# What is Maxwell's equations: Definition and 245 Discussions

Maxwell's equations are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. They describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. The equations are named after the physicist and mathematician James Clerk Maxwell, who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon.
An important consequence of Maxwell's equations is that they demonstrate how fluctuating electric and magnetic fields propagate at a constant speed (c) in a vacuum. Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum of light from radio waves to gamma rays.
The equations have two major variants. The microscopic equations have universal applicability but are unwieldy for common calculations. They relate the electric and magnetic fields to total charge and total current, including the complicated charges and currents in materials at the atomic scale. The macroscopic equations define two new auxiliary fields that describe the large-scale behaviour of matter without having to consider atomic scale charges and quantum phenomena like spins. However, their use requires experimentally determined parameters for a phenomenological description of the electromagnetic response of materials.
The term "Maxwell's equations" is often also used for equivalent alternative formulations. Versions of Maxwell's equations based on the electric and magnetic scalar potentials are preferred for explicitly solving the equations as a boundary value problem, analytical mechanics, or for use in quantum mechanics. The covariant formulation (on spacetime rather than space and time separately) makes the compatibility of Maxwell's equations with special relativity manifest. Maxwell's equations in curved spacetime, commonly used in high energy and gravitational physics, are compatible with general relativity. In fact, Albert Einstein developed special and general relativity to accommodate the invariant speed of light, a consequence of Maxwell's equations, with the principle that only relative movement has physical consequences.
The publication of the equations marked the unification of a theory for previously separately described phenomena: magnetism, electricity, light and associated radiation.
Since the mid-20th century, it has been understood that Maxwell's equations do not give an exact description of electromagnetic phenomena, but are instead a classical limit of the more precise theory of quantum electrodynamics.

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1. ### I Where did I make a mistake in simplifications of equations of EM field?

All tensors here are contravariant. from maxwell equation in terms of E-field we know that: $$\rho=\frac{\partial E_1}{\partial x_1}+\frac{\partial E_2}{\partial x_2}+\frac{\partial E_3}{\partial x_3}$$ from maxwell equation in terms of magnetic 4-potential in lorenz gauge we know that...
2. ### I Existential problem on Electromagnetism and the combination of Relativity and Quantum Mechanics

In Feynman's famous Physics book, in a discussion of the generality of Maxwell's equations in the static case, in which he addresses the problem of whether they are an approximation of a deeper mechanism that follows other equations or not, he says: I was wondering first of all if this was a...
3. ### What happened to "the Ballad of Maxwell's equations" sung to the tune of A-Ha, "take on me"?

Hi folks, I know that this exists. I used to be able to find it. But not anymore. It goes something like "Gauss and Faraday, had a lot of thing to say about magnetic fields" and on and on. I hope that it still exists but I have not been able to find it anywhere. Thanks Tom
4. ### I Why are Maxwell's equations and the Lorentz force "so different"

Hi. Maxwell's equations tell us how charges and currents act on electric and magnetic fields. However, when we conversely want to investigate how EM fields act charges and currents, we need this very different thing called Lorentz force. This all looks so asymmetric to me because those laws...
5. ### I Exploring the Retarded Wave Solution of Maxwell's Equations for Gravitation

Supposedly, the retarded wave solution to Maxwell's equations applies to gravitation as well as electrodynamics. The space station doesn't fly off into the distance because every object in the universe (at whatever distance) focuses gravity through the mass of the station. Every object on the...
6. ### I Are spherical transverse waves exact solutions to Maxwell's equations?

In this paper in NASA https://www.giss.nasa.gov/staff/mmishchenko/publications/2004_kluwer_mishchenko.pdf it claims (at page 38) that the defined spherical waves (12.4,12.5) are solutions of Maxwell's equations in the limit ##kr\to\infty##. I tried to work out the divergence and curl of...
7. ### Critique my Physics History Primer (Maxwell's Equations)

I gave a short Maxwell's equation history lesson and included a quick explanation of the connection to Maxwell's predecessors. Just wanted to see if I hit those points right. I don't think I made any physics mistakes, but this was a little more conceptual with some calculus flavor as the student...
8. ### I Induced Electric and Magnetic Fields Creating Each Other

Hi, We know that a varying magnetic field creates and induced electric field, and a varying electric field creates an induced magnetic field. If there is a varying electric field (let's say sinusoidal), then this electric field creates an induced magnetic field. And if this produced magnetic...
9. ### I Show Maxwell's Eqns. on a Cauchy Surface (Wald Ch. 10 Pr.2)

This problem is Wald Ch. 10 Pr. 2.; it asks us to show that ##D_a E^a = 4\pi \rho## and ##D_a B^a = 0## on a spacelike Cauchy surface ##\Sigma## (with normal vector ##n^a##) of a globally hyperbolic spacetime ##(M, g_{ab})##. Using the expression ##E_a = F_{ab} n^b## for the electric field gives...
10. ### I Solving Spinorial Maxwell's Equations with Wald

I'm trying to figure out how to do these sorts of calculations but I'm having a lot of trouble figuring out where to start. Take problem 3) of Chapter 13 of Wald, i.e. given that a real antisymmetric tensor ##F_{ab}##, corresponding to the spinorial tensor ##F_{AA' BB'}## by the map...
11. ### B Charged Particle on Earth's Surface: Will It Emit Radiation?

General relativity tells us that an object in free-fall is actually inertial, following a geodesic through curved spacetime, and not accelerating. Instead, it's objects like us, on the surface of a large body, that are accelerating upwards. Maxwell's equations also tell us that accelerated...
12. ### Doubt related to notation used in writing Maxwell's equations

What does ##S=\partial V## and ##C=\partial S## signify, usually I have only seen books writing ##C## when evaluating a line integral over a curve ##C## and ##S## when evaluating a surface integral over a surface ##S##. Could someone clarify what ##\partial S## and ##\partial V## mean?
13. ### The back way for deriving Maxwell's Equations: from charge conservation?

I found one article in 1993 talking about it.[Unacceptable reference deleted by the Mentors]
14. ### Polarized wave in an anisotropic medium

Calculate the wavelength for an ##E_x## polarized wave traveling through an anisotropic material with ##\overline{\overline{\epsilon}}=\epsilon_0diag({0.5, 2, 1})\text{ and }\overline{\overline{\mu}}=2\mu_0## in: a. the y direction b. the z direction Leave answers in terms of the free space...

21. ### Fourier transform of Maxwell's equations

Hello, I am unfamiliar with Maxwell's equations' Fourier transform. Are there any materials talking about it?

43. ### The D field in Maxwell's equations

I recently saw that in the solution of a problem the following assumption was made - "there are no free charges in the problem, therefore the D field must be equal to 0 ". however if we use that logic to calculate the field of a polaraized sphere we get a wrong result (E=-P/e0 instead of E =...
44. ### I What is the relationship between Maxwell's equations and quantum field theory?

Hello,I have been wondering about the validity of Maxwell's equations in quantum physics. I looked in the internet and it seems from what I understood that: Maxwell's equations are valid for any situation, classical or quantum. In fact, maybe it holds more legitimacy than Schroedinger equation...
45. ### I Interesting Derivation of Maxwell's Equations

I really love seeing derivations of the EFE's, Maxwell's equations, Schrodinger equation etc. I have seen a number of derivations of Maxwell's Equations but this is the shortest, most illuminating and best I have come across - it basically just uses covarience - and as it says - a little bit...
46. ### Who contributed to Maxwell's Equations and how?

The four equations carry the names of Gauss, Faraday and Ampere, however, I cannot seem to find any information regarding each's involvement in them. Are those names solely due to physical observations made by the person (in the case of Faraday for example) or have they contributed towards the...
47. ### Macroscopic Maxwell's equations and speed of light in media

So I followed the derivation of the Macroscopic Maxwell's equations by averaging the fields / equations and doing a taylor series to separate the induced charges and currents from the free ones. But why does light now "suddenly" travel slower in dielectric media? I mean, sure, it comes out from...
48. ### A Leonhard Euler's 4-Squares Identity & Maxwell's Equations

The 4-Squares-Identity of Leonhard Euler (https://en.wikipedia.org/wiki/Euler%27s_four-square_identity) : has the numeric structure of Maxwell’s equations in 4-space: Is somebody aware of litterature about this?
49. ### I Maxwell's Equations, Hodge Operators & Tensor Analysis

Hello! I am reading this paper and on page 18 it states that "in (2 + 1)D electrodynamics, p−form Maxwell equations in the Fourier domain Σ are written as: ##dE=i \omega B ##, ##dB=0##, ##dH=-i\omega D + J##, ##dD = Q## where H is a 0-form (magnetizing field), D (electric displacement field)...
50. ### Does this look like a possible solution to Maxwell's equations?

Using two planes of sheet plastic I sketched electric and magnetic field lines that have zero divergence. Each plane was then slit half way down the axis of symmetry and then slid together. Holding them roughly perpendicular they were photographed. On the two planes do the field lines look like...