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. olgerm

    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. Hak

    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. M

    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. G

    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. paige turner

    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. Delta2

    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. kq6up

    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. B

    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. ergospherical

    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. ergospherical

    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. Amaterasu21

    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. Hamiltonian

    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. T

    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. L

    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...
  15. Delta2

    Maxwell's equation in microscopic formulation and speed of EM-waves

    Starting from the microscopic form of Maxwell's equations and following standard mathematical procedure outlined in Inhomogeneous electromagnetic wave equation - Wikipedia we can have as end result the following equations: $$(\nabla^2-\frac{1}{c^2}\frac{\partial^2}{\partial...
  16. dlang99099

    Relating Maxwell's equations to parasitic inductance due to layout

    I am trying to show how parasitic inductance is increased or decreased based on coupling between adjacent parallel capacitors. These capacitors are DC link capacitors so I'm not sure if that means it is not time varying and maxwells equations is no longer relevant?
  17. LCSphysicist

    About Maxwell's equations and the attempt to insert a magnetic monopole

    I was reading somethings about magnetic monopoles, and how, if it were discovered its existence, it would changes the Maxwell equations, in summarizing, is not the first time i see this: It is the consequences of the existence of a magnetic monopole? That is, IF it exist, SO the imagem is true...
  18. Haorong Wu

    Prove that phase-conjugate waves satisfy Maxwell's equations

    This is the second part of a problem. In the first part of the problem, I have proven that ##\mathbf E^* =\mathbf E_0^* ( \mathbf r) e^{i \omega t}## satisfies the Maxwell equations. Then, in this part of the problem, I tried to first prove that ##\mathbf E^{'} =\mathbf E_0 ( \mathbf r) e^{i...
  19. fisher garry

    Frequency in Maxwell's equations

    $$\textbf{F} \cdot d\textbf{l}=q(\textbf{E}+\textbf{v} \times \textbf{B})\cdot \textbf{v} dt$$ If we denote $$q=\rho d \tau$$ and $$\rho \textbf{v}=\textbf{J}$$ $$\frac{dW}{dt}=\int_{V} (\textbf{E} \cdot \textbf{J}) d \tau.$$ From maxwell law's $$\textbf{E} \cdot \textbf{J}=\frac{1}{\mu _0}...
  20. thaiqi

    Fourier transform of Maxwell's equations

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

    Maxwell's equations for TEM mode

    Dear All, I'm confused after reading of some chapter in a book, in which equations related to TEM mode have been derived. I want to prove mathematically, that Electric and Magnetic fields are ortogonal to each other. Thus, I use well known Maxwell equation: $$\nabla \times \overrightarrow{E} +...
  22. F

    Maxwell's equations and time-harmonic solutions

    Hello, For a linear medium, the two curl Maxwell's equations $$\nabla \times \bf{E} = - \frac {\partial \bf{B}} {\partial t}$$ $$\nabla \times \bf{H} = \frac {\partial \bf{D}} {\partial t}$$ change to $$\nabla \times \bf{E} = i \omega \bf{B}$$ $$\nabla \times \bf{H} = - i \omega\bf{D}$$...
  23. F

    Maxwell's Equations, Dielectrics, Auxiliary Fields D and H

    Hello, I am reviewing how classical EM problems are treated when dielectric materials are involved. Maxwell's equations relate the following vector fields: ##E(r,t)##, ##B(r,t)##, ##D(r,t)##, ##H(r,t)##, ##J(r,t)## and scalar field ##\rho(r,t)##. The two constitutive equations are also needed to...
  24. snoopies622

    Arriving at the differential forms of Maxwell's equations

    In college I learned Maxwell's equations in the integral form, and I've never been perfectly clear on where the differential forms came from. For example, using \int _{S} and \int _{V} as surface and volume integrals respectively and \Sigma q as the total charge enclosed in the given...
  25. M

    The relationship of one of Maxwell's equations to Coulomb's Law

    My textbook tells me that one of Maxwell's equations, namely divergence of E = 4pi * charge density (in cgs) or divergence of E = pi / epsilon nought (in SI) is exactly equivalent to Coulomb's Law. How in the world is that so? Any ideas would be appreciated.
  26. G

    How is the speed of light derived in Maxwell's equations?

    I know we end up with ##c=\sqrt{\frac{1}{μ _0.ε _0}}## The reason I would like a bit of help is that I understand that the value of c as deduced from Maxwell's equations is independent of any frame of reference. I can see that this is the case from the above equation involving the...
  27. Delta2

    Do Maxwell's equations in integral form imply action at a distance?

    Lets take for example Gauss's law in integral form. Suppose at time ##t## we have charge ##q(t)## (at the center of the gaussian sphere) enclosed by a gaussian sphere that has radius ##R>>c\Delta t##. At time ##t+\Delta t## the charge is ##q(t+\Delta t)## and if we apply gauss's law in integral...
  28. A

    I Maxwell's equations with differential forms

    Hello! I was not quite sure about posting in this category, but I think my question fits here. I am wondering about Maxwell equations in vacuum written with differential forms, namely: \begin{equation} \label{pippo} dF = 0 \qquad d \star F = 0 \end{equation} I know ##F## is a 2-form, and It...
  29. guiablo

    Classical Which book should I buy to understand Maxwell's Equations deeply

    I've already done Maxwell's equations in class but I would like another point of view, and to have a physical reference to check anytime I have a doubt, so I would like a quite high lvl electrodymacis book but that does not focus only in concrete subjects, I mean a book that covers al the...
  30. K

    Which Form of Maxwell's Equations is More Useful? (Integral versus Differential)

    There are two forms of the Maxwell equations, one is the differential form, the other is the integral form. Which one is more useful?
  31. T

    Instantaneous solutions to Maxwell's equations' potentials conversion?

    This page shows solutions for Maxwell's equations of the electric and magnetic potentials (Eqn.s (509) and (510)): http://farside.ph.utexas.edu/teaching/em/lectures/node50.html They are derived with the aid of a Green's function: http://farside.ph.utexas.edu/teaching/em/lectures/node49.html...
  32. Charles Link

    If Maxwell's equations are linear....

    I posted previously about this topic a couple of years ago, and it really came across like a ton of bricks, but now that I have established some credibility, perhaps it will be read with some interest. In the course of my career, (in the years 1986-1990), on two occasions I discussed with...
  33. Tan Thom

    Maxwell's Equations and Divergence

    Homework Statement I was working on a problem from Maxwell Equations. Why is the below zero? Homework EquationsThe Attempt at a Solution
  34. K

    I Maxwell's Equations in Different Lorentz Frames: Are They Equivalent?

    Do the Maxwell equations in the usual 3-vector form have the same form in any Lorentz frame? For example, the one that says ##\nabla \cdot \vec B = 0## will be valid in another, primed Lorentz frame? That is ##\nabla' \cdot \vec {B'} = 0##?
  35. H

    Magnetic field of a moving charge and Maxwell's equations

    How can we calculate the magnetic field of moving charge by the Maxwell equation ##\nabla \times H=J+\frac {\partial D} {\partial t}##? I mean which term, ##J## or ##\frac {\partial D} {\partial t}##, should be taken into account in calculations? The first, second, or both? Can we deal with the...
  36. W

    Maxwell's equations and exterior algebra

    Maxwell's equations in differential form notation appeared as a motivating example in a mathematical physics book I'm reading. However, being a mathematical physics book it doesn't delve much into the physical aspects of the problem. It deduces the equations by setting dF equal to zero and d(*F)...
  37. iVenky

    From Maxwell's equations to EM waves

    Hi, I just finished studying Maxwell's equations. Based on my understanding, when you solve maxwell's equation, you get the wave equation and it simplies to in a charge and current-free region. I understand that these two equations are similar to an equation of a wave in space. What I am...
  38. I

    Maxwell's Law of Induction - How Does it Work?

    So, I was studying Maxwell's equations and I don't really understand the last one - Ampere's Law (with Maxwell's extra term added in). The bit I'm not able to understand is the term Maxwell added. How exactly does a changing electric field through a closed loop induce a magnetic field along that...
  39. P

    Using Maxwell's equations with realtime results

    I have spent a couple of hours watching an explanation of Einstein Field equations and Maxwells equations re EM on you tube. Found it very enlightening and interesting. Only thing was that when lecturer had screens full of equations never at any time did he actually show how to "plug in" actual...
  40. omega_minus

    I Deriving Maxwell's Equations from Field Tensor (Griffith 4ed)

    Hello, I am reading Griffith's "Introduction to Electrodynamics" 4ed. I'm in the chapter on relativistic electrodynamics where he develops the electromagnetic field tensor (contravariant matrix form) and then shows how to extract Maxwell's equations by permuting the index μ. I am able to...
  41. W

    Lagrangian Field Theory - Maxwell's Equations

    Homework Statement $$ L = -\frac{1}{2} (\partial_{\mu} A_v) (\partial^{\mu} A^v) + \frac{1}{2} (\partial_{\mu} A^v)^2$$ calculate $$\frac{\partial L}{\partial(\partial_{\mu} A_v)}$$ Homework Equations $$ A^{\mu} = \eta^{\mu v} A_v, \ and \ \partial^{\mu} = \eta^{\mu v} \partial_{v}$$ The...
  42. QuasarBoy543298

    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 =...
  43. Phylosopher

    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...
  44. bhobba

    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...
  45. A

    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...
  46. R

    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...
  47. E

    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?
  48. S

    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)...
  49. Spinnor

    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...