Maxwell’s Equations

Based on the studies of Michael Faraday, Maxwell unified, in 1864, all observable electrical and magnetic phenomena in a work that established connections between the various theories of the time, deriving one of the most elegant theories ever formulated.

Maxwell demonstrated, with this new theory, that all electrical and magnetic phenomena could be described in just four equations, now known as Maxwell’s equations.
These are the basic equations for electromagnetism, just as the law of universal gravitation and Newton’s three laws are fundamental to Classical Mechanics.
The mathematical deductions of Maxwell’s equations will not be presented in this article, since they require knowledge of Differential and Integral Calculus, which is only fully studied in higher education courses.

Maxwell’s equations for electromagnetism consist of the unification between Gauss’s Laws for electricity and magnetism, generalized Ampere’s Law and Faraday’s Law for electromagnetic induction.

Then follow Maxwell’s equations:

1) Gauss’ law for electricity:

This is the first of Maxwell’s four equations, originally proposed by the German mathematician Carl Friedrich Gauss (1777-1855), it is the equivalent of Coulomb’s law in static situations. It relates electric fields and their sources, electric charges,

2) Gauss’s law for magnetism:

This law is equivalent to the first, but applicable to magnetic fields and also showing the non-existence of magnetic monopoles (there is no isolated south or north pole). According to this law, magnetic field lines are continuous, unlike lines of force in an electric field that originate at positive electric charges and end at negative electric charges.

3) Ampere’s Law:

Ampere’s law describes the relationship between a magnetic field and the electric current that causes it. It states that a magnetic field is always produced by an electric current or a changing electric field.This second way of obtaining a magnetic field was predicted by Maxwell himself, based on the symmetry of nature: if a changing magnetic field induces an electric current, and consequently an electric field, then a changing electric field must induce a magnetic field.

4) Faraday’s Law:

The fourth of Maxwell’s equations describes the characteristics of the electric field giving rise to a variable magnetic flux. The generated magnetic fields are time-varying, thus generating rotational electric fields.

Until the end of the 19th century, it was believed that with these equations there was nothing more to be discovered in physics. However, in 1900, Max Planck initiated the so-called quantum physics, with his postulates about blackbody radiation.

In 1905, Albert Einstein revolutionized once and for all the knowledge of science, launching the Theory of Relativity and the Photoelectric Effect, paving the way for the greatest scientific development in history.
Maxwell’s equations are considered the final milestone of what we call Classical Mechanics.
Maxwell was the first physicist to find the speed of electromagnetic waves through mathematical calculations, all thanks to his famous equations.

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