Electric Force between Two Point Charges

To address the electric forces among electrically charged particles, first consider two particles with electric charges q 1 q 1 and q 2 q 2 , separated in empty space by a distance r r. The electric force between such particles was experimentally determined by Charles Augustin de Coulomb (1736 - 1806). The result he obtained is called the Coulomb law, and is written here in a more modern vector notation that was not known to Coulomb.

The symbol k k in this equation is called the Coulomb constant, and has the numerical value k ≈ 8.988 × 10 9 ⁢ N · m 2 C 2 k ≈ 8.988 × 10 9 ⁢ N · m 2 C 2

It's important to understand that we have not derived Equation (1) from any more basic principle. The Coulomb law is an experimental result. Coulomb asked nature how the electric force works (by doing experiments to measure this force), and nature replied with experimental results leading to Equation (1). From our point of view in introductory physics, and from Coulomb's point of view, Equation (1) is an empirical result.

In using this equation, it is important to have clearly in mind the meaning of the symbols that appear in it, especially that of the unit vector r ^ r ^ . In the basic relationships of electricity and magnetism, the unit vectors that appear always point from the location of source points to the location of field points. Field points are where your calculations apply. Source points locate the electric charges (or currents) used in your calculation. If we use Equation (1) to find the force on the charge q 2 q 2 due to the presence of the charge q 1 q 1 , then the location of q 2 q 2 is the field point, the location of q 1 q 1 is the source point, and r ^ r ^ points away from q 1 q 1 toward q 2 q 2 . If we choose instead to calculate the force on q 1 q 1 , we reverse our choice of field and source points, and the unit vector r ^ r ^ points away from q 2 q 2 toward q 1 q 1 . In the basic equations, distances (such as r r in Equation (1)) always mean distances from source points to field points.

So we can use Equation (1) whenever there are two and only two point charges, only one of which is a source charge. Thus this equation works in the situation diagramed below.

Figure 1

Now we plug this situation into Equation (1), where we calculate the force on the charge q 2 q 2 (so that r ⇀ 1 r ⇀ 1 is the source point and r ⇀ 2 r ⇀ 2 is the field point). Note that the displacement vector from the source q 1 q 1 to the field point at q 2 q 2 is ( r ⇀ 2 - r ⇀ 1 ) ( r ⇀ 2 - r ⇀ 1 ) , so the distance squared is r 2 = ( r ⇀ 2 - r ⇀ 1 ) · ( r ⇀ 2 - r ⇀ 1 ) = ∥ ( r ⇀ 2 - r ⇀ 1 ) ∥ 2 r 2 = ( r ⇀ 2 - r ⇀ 1 ) · ( r ⇀ 2 - r ⇀ 1 ) = ∥ ( r ⇀ 2 - r ⇀ 1 ) ∥ 2 and the unit vector is r ^ = ( r ⇀ 2 - r ⇀ 1 ) ∥ ( r ⇀ 2 - r ⇀ 1 ) ∥ r ^ = ( r ⇀ 2 - r ⇀ 1 ) ∥ ( r ⇀ 2 - r ⇀ 1 ) ∥ Substitution results in the following.

In this equation, we added a subscript "2" on the force to make explicit that this is the force acting on the point charge q 2 q 2 .

We can simplify in this case just by being smart in choosing our coordinate system. If we place the origin on the source charge, we make r ⇀ 1 = 0 r ⇀ 1 = 0 , and if we choose a coordinate axis, say the x x axis, to run through the position of the field charge, then we make r ^ = x ^ r ^ = x ^ and ( r ⇀ 2 - r ⇀ 1 ) = r ⁢ x ^ ( r ⇀ 2 - r ⇀ 1 ) = r ⁢ x ^ . With this coordinate system, Equation (1) becomes

Graphically, we have arranged the coordinate system so that the application looks like this:

Figure 2

Note that the choice of the coordinate system used makes absolutely no difference in the problem or its solution. It only changes the way the problem and its solution are stated. Clearly, the second choice of coordinates also makes the mathematics easier, so it is the obvious better choice.