Maxwell equations Definition and 58 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. JH_1870

    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...
  2. JH_1870

    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...
  3. Maxicl14

    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...
  4. baby_1

    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...
  5. 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} +...
  6. Jaffer2020

    Refraction of light -- What is the best description for 10th grade physics students?

    In my latest 10th grade physics lesson, we were learning about the refraction of light. I decided to share what I knew about why light slows down in a vacuum, which is, in short, because the electric field of the electromagnetic wave exerts a force on the charged electrons of a medium, which in...
  7. christang_1023

    Understand Faraday's Law and Lenz' Law

    Above is an example figure. 2. When a ring in a changing magnetic field is not complete (i.e. open circuit with a small gap), how to analyze the emf of the ring? According to the general form of Faraday's law, ## \oint \vec{E} \cdot d \vec{s} = -\frac{d \Phi}{dt} ##, I deduce that although it...
  8. G

    Conceptual difference between a motor and a generator

    Hi. In order to explain the motion of an (accelerating) electromotor, we need the Lorentz force which itself is not one of Maxwell's equations. Conversely, if we use the same electromotor inversely to generate electricity, Faraday's law (which is a Maxwell's equation) and the resistance of the...
  9. J

    I Feynman-Heaviside formula and Mach's principle

    I was wondering if the [Feynman-Heaviside formula]( for the electric field of a moving charge could be used to write down the force/reaction force between charges ##q_1## and ##q_2## in a Machian purely relational way. The retarded electric...
  10. 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...
  11. Samuel Williams

    Are longitudinal magnetic waves possible?

    Homework Statement Are longitudinal magnetic waves possible? Give reasons for your answer. Homework Equations Working with Maxwell's equations, Lorentz force, electrostatic and electromagnetic waves in plasma. The Attempt at a Solution No idea whatsoever. I believe it is possible based on...
  12. Alettix

    Magnetic field decay in cylinder coaxial with solenoid

    Homework Statement For a medium of conductivity ##\sigma##: $$ \nabla^2 \vec{B} = \sigma \mu \mu_0 \frac{\partial \vec{B}}{\partial t} + \mu \mu_0 \epsilon \epsilon_0 \frac{\partial^2 \vec{B}}{\partial^2 t} $$ A long solenoid with ##r=b## has n turns per unit length of superconducting wire anc...
  13. Wrichik Basu

    Classical Book(s) with problems on classical electromagnetism

    I started studying the book "A Student's Guide to Maxwell's Equations" by Daniel Fleisch some time back. It is an excellent book, giving a very good idea about the main laws of electromagnetism. I will soon finish the book. Now I need some book(s) which has problems on all the laws in classical...
  14. A

    Verification of Ampere-Maxwell Law

    Homework Statement A current I flows along a wire toward a point charge, causing the charge to increase with time. Consider a spherical surface S centred at the charge, with a tiny hole where the wire is – see figure below. The circumference C of this hole is the boundary of the surface S...
  15. N

    A Fields from an AC wire

    I'm having trouble figuring out how to solve Maxwell's equations for the electric field of an AC wire. I assume the Voltage waveform in the wire is 120sin(60t). This circuit only has a 14ohm heater in it, according to Ohm's law I=V/R. The current is equation to I(t)=120sin(60t)/14 It is...
  16. ajeet mishra

    I Solution of Poission eqution.

    My professor told that poission equation has a unique solution even for mixed boundary conditions( i.e. Dirichlet bc for some part and Neumann for the remaining part). But how is this possible? As different boundary conditions for the same problem will give different solutions.
  17. Marcus95

    Finding Total Charge from E-field

    Homework Statement A static charge distribution has a radial electric field of magnitude ##E = \alpha \frac{e^{-\lambda r}}{r} ## where λ and α are positive constants. Calculate the total charge of the distribution. Homework Equations Gauss's law ##Q/\epsilon_0 = \int \vec{E} \cdot d\vec{S}##...
  18. vanhees71

    The Homopolar Generator: An Analytical Example - Comments

    Greg Bernhardt submitted a new PF Insights post The Homopolar Generator: An Analytical Example Continue reading the Original PF Insights Post.
  19. E

    A Euler's 4-squares-identity and Maxwell's equations

    The 4-Squares-Identity of Leonhard Euler ( : has the numeric structure of Maxwell’s equations in 4-space: Is somebody aware of litterature about this?
  20. M

    Effect of electrical permittivity on the speed of light?

    Why does an increased electrical permittivity reduce the phase velocity of light in a medium? Furthermore, what interactions do we see on an atomic level? I am aware of the equation that defines the speed of light in terms of the electrical permittivity and magnetic permeability, but I do not...
  21. C

    I Waveguide: get I and II M.eq. from III and IV M.eq.

    Consider a waveguide with axis parallel to axis ##z##. Using cartesian coordinates the fields inside the waveguide can be written as Where ##\alpha## is the wavenumber and ##k=\frac{\omega}{c}## . The maxwell equations ##\nabla \times E=-\frac{\partial B}{\partial t}## and ##\nabla \times...
  22. P

    Does the first Maxwell equation work also for moving charges?

    My professor said that the first two maxwell equation's are about static fields Another fast question: Do a single charge that is moVing generate a magnetic field? I saw a demonstration that explains the magnetic field as an electric field using only special relativity, there is a free charge...
  23. snate

    I Why μ*ε=1/c^2

    I've been following the derivations in the following video up until that point. I don't quite understand why does it imply that μ*ε=1/c^2. Thanks.
  24. R

    The strength of induced electric field

    Homework Statement A thin and straight metallic rod of length L is rotating about its middle point with angular velocity ##\omega## in a uniform magnetic field B. The axis of rotation is perpendicular to the length of the rod and parallel to the magnetic field. The strength of the induced...
  25. F

    Physical degrees of freedom of an Electromagnetic field

    As I understand it, the classical source-free electric, ##\mathbf{E}## and magnetic, ##\mathbf{B}## wave equations are solved by solutions for the electric and magnetic fields of the following form: $$\mathbf{E}=\mathbf{E}_{0}e^{i (\mathbf{k}\cdot\mathbf{x}-\omega t)}$$...
  26. N

    What do electric and magnetic fields look like around a dipole antenna?

    I am studying to about maxwell's equations on the electric and magnetic field. First of all, just do the example and exercise: [Broken] I have this dipole antenna...
  27. V

    Getting the magnetic field as a function of current

    Howdy, So, I'm curious, is there a general relationship between current input into an electromagnetic and the magnetic field that it generates in space? The trivial example is wrapping a wire around a rod, then sending a current through it which causes a magnetic field. My set up is a simple...
  28. L

    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.
  29. G

    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...
  30. F

    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...
  31. G

    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...
  32. A

    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...
  33. Z

    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...
  34. V

    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...
  35. lonewolf219

    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...
  36. S

    Spatial derivative of Electric Field in Faraday's Law?

    According to Faraday's Law, Time-Changing magnetic field creates an induced current in a closed conducting loop. This is the equation: ##\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}## 1-) Does this current (##\nabla \times \mathbf{E} ##) have to be an alternate...
  37. M

    Satisfying Maxwell's Equations

    Homework Statement Show whether or not the following functions satisfies Maxwell's Equations in free space. (That is, show whether or not they represent a valid electromagnetic wave). E(x,y,t)=(0,0,E_0 sin(kx-ky+\omega t)) B(x,y,t)=B_0 (sin(kx-ky+\omega t),sin(kx-ky+\omega t),0) Homework...
  38. D

    Determine the direction of magnetic field from Maxwell eqs?

    Hi! My high school physics tells me using right hand grip rule to determine the direction of magnetic field induced by a current carrying wire, but I wonder whether I can deduce the direction merely from Maxwell's Equations? Suppose now we have a current density in cylindrical...
  39. F

    Maxwell's eqs. & unification of electric & magnetic fields

    Maxwell's equations reveal an interdependency between electric and magnetic fields, inasmuch as a time varying magnetic field generates a rotating electric field and vice versa. Furthermore, the equations predict that even in the absence of any sources one can have self propagating electric and...
  40. Diego Berdeja

    I Lorentz Transformations in the context of tensor analysis

    Hello everyone, There is something that has been bugging me for a long time about the meaning of Lorentz Transformations when looked at in the context of tensor analysis. I will try to be as clear as possible while at the same time remaining faithful to the train of thought that brought me...
  41. M

    Over what frequency range are Maxwell's equations valid?

    Homework Statement I am studying for an Optics exam and in one of the practise tests is the following question: "Over what frequency range are Maxwell's equations valid?" Homework Equations Maxwell's Equations The Attempt at a Solution I've searched through my Griffiths Intro to...
  42. TheSodesa

    Electromagnetic wave in glass

    Homework Statement A harmonic EM-wave is propagating in glass in the +x-direction. The refractive index of the glass ##n = 1.4##. The wave number of the wave ##k = 30 \ rad/m##. The magnetic portion of the wave is parallel to the y-axis and its amplitude ##H_0 = 0.10A/m##. At ##t=0## and ##x =...
  43. C

    Electric field in the vicinity of an antenna

    I don't know why I was persuaded that in the free space, the electric field of an EM wave is always orthogonal to the direction of propagation. I've recently read my old textbook, and found that this is true only when the wave is far from the emitting source. But if I've understood right the...
  44. P

    EM Wave: Phase of the electric and magnetic waves?

    In a vacuum, the plane wave solutions to Maxwell's Equations are... E=E0*cos(wt-kr) B=B0*cos(wt-kr) ie they are in phase. (See for example ) I don't...
  45. E

    Static magnetic field from time-varying electric field

    Hello! In this thread, in this answer, my statement "A time-varying electric field creates a magnetic field which is time-varying itself" was refuted. Because I never observed this before, I would like to discuss about it. As far as I know, Maxwell's equations are valid always together, that...
  46. DavideGenoa

    Second derivatives of magnetic potential

    Hello, friends! I have been told that, if ##\mathbf{J}## is of class ##C^2## and ##V\subset \mathbb{R}^3## is a ##\mu##-measurable and bounded set, where ##\mu## is the ordinary Lebesgue measure on ##\mathbb{R}^3##, then, for all ##\mathbf{x}\in\mathbb{R}^3##...
  47. DavideGenoa

    Commutations and delta in deriving Ampère's law

    Hi, friends! I have been able to understand, thanks to Hawkeye18, whom I thank again, that, if ##\mathbf{J}## is measurable according to the usual ##\mathbb{R}^3## Lebesgue measure ##\mu_{\mathbf{l}}## and bounded, a reasonable hypothesis if we consider it the density of current, if...
  48. DavideGenoa

    Biot-Savart law from Ampère's (with multivariate calculus)

    Let us assume the validity of Ampère's circuital law\oint_{\gamma}\mathbf{B}\cdot d\mathbf{x}=\mu_0 I_{\text{linked}}where ##\mathbf{B}## is the magnetic field, ##\gamma## a closed path linking the current of intensity ##I_{\text{linked}}##. All the derivations of the Biot-Savart law for a...
  49. E

    Field form in the optic fibers from Maxwell's equations

    Hello! In this document a solution of Maxwell's equations in cylindrical coordinates is provided, in order to determine the electric and magnetic fields inside an optic fiber with a step-index variation. The interface between core and cladding is the cylindrical surface r = a. For example, the...
  50. E

    Independent fields' components in Maxwell's equations

    In a source-free, isotropic, linear medium, Maxwell's equations can be rewritten as follows: \nabla \cdot \mathbf{E} = 0 \nabla \cdot \mathbf{H} = 0 \nabla \times \mathbf{E} = -j \omega \mu \mathbf{H} \nabla \times \mathbf{E} = j \omega \epsilon \mathbf{E} If we are looking for a wave...