# What is Maxwell equations: Definition and 143 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 Are Maxwell's equations linearly dependent?

HI, consider the 4 Maxwell's equations in microscopic/vacuum formulation as for example described here Maxwell's equations (in the following one assumes charge density ##\rho## and current density ##J## as assigned -- i.e. they are not unknowns but are given as functions of space and time...
2. ### I Confused about electromotive forces

Hi, reading Griffiths - Introduction to Electrodynamics I'm confused about his claims in section 7.1 My point is that the job of electromotive force ##f## is actually produce the "movement/drift" of free charges against the electromagnetic field, so ##f## should not be given by the Lorentz...
3. ### Total Internal Reflection and Transmitted Wavelength

In my electrodynamcis assignment I'm being asked to derive the wavelength of the normally polarised wave transmitted through a glass/air interface as a function of ##n_1## (the refractive index of the first medium) using the concept of phase continuity and the fact that maxima should be equal at...
4. ### I "Strange contradiction" that Maxwell found and resolved

In "The Strange Story of the Quantum", Banesh Hoffmann writes: What was that contradiction?
5. ### I Maxwell's equations in the presence of matter -- Derivation

I want to calculate ##\int \vec{P}\left(\overrightarrow{r^{\prime}}\right) \cdot \vec{\nabla}_{\overrightarrow{r^{\prime}}} \frac{1}{\left|\vec{r}-\overrightarrow{r^{\prime}}\right|} d^{3} \overrightarrow{r^{\prime}}## with macroscopic polarization...
6. ### A Derivation of the Maxwell and Proca Field Equations

I am back to my writing desk and I was looking up different and (hopefully) relatively "basic" derivations of the field equations. I found a nice little derivation of the Proca (and Maxwell) equations by Gersten (1998, PRL, 12, 291-298 is the reference, but I got it from the web so what I have...
7. ### A A question about current density in finite element analysis

I know that if there is only one conductor providing the current density, then the current density can be used. But if you apply Maxwell's equation when there are multiple current sources, I don't know which value to use. This is not an analysis using a tool, but a problem when I develop the...
8. ### A Relationship between magnetic potential and current density in Maxwell

I am currently studying to solve Maxwell's equations using FEM. I have a question about Maxwell's equations while studying. I understood that the magnetic potential becomes ▽^2 Az = -mu_0 Jz when the current flows only in the z-axis. I also understood the effect of the current flowing in a...
9. ### I EM equations - am I missing something?

Summary:: There seems to be a mismatch, in the "Maxwell's" equations, between the number of equations and number of variables. I was trying to play around with the equations for Electromagnetism and noticed something unusual. When expanded, there are 8 equations, 6 unknown variables, and 4...
10. ### Phase velocity in oblique Angle propagation (Plane wave)

Hello, Regarding the wave oblique angle propagation and based on Balanis "Advanced engineering Electromagnetic" book on page 136 ( it has been attached) I need to know why the phase velocity in x direction is not important to keep in step with a constant phase plane( Just equation 4-23). I...
11. ### Maxwell Equations, Lorentz Force and Coulumb Force

Here the 3 set of equations we know, the Maxwell Equations, Lorentz Force, and Coulumb Force, actually I doubt a lot what set of equations represent all the electromagnetic aspects, I try research over the internet and I found a lot of contradictions in the answers, someone says we can get the...
12. ### Confusion when dealing with loops and surfaces with Maxwell equations

DISCLAIMER: in Italy, we talk about "circuitazione" of a field through a closed loop ##\gamma##, for the physical quantity $$\Gamma_\gamma(\overrightarrow{E}) = \sum_{k}\overrightarrow{\Delta l_k} \cdot \overrightarrow{E_k}$$ but after some research, I haven't managed to find the correspondent...
13. ### Can the charge conservation law be derived from Maxwell equations?

From Maxwell's equations \partial_\nu F^{\mu\nu}=J^{\mu}, one can derive charge conservation. The derivation is 0\equiv \partial_\mu \partial_\nu F^{\mu\nu}= \partial_\mu J^{\mu} { \Rightarrow}\partial_\mu J^{\mu}=0. However, a circular reasoning exists in it. For the sake of better...

42. ### How do I apply Maxwell's equations?

For example, if I have a magnetic field perpendicular to some surface and I change this magnetic field with constant speed, how do I calculate the Electric field at any point on this surface, since ∫E⋅ds=k, where k is some constant, could be done with many different vector fields.
43. ### Identities of fields in Maxwell's equations

Hello. I would like to ask one simple question. Do we need to distinguish E-field (Electric field) in Gauss's law from those in Maxwell-Faraday equation and Ampere's circuit law? I firstly thought that E-field in Gauss's law is only for electrostatics so I need to distinguish it from E-field in...
44. ### Plasma Book with Maxwell equations in a plasma?

My main goal is to understand plasmons, for that I have been told that the theoretical background is Maxwell equations in a plasma. Do you know any book with these equations? Thanks!
45. ### Deriving Magnostatics equations from steady currents

I've always heard that maxwell's equations contains essentially all of eletromagnetic theory. However, there's one thing I'm having trouble doing for myself: deriving the magnestatics equations from the maxwell's equations. Of course: it's clear that if you put ∂[t]E=∂[t]B=0 (partial derivative...
46. ### A Assumptions behind Maxwell's equations for constant speed

I need some help in defining what are the assumptions needed to derive a constant speed of light from Maxwell equations. Is it correct to say that this result applies to a sinusoidal wave as an assumption? In my understanding that is (more or less) equivalent to planar waves in vacuum: is it...
47. ### Finding E and B field of a weird charge distribution

Homework Statement Initially there is a spherical charge distribution of with a radius ##R_0## and uniform charge density ##ρ_0##. Suppose the distribution expands spherically symmetrically such that its radius at time t is ##R_0 + V t##, where V is the velocity. Assuming the density remain...
48. ### Prove that a capacitor driven by an AC voltage radiates EM

Homework Statement When the capacitor driven by DC voltage ##V_0##,it has the electric field distribution ##f(x,y,z)## When ##V=V_0e^{iwt}##,how to show the EM travel in the space forever like the light? Homework Equations ##-{\nabla}^2E-u{\epsilon}\frac{{\partial}^2{E}}{{\partial}t^2}=0...
49. ### Boundary condition for electrostatics problem - found issue?

Hey everyone Just a picture of my configuration. The assumption here is $$\epsilon_a,\epsilon_b,\epsilon_c$$ are different from one another. Really the interest of this problem is to find the scalar potential $$\phi$$, such that $$\nabla^2 \phi = 0$$. So now my question, about jump...
50. ### Help with tensor formulation of special relativity

Homework Statement Hi, I can't seem to understand the following formula in my professor's lecture notes: F_αβ = g_αγ*g_βδ*F^(γδ) Homework Equations Where g_αβ is the diagonal matrix in 4 dimensions with g_00 = 1 and g_11 = g_22 = g_33 = -1 and F^(γδ) is the electromagnetic tensor with c=1...