### Coulomb's Law

Coulomb discovered the mathematical dependence that governs the force acting between  pairs of point charges. This mathematical expression is called Coulomb's Law. Coulomb's Law, as any law of physics, is the result of experimental observations.

 In the diagram, represents the distance between the two charges and . The force is the force on the charge one due to the charge two. Similarly, the force is the force on the charge two due to the charge one. Coulomb's law states that the force between the charges is an action and reaction pair of forces whose magnitude is proportional to the product of the magnitudes of the two charges and inversely proportional to the distance square between the charges. The direction of the forces are along the line going from one charge to the other. These forces point toward the charges when the forces are attractive (opposite charges) and the forces "pull" away the charges when the forces are repulsive (same type of charge).  where   is called Coulomb 's constant and represents the magnitude of the Coulomb's force between the charges. The arrows on top of a symbol indicate that the quantity is a vector and; therefore, the magnitude and direction of this quantity are needed for their full identification. In the case of the top diagram, one of the charges is positive (red) and the other is negative (blue). Therefore, the forces between the charges are attractive. The bottom diagram shows two charges with the same type of charge; in which case, the forces are repulsive. As with any action and reaction pair, the magnitudes of the two forces are the same; the directions of the forces are opposite to each other; and, the two different forces act on different objects (charges). Therefore, on each individual charge, there is a net force acting on it that will make the charge move along the direction pointed by the force. In the electrostatic case, it is assumed that the charges are somehow "nailed" to their location by forces other than electric forces.

### Superposition Principle

The superposition principle for the forces among electric charges expresses that if more than two charges are present, the resultant force acting on one of the charges is the result of the vector addition  of the individual Coulomb forces exerted on this charge by all the other charges.

 The drawing sequence on the left illustrates the use of the superposition principle in electrostatic. The first drawing shows an arbitrary charge distribution . All these charges will interact electrically with each other. Nevertheless, to analyze the problem, the study focuses the attention on one of the individual charges (later the study could focus on one of the other charges). In this case, the fourth charge has been individualized for the study. The second drawing, , emphasizes this by showing the line of action of the forces exerted by the other charges on  . The third drawing shows the actual vectors representing the forces acting on this charge . The size of this vector is related to the magnitude of the Coulomb's force between the individual charges and . For the purpose of these drawings, those vector magnitudes have been chosen arbitrarily and without justification. The fourth drawing, , shows those vectors added using the graphical method of adding vectors by connecting "tip to tails". In addition, the fifth drawing shows the resultant of such vector addition . The final drawing in the sequence shows the original charge distribution with the vector representing the net force acting on the charge . Thus, the sequence represents the graphical addition of . The same procedure can be repeated in order to obtain the net force acting on one of the other charges.

Accordingly with the previous sequence of drawings, in order to obtain the net force on one of the charges (), the following steps should be followed:

1. Use Coulomb's law to calculate the magnitudes of the individual forces.

2. Based on the signs of the charges, determine the direction of the individual forces.

3. Calculate the net force acting on the charge using the vector addition of the individual forces.

How should we modify the previous result if the charge is changed? What calculations need to be re-done?

### Understanding Coulomb's Law

The following questions can help to understand the dependency of Coulomb's law on the different variables.

 1) J Q 29
 In reference to the Figure, if d2 is twice d1, and q1 is the same as q4, the magnitude of the forces exerted by the charge q0 over the charges q1 and q4 compare as

 a) the force in q4 is twice the force on q1. b) the force in q4 is half the force on q1. c) the same for both N d) the force in q4 is one-fourth the force on q1. e) None of the above.

 2) J Q 30
 In reference with the Figure, if d2 is twice d1, and q1 is the same as q4, for the forces exerted by q0 on both q2 and q3 to be the same

 a) the value of the charge q3 must be half the value of the charge q2. b) the value of the charge q3 must be twice the value of the charge q2. c) the value of the charge q3 must be one fourth of the value of the charge q2. N d) the value of the charge q3 must be four times the value of the charge q2. e) None of the above.

 3) J Q 31
 In reference to the Figure, if d2 is twice d1, and all the charges have the same value, the force exerted by q0 in the individual charges is the same for

 a) q1 and q4 N b) q3 and q4 c) q1 and q3 d) q2 and q4 e) None of the above.

 4) J Q 32
 With reference to the Figure, if d2 is twice d1, and if individually q1, q2, q3 are the same as q4; then, the forces exerted by q0 on the other charges are

 a) the same for all of them b) greater for q2 and q3 than for q1 and q4 c) smaller for q1 and q4 than for q2 and q3 N d) greater for q1 and q2 than for q3 and q4 e) None of the above.

 by Luis F. Sáez, Ph. D. Comments and Suggestions: LSaez@dallaswinwin.com