# What is Maxwell's equation: Definition and 25 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. ### 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...
2. ### Why am I getting Maxwell's second equation wrong?

While going through an article titled "Reflections in Maxwell's treatise" a misunderstanding popped out at page 227 and 228. Consider the following equations ##(23\ a)## and ##(23\ c)## in the article (avoiding the surface integral): ##\displaystyle \psi_m (\mathbf{r})=-\dfrac{1}{4 \pi} \int_V...
3. ### Determining D, H, and β

In a certain region for which σ=0, μ=2μ0, and ε=10ε0, J= 60sin(109t - βz). Determine D, H, and β.
4. ### 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 =...
5. ### Small question about maxwell's equation curl of H

The first page of this short pdf from MIT sums the starting point to formulate my question: https://ocw.mit.edu/courses/physics/8-022-physics-ii-electricity-and-magnetism-fall-2006/lecture-notes/lecture29.pdf We can see that ∇xH = Jfree and ∇xB =μo (Jfree +∇xM) ∇xB =μo (Jfree +JB) And now my...
6. ### Faraday's Unipolar Induction Equation

Hello all, I have a question regarding Maxwell's Equations and Faraday's unipolar induction equation. If we study the case of a cylindrical magnet with a radius of r which is rotating about its axis with angular velocity w. The electrons within the magnet collide with the moving atoms, causing...
7. ### Derivation: Maxwell's equation only from the Lorenz gauge

Hi. Here, somebody apparently derives Maxwell's equations using only symmetry of second derivatives and the Lorenz gauge condition. Unfortunately it's in German, but I think the basic ideas are clear from the maths only. In this derivation, the magnetic field turns out to be divergence-free...
8. ### Maxwell's equation has well defined divergence?

Homework Statement How to I explain that maxwell's equation has well defined divergence Homework Equations All four EM Maxwell's equation The Attempt at a Solution I discussed it by showing one of the property of Maxwell's equation that is the Divergence of a Gradient is always zero (With...
9. ### Maxwell's equation of magnetic field

Hi guys! Im doing a Project about maxwell equations and need help with something. Let's say you have a current going through a copper wire with an insolation on, and ofcourse there is a magnetic field. I've wondering if the magnetic field have a temperatur impact on the cabelinsolation. I...
10. ### Trying to understand Maxwell's equation in a medium

So I believe I understand Maxwell's equation in vacuum pretty well and I feel like I understand them in different medium when I read in a textbook, but when I have to apply it to exercises I get thrown off quite a bit. For example, I cam across a PhD qualifying exam that had split Euclidean...
11. ### Maxwell's equation in reciprocal space

I found this http://people.sissa.it/~benassi/capitolo1/node2.html [Broken] in the vacuum the equations would be q B = q E = 0 ##q \times E = (\omega / c) B## ##q \times B = -(\omega / c) E## Is there a typo? there is no t derivative. If E = 0 B would have to be null. Has B to be allways...
12. ### Finding Maxwell's Equation parameters problem

Homework Statement Homework Equations http://esclab.tw/wiki/images/math/0/a/2/0a24f9d68ba8fd1a1a8e6c6b36a00be3.png [Broken] The Attempt at a Solution [/B] Hello As you see the answers of 19 and 20 problems we should try to find K or B(wave number) in wave equation.both of equations are in...
13. ### Maxwell's equation which convective derivative

http://arxiv.org/pdf/physics/0511103.pdf I was wondering what people thought of this paper. Please read up to at least page 3 before responding. I find it to be pretty convincing up to page 4. Thanks for any response.
14. ### Simplified Maxwell's Equation Proof

Problem: I'm trying to crudely prove the following: \frac{{\partial}B}{{\partial}x}=-\mu_{o}\epsilon_{o}\frac{{\partial}E}{{\partial}t} Solution (so far): I can get the derivation, but the minus sign eludes me somehow... Integrating over a thing rectangular loop of length l and width dx...
15. ### A Maxwell's equation in differential forms formalism

Homework Statement This is not actually a homework but a personal work. Here it is: Using the differential forms: F=\tfrac{1}{2!}{{F}_{\mu \nu }}d{{x}^{\mu }}\wedge d{{x}^{\nu }} and J=\tfrac{1}{3!}{{J}^{\mu }}{{\varepsilon }_{\mu \alpha \beta \gamma }}d{{x}^{\alpha }}\wedge d{{x}^{\beta...
16. ### Exploring the Magnetic Field: Applying Maxwell's Equation and Stokes' Theorem

One of Maxwell's equations says that \nabla\cdot\vec{B}{=0} where B is any magnetic field. Then using the divergence theore, we find \int\int_S \vec{B}\cdot\hat{n}dS=\int\int\int_V \nabla\cdot\vec{B}dV=0 . Because B has zero divergence, there must exist a vector function, say A...
17. ### Maxwell's Equation and Special Relativity

I've got a couple of question about Maxwell's Equation and its relation to Special Relativity. 1. Why do we think that free space has permittivity and permeability? For me, due to quantum effects free space should have permittivity and permeability, but how did Maxwell accept that the free...
18. ### Solving Maxwell's Equation

Homework Statement A sinusoidally-varying voltage V(t) = Vosinωt with amplitude Vo = 10 V and frequency of f = ω/(2π) = 100 Hz is impressed across the plates of a circular-shaped parallel plate air-gap capacitor of radius a = 1.0 cm and plate separation d= 0.01 mm. The amplitude of Maxwell's...
19. ### How to derive Lienard-Wiechert potential from Maxwell's equation?

I've seen one derivation on Feynman Lectures on Physics, but the derivation is not really rigorous(he took a very special case for the derivation),I googled about the topic and couldn't find a satisfactory one. So can anybody give me a rigorous one? Thanks in advance
20. ### Maxwell's equation and Helmholtz's Theorem

I am trying to find the magnetic field around a moving point particle. I have already found the curl. The only step remaining is to use Helmholtz's theorem. I am using http://farside.ph.utexas.edu/teaching/em/lectures/node37.html" [Broken]. I am going to use equation 300, but I am not sure...
21. ### Maxwell's Equation HELP

Maxwell's Equation HELP! This is one of the Maxwell's Equations that I can't understand: \oint E.dl = -\int dB/dt .dS From what I understand, \intE.dl gives the potential difference along the line. Therefore \ointE.dl should always equal to zero because it is the potential difference of a...
22. ### Maxwell's Equation

Hi, I'm new here. I have a question. How do you verify that: E = E(max)cos(kx-wt) is a solution to Maxwell's derived equation: ((d^2)E/dx^2) = e(epsilon nought)u(permitivity of free space) x (d^2)E/dx^2. Thanks. What I first did was to substitute k = 2pi/lambda, w = 2pi(f). Then I set 2pi(f)...
23. ### Maxwell's equation and E&M

hi everyone can anyone explain to me or show me some site that explain Maxwell's equation in simple language... i have been reading most of my teacher's books but i still don't get it. (my teacher doesn't want to teach me... sigh) and where can i find a good introduction of E&M? thx a lot!
24. ### Tensor gradient and maxwell's equation

Hi everyone, I now have difficulties in using formula for gradient of tensor. The following is tensor for field strength Fsubscripts_alpha_beta = [ 0 -Ex -Ey -Ez Ex 0 Bz -By Ey -Bz 0 Bx Ez By -Bx 0 ] My question is, how do we derive the Maxwell equations...
25. ### Maxwell's Equation: Proving the Speed of Light

How exactly did Maxwell prove speed of light is \frac{1}{\sqrt{\mu\epsilon}}