From the long perspective of human history, for example, ten thousand years from now, there is no doubt that the most important event of the 19th century will be Maxwell's discovery of the laws of electrodynamics. The American Civil War will be insignificant compared to this important scientific event of the same year. —Richard Feynman, Volume II; Lecture 1, Electromagnetism.
Partial differential equations play a huge role in theoretical and mathematical physics. It all began in the 19th century, when the wave theory of light was established with the observation and analysis of good experimental results. Here, for the first time, partial differential equations emerge as a natural expression of fundamental physical reality. A new era of theoretical physics began with Maxwell, Faraday, and Hertz. In this revolution in physics, Maxwell was the absolute leader. Maxwell expressed in his differential equations the whole idea of light and electromagnetic phenomena at the time. Maxwell showed that electric and magnetic fields are actually dependent variables, redefining previous concepts of electric and magnetic fields.
We can say this: before Maxwell, people regarded physical reality (in terms of what it was thought to represent events in nature) as points of matter whose changes consisted entirely of motion and were subject to differential equations. After Maxwell, physical reality was seen as represented by continuous fields that were mechanically unsolvable and subject to partial differential equations. This change in the concept of reality is the most profound and productive change in the physics community since Newton. —A. Einstein's speech on the 100th anniversary of Maxwell's birth. Published in 1931.
Faced with faraday's experimental results, as well as the theory of other early eminent physicists, such as André-Marie-Ampère, Maxwell was puzzled by the mathematical form of the equations for electric and magnetic fields and made some modifications to their equations with extraordinary insight. Today, Maxwell's theory can be summarized by four equations. But his formula takes the form of 20 joint cubic equations, containing 20 variables. The dimensional parts of its equation (X, Y, and Z directions) must be listed separately. He also used some counterintuitive variables.
Here, E and B and J describe the vector fields of electric field intensity, magnetic flux density, and current density, respectively, ρ describe charge density, D is the electrical displacement, H represents the magnetic field strength, and t is time.
Maxwell's first equation was:
By integrating on an arbitrary volume V, we get:
But according to Gauss's theorem, we get:
Here, q is the net charge contained in volume V. S is the surface that surrounds volume V. Maxwell's first equation is represented. The total electrical displacement by the surface that surrounds the volume is equal to the total charge within that volume.
Maxwell's second equation is:
Using Gaussian divergence theorem to turn the volume integral into a surface integral, we get:
Maxwell's second equation states that the total outward magnetic induction flux B through any enclosed surface S is equal to zero.
Maxwell's third equation is:
According to Stoke's theorem, converting the surface integral on the left-hand side into a linear integral, we get:
Maxwell's third equation marks it. The electromotive force around the closed path (e.m.f. e = ∫c E.dI) is equal to the negative rate of change of the magnetic flux connected to the path (since the magnetic flux Φ = ∫s B.dS).
Maxwell's fourth equation is:
On the surface S bounded by curve C, the curved area fraction is obtained:
Using Stoke's theorem to convert the surface integral on L.H.S. in the above equation into a linear integral, we get:
Maxwell's fourth equation states that the magnetic force around a closed path is equal to the conducted and displacement currents of any surface that passes through the boundary of that path.
The general equation is then applied to cases where magnetic perturbations propagate through non-conductive fields, which show that the only perturbations that can propagate in this way are those that are transverse in the direction of propagation at a speed of v, which was discovered from Weber's experiments and represents the number of electrostatic units contained in an electromagnetic unit. This speed is so close to the speed of light that we seem to have good reason to conclude that light itself (including radiant heat and other radiation, if any) is an electromagnetic perturbation propagating through an electromagnetic field in the form of waves. — James Clark Maxwell, Dynamic Theory of Electromagnetic Fields (1864), Introduction.
Original manuscript of Maxwell's seminal paper photograph. Royal Society
Maxwell's equations are very similar to Hamilton's equations in that they show the rate of change of correlation quantities over time, their values at any given time, and in Maxwell's equations, these quantities are electric and magnetic fields. However, there is an important difference between Maxwell's equations and Hamilton's equations. Maxwell's equations are field equations, while Hamilton's equations are particle equations, which means that in Maxwell's propositions, describing the state of the system requires an infinite number of parameters, while for Hamilton's propositions a finite number of parameters (three-point coordinates) is required.
Maxwell's equations of electromagnetism describe how electric and magnetic fields arise from charges and currents, how they propagate, and how they affect each other. Not only are these equations significant in helping to formulate and interpret theories and multiple areas of mathematical physics, but together with the Lorentz force, they quantify most of the physical processes we experience in our daily lives.
Maxwell's equation outside the University of Warsaw, Poland.
Maxwell's equation can be considered one of the fundamental pillars of quantum mechanics and modern physics because it explains well the fact that the propagation of light does not require a medium. In the 19th century, theoretical physicists realized that There was some solution to Maxwell's equations, in which both an electric and magnetic field could exist simultaneously without charge. This solution is an oscillating traveling wave that moves at a speed of 299792458 meters per second. Some experiments conducted later showed that the light itself also moved at exactly the same speed. It's not a coincidence, they're the same thing. Obviously, light is not a magical entity, and we can create light by manipulating the charge. This has led to the creation of artificial light sources such as radios (radio waves are a low-energy form of light), lasers, and synchrotrons. Maxwell's theory of electromagnetism produced nothing more than the idea that energy could be transmitted from one place to another through electromagnetic waves, which proved to be surprisingly fascinating and extremely thought-provoking in the physics community.
Statue of James Clark Maxwell, erected on George Street, Edinburgh in 2008.
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