July 26, 2022

Electrostatics is the branch of electrodynamics that concerns itself with the study of charge at rest. Charge in itself is a hard thing to describe. The usual definition of it is as follows, “Charge is a fundamental property of matter, like charges repel each other while unlike attract each other. Charge is a conserved quantity both globally (the net charge remains the same for the whole universe…) and locally (…but this does not mean that it can teleport from one place to another). Charge is quantized”. These properties can be summarized as follows

$$ \vec F=\frac1{4\pi}\frac{qq'}{r^2}\hat r \tag{Force Law} $$

$$ \vec\nabla\cdot\vec J+\frac{\partial\rho}{\partial t}=0 \tag*{(Conservation of Charge)} $$

$$ Q=ne,\, n\in \Z \tag*{(Quantization of Charge)} $$

We will learn much more about these equations in due course.

Though of course there nothing is wrong with this definition, it's just that it does not bring us any closer to the abstraction of the concept of charge. And thus since I myself don’t have a definition of charge I will not try to give one.

It is important to appreciate that charge is a conserved quantity not just because it makes electrodynamics easier but also because it fundamentally defines how our universe will behave. The ratio of repulsion due to the electric charge between two electrons and the attraction between them due to their masses is of order $10^{42}$. Thus it becomes plausible to ignore the gravitational force altogether of electrons and indeed for any charged particles in general. However clearly this is not the case, after all, it is quite hard to experience the effects of electrical forces in our daily lives on there own (well ignoring the things that were running in the background like our gadgets). The reason for this is that the effects of the charges can be canceled to a great approximation due to two types of charges being there. The positive repels other positives from it as far as possible while attracting the negative ones, in such a manner that they effectively cancel out. But this isn't the case with gravity, which is always additive in nature, thus its effects pile up enough to keep us fixed on our chairs on a rotating planet. If there was a charge imbalance then the repulsive forces would never allow the matter to collapse into huge concentrations like stars. But since stars exist in our universe this balance must be very delicate, in fact, most physicists believe that it must be perfect such that the net charge of the universe is zero. Also, the charge has to be conserved, that is if by some process the net charge of the universe increased, no matter how slowly, it would be enough to show its effects on cosmic scales, but since there are no effects we propose that charge is conserved, as verified by every experiment done till date.

We are not dealing here with the electrical phenomenon happening at the atomic scales, that is the phenomenon that causes the effects of electrical forces to cancel out on large scales, but instead are interested in what happens when this balance is disturbed. That is we are going to study Classical Electrodynamics and not Quantum Electrodynamics.

The fundamental relation in whole electrodynamics (yes electrodynamics, not just electrostatics, as we will see in chapter 4) is the expression of Coulomb’s law

$$ \vec F=\frac1{4\pi}\frac{qq'}{ r^3}\vec r\equiv\frac1{4\pi}\frac{qq'}{r^2}\hat r \tag{1.1} $$

Here $\vec F$ is the force between two point charges $q$ and $q'$, and $\vec r$ is the displacement vector between them. The unit system used in the whole section of electrodynamics is the Heaviside-Lorentz system.

One another fact that we have to take at face value is that Coulomb’s law follows the principle of superposition. This principle states that the force caused by one particle is not altered by the introduction of the third particle. This principle is not to be treated as equivalent to the law of vector addition. Consider for example that due to some property $K$, the force between the two particles is given by $\vec F=K_1K_2\vec r_{12}$, where $\vec r_{12}$ is the displacement vector between them. Now introduce another particle in the system and remove the second one, this would give us a force of $\vec F= K_1K_3\vec r_{13}$. Next, if we introduce both of the particles at the same time and get a force of the form $\vec F=K_1K_2\vec r_{12}+K_1K_3\vec r_{13}+K_2K_3\vec r_{23}$, then clearly this force does not follow the principle of superposition due to the extra term of $K_2K_3\vec r_{23}$. The additional term is caused solely due to all particles being in the system at the same time. Thus even though vector addition is followed, the principle of superposition is not. Therefore in short if we have $n$ charges each applying a force on some charge $q_1$, then according to Coulomb’s Law and the principle of superposition we get

$$ \vec F=\frac1{4\pi}\bigg(\frac{q_1q_2}{r_{12}^2}\hat r_{12} +\frac{q_1q_3}{r_{13}^2}\hat r_{13}+...+\frac{q_1q_n}{r_{1n}^2}\hat r_{1n}\bigg) \\ ‎\\ \vec F=\frac1{4\pi}q_1\sum_{i\neq 1}^n\frac{q_i}{r_{1i}^2} \hat r_{1i} \tag{1.2} $$