The study of electromagnetism begins with the knowledge that there is a fundamental property of particles, called electric charge. Early experiments in electrostatics found that this charge can be negative or positive. Like charges attract while unlike charges repel:

Electrostatics-like charges repel, unlike charges attract

So far so good. But one may ask the question: how much is that attraction(or repulsion) given the position of the two charged particles in question? What if there are more than 2 charges?

These questions are easy to ask, but do not have straight forward answers. For example, let’s say there are 5 charged particles under consideration. For the sake of simplicity we assume that they are all positively charged.

five_charges

Let’s call particle 1 as the test charge. By test charge we mean that we will only consider forces on this particle due to other particles. Of course, if we can find force on this particle, we can follow similar procedure to find forces on others. The first assumption(rather obvious) is that this particle does not exert force on itself. Hence, we need to consider 4 forces, one each from rest of the particles. We call them as source charges collectively.

five_charges1

The fundamental problem of Electromagnetism is that given the time dependent positions of some source charges, calculate the trajectory(path followed) by the test charge.

Principle of Superposition

The first problem is, we do not know how the charges affect each other and hence we do not know whether they interfere with each other. In other words, does the force applied by charge 2 on 1 changes depending on the presence of other charges?

It is as if two friends were talking on the phone and a mutual friend calls one of them. If they like, they can take him in and start a conference call. In that case, conversation among them depends on all three of them. If, however, the third person can communicate only through email, then he won’t get the gist of what other two are talking about, and the one on the phone won’t get what the other two are emailing about. However, the one who’s connected to them both will interact with them both, separately. In the end, he has information from the friend on phone as well as from the friend on email.

For charges, we observe experimentally that it is the latter case. The presence of other charges does not affect the individual interaction between any two charges. However, the net force felt by the test charge is the sum total of (vectorially) the forces applied by all the source charges. This principle is known as the principle of superposition and is expressed as:

Electrostatics-principle of Superposition

(F12 = Force on charge 1 due to charge 2)

Okay, so that simplifies our task greatly, we simply compute the forces between every source charge and the test charge individually and add them vectorially.


It sounds good in principle and is actually very helpful but the task is complicated by another experimental observation: the force between any two charge depends on the distance between them as well as their velocities and acceleration. But since charges are moving under the influence of each other, we can’t pinpoint the actual velocities, positions and accelerations for all the charges at a given instant of time.

Even if we could, the problem is that the electric force is not instantaneous, because the interaction is mediated by electromagnetic waves, which take some time to arrive from one charge to the other, again depending on their separation and thus, we need to know the position and velocity of the source charge when the electromagnetic wave left it to interact with the test charge.

Coulomb’s Law

Given these three problems, the simple task of answering the question of electric force experienced by a test charge by virtue of some source charges seems hopeless. However, there are certain limits to which we can predict. For example, if the source charges are held in place somehow and are not too distant from the test charge, the interactions can be approximated as instantaneous because electromagnetic waves travel at very high speed, the speed of light. These limits are usually within our everyday experiences and if the velocity factor goes away (because source charges are stationary), we find that the interaction is a simple inverse square relationship which also depends on the amount of charges:

Electrostatics-Coulomb's Law

Thus, Coulomb’s law applies when the source charges are fixed and are relatively close to the test charge. Then we can compute this law to calculate individual forces on test charge due to each of the source charges and add them to get the net force using the principle of superposition:

superposition

All of the domain of Electrostatics deals within these approximations. We assume that only test charge is moving and the distances are within the limits of Newtonian Mechanics. All that remains is then calculating individual forces on the test charge(using Coulomb’s law) and adding them to get net force(using principle of superposition). Once you have the net force and the position(and velocity) of the test charge, you can easily calculate the trajectory using Newtonian Mechanics.

Electrostatics: Coulomb’s Law and Principle of Superposition

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