let's consider the vector F to be the force on q due to q1. We change the equation slightly:

where is a unit vector pointing from q1 to q2. We define the quantity

to be the electric field due to q1 and can now write F = qE.

If there are several charges, then to find the force on charge q, we add together (as vectors!) the forces produced by all the other charges:

where ri is the distance from qi to q. Again we can write this as

where the sum is a vector sum.

What if we don't have a bunch of discrete charges qi, but instead have a continuous distribution of charge? We can break up the source distribution into many small pieces of charge, and add up all their contributions to the electric field. In the language of calculus, we are integrating:

where r is the distance from the piece dq to the point where the field is measured.

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