October 17, 2022

At this point, there are mainly three ways to proceed, first is to take the conventional route and start with Ampere’s experiments and Lorentz force law. The second is to take Maxwell’s equations as postulates and proceed from there. Finally, we can start directly with Coulomb’s law and apply special relativity to it, which gives us Maxwell’s equations in their entirety including time-dependent cases and Lorentz law. With this approach, we can reduce whole Electrodynamics to a single four-tensor equation. But the problem with this approach is that it's pretty sophisticated. It would be much better to follow the conventional approach first, understand the meanings of all terms and equations, and then after this understanding fall back to the relativistic approach. This is what we’ll be doing for this and the next chapter. In the fourth chapter, we will put finishing touches to our basic formulation of electrodynamics.

Ampere’s Experiment:

Ampere after learning about Ørsted’s experiments did some of his own with straight long wires carrying currents. These currents were special in such a way that the charge leaving the battery always equated to the charge entering the battery at each instant. If we create a circuit with a very long wire carrying current $I$ and only restrict our analysis to the case where we are far away from the ends of this wire and other wires in the circuit, then if we bring another straight but short wire carrying another current $I'$ then Ampere found that the force per unit length on the small wire is given by

$$ \vec F=\vec I'\times\vec B\tag{2.1} $$

Here $\vec I'$ points in the same direction as the current in the small wire, while the vector $\vec B$ is called the magnetic field. In cylindrical coordinates, Ampere found out that the $\vec B$ point in the angular direction with the magnitude

$$ B=\frac1c\frac{I}{r} \tag{2.2} $$

Where $c$ is the speed of light while $r$ is the distance from the wire to the point at which the field is calculated.