Coulomb's Law and Superposition Principle

1. Coulomb's Law: The Force Between Charges

The French physicist Charles-Augustin de Coulomb experimentally determined the law governing the force between two stationary point charges in the 1780s. In its simplest (scalar) form, it states that the magnitude of the force $F$ is directly proportional to the product of the magnitudes of the charges ($|q_1|$ and $|q_2|$) and inversely proportional to the square of the distance $r$ between them.

The constant $k$, known as Coulomb's constant, is approximately $8.99\times 10^9\text{N}\cdot {\text{m}}^2/{\text{C}}^2$.

While this formula gives the magnitude, it doesn't specify the direction. Force is a vector, and a complete description requires vector notation. This is where the vector form of Coulomb's Law becomes indispensable.

The Vector Form

To express Coulomb's Law as a vector, we need to define the positions of our charges. Let's say charge $q_1$ is at a position given by the vector ${\vec{r}}_1$ and charge $q_2$ is at position ${\vec{r}}_2$, both relative to some origin. The force exerted by $q_1$ on $q_2$, which we denote as ${\vec{F}}_{12}$, depends on the displacement vector pointing from $q_1$ to $q_2$, which is ${\vec{r}}_{12}={\vec{r}}_2-{\vec{r}}_1$.

The vector form of Coulomb's Law is:

Here, $r_{12}$ is the magnitude of the displacement vector (the distance), and ${\hat{r}}_{12}$ is the unit vector that points from $q_1$ to $q_2$. It's defined as ${\hat{r}}_{12}=\frac{{\vec{r}}_{12}}{|{\vec{r}}_{12}|}$.

One of the most elegant aspects of this formulation is that the signs of the charges automatically determine the correct direction.

• If $q_1$ and $q_2$ have the same sign (both positive or both negative), their product $q_1{q}_2$ is positive. The force ${\vec{F}}_{12}$ is then parallel to ${\hat{r}}_{12}$, pointing away from $q_1$. This is repulsion.
• If they have opposite signs, $q_1{q}_2$ is negative. The force ${\vec{F}}_{12}$ is antiparallel to ${\hat{r}}_{12}$, pointing back toward $q_1$. This is attraction.

The following video provides an excellent animated breakdown of how to construct the vector form of Coulomb's Law from these principles.

As the video shows, substituting the definition of the unit vector gives an alternative and very useful form:

This form is often more convenient for calculations, as you can work directly with the components of the displacement vector ${\vec{r}}_{12}$. Also, note that by Newton's third law, the force exerted by $q_2$ on $q_1$ is simply ${\vec{F}}_{21}=-{\vec{F}}_{12}$.

2. The Principle of Superposition

Coulomb's Law describes the interaction between two point charges. But what happens if we have three, four, or a million charges? The answer lies in the Principle of Superposition.

The principle states that the net electrostatic force on any given charge in a system of charges is the vector sum of the individual forces exerted on it by all the other charges.

Crucially, the force between any two charges ($q_i$ and $q_0$) is calculated using Coulomb's Law as if the other charges were not there. This is a remarkable feature of the electric force.

To solidify your understanding of the formal definitions of both Coulomb's Law and the superposition principle, please review the following text.

3. Problem Solving with Superposition

The superposition principle provides a clear strategy for calculating the net force on a charge. Since you are comfortable with vector algebra, this process should feel systematic.

The Strategy:

1. Draw a Diagram: Sketch the positions of all charges and label them. Identify the charge on which you want to find the net force.
2. Calculate Individual Forces: For each other charge in the system, determine the force vector it exerts on your target charge using the vector form of Coulomb's Law. It's often easiest to find the magnitude first, then determine the direction (or vector components).
3. Decompose into Components: Resolve each individual force vector into its x, y, and (if necessary) z components. This is the crucial step where you will use trigonometry.
4. Sum the Components: Add the x-components of all the force vectors to get the x-component of the net force ($F_{\text{net},x}$). Do the same for the y- and z-components.
5. Find the Resultant: Combine the components of the net force to find its final magnitude (using the Pythagorean theorem) and direction (using the arctangent function).

The following video provides an excellent, detailed walkthrough of this entire process for a common exam-style problem.

To challenge yourself further, you can apply this method to more complex geometries. The following reading provides worked examples for different arrangements of charges.

Conclusion

In this lesson, we have developed the tools to precisely calculate the forces within a system of static charges. The key takeaways are:

• Coulomb's Law in vector form (${\vec{F}}_{12}=k\frac{q_1{q}_2}{r_{12}^2}{\hat{r}}_{12}$) provides a complete description of the electric force, automatically handling both magnitude and direction.
• The Superposition Principle allows us to find the net force on a charge from a group of other charges by simply taking the vector sum of the individual forces.
• The practical method for solving these problems involves a systematic process of decomposing each force vector into components, summing the components, and then reconstructing the resultant vector.

We have described the electric force as an "action at a distance." In our next lesson, we will introduce a more powerful and profound concept: the electric field. We will see that charges don't interact directly; instead, they create an electric field in the space around them, and it is this field that exerts a force on other charges. This shift in perspective will be crucial for understanding all subsequent topics in electricity and magnetism.