# What is Maxwells equations: Definition and 28 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. ### An Interesting Question on Faraday's Law of Electromagnetic Induction

On examining Maxwell's third equation which is about time varying magnetic fields (Faraday's electromagnetic induction) we find that time varying magnetic fields produce loops of electric fields in space irrespective of whether a coil is present or not, if any coil is present then these loops of...
2. ### Solving Charge Across Plates of a Capacitor w/ Negative Ions

Homework Statement I have a material placed between parallel plates depleted of free electrons and contain negative ions. What would happen to the charge stored across the plates? Would it still be similar to placing a capacitor with a di-electric constant between them? Homework Equations Q=CV...
3. ### Determining the B-field in center of current carrying loop

Homework Statement Determine the B-field inside the middle of a circular loop of current. Homework Equations Attempt at using Ampere's law: ##\oint \vec{B} \cdot d \vec{l} = \mu_0 i## The Attempt at a Solution ##\oint B \cdot R d \theta = \mu_0 i \Rightarrow BR(2 \pi) = \mu_0 i \Rightarrow B...
4. ### Electromagnetism, Auxiliary field

Homework Statement I only need help with part C and D Homework Equations Curl of B = mu0 J J = J free + J bound H = B/mu0 - M The Attempt at a Solution For C i, I said they're the same because free current is the same at both points. My argument is that because curl of B (at L) = mu0 Jfree...
5. ### Magnitude of the induced magnetic field in a circuit

Homework Statement The circuit in Fig. 32-31 consists of switch S, a 12.0 V ideal battery, a 20.0 M resistor, and an air-filled capacitor. The capacitor has parallel circular plates of radius 5.00 cm, separated by 3.00mm. At time t=0, switch S is closed to begin charging the capacitor. The...
6. ### Recommended Books for Studying Relativistic Electrodynamics

Hey guys, Can you please refer some good books to refer to in studying relativistic Electrodynamics (introductory parts), covering the Maxwell's equations in tensor form the L-W potentials and other aspects. FYI am just a beginner in relativistic Electrodynamics. Thanks for the help.
7. ### Deriving capacitor and inductor models from Maxwell's eqs

I'm sure the inductor model, i.e. vL(t) = iL'(t)*L follows without directly from Faraday's eq. But even there, with Faraday's equation we think of the changing magnetic field as inducing the voltage in the loop, where in the model it seems the other way around, that is, the voltage increases...
8. ### EM maxwells equations problem infinite cylinder

Homework Statement Suppose we have a infinite cylinder of radius=R and with uniform volume charge density ρ. Use Maxwell's Equations or relationships from them to find E, B, V, and A everywhere. Pretty easy. But how do you approach the problem when you bring an angular vel into the mix...
9. ### Are plane waves real/physical?

When finding solutions to Maxwells equations we always cosider the case of a plane wave. But are plane waves real/physical solutions we can consider in real life? My guess is not because it is required to propagate infinitely. So why do we use plane waves to solve Maxwell's equations?
10. ### What is this version of the wave equation?

I came across this expression for the wave equation: \nabla^2E + \mu\sigma\frac{\partial{E}}{\partial{t}} - \frac{n^2}{c^2}\frac{\partial{E}}{\partial{t^2}} = 0 My question is what kind of medium is it for/where did it come from?
11. ### Why do changing magnetic fields produce electric fields?

Zahid Iftikhar asked why charges get separated in a changing magnetic field over in the EE forum. I pointed him to Maxwell's equations and also pointed out we took them to be observational and axiomatic. Yet it occurred to me there might be an reason in quantum probability. So is there a...
12. ### Understanding Maxwell's Equations: Integral and Differential Forms

I am getting really confused with the millions of different versions/forms of Maxwell's equations. I know there is differential form and integral form, but sometimes there is a B for magnetic field...other times there is an H. Sometimes there is dependence on \rho or J (current density) and...
13. ### Electromagnetic wave from Maxwells equations in free space

My textbooks says in a region where there is no charge or current Maxwell's equations read divergence of E=0 Curl of E=-dB/dt all d are partial Divergence of B=0 Curl of B=ue(dE/dt) I get the math of showing that there are waves, but I don't get some of these conditions. 1st don't you...
14. ### Maxwells equations from variational principle

1. Hey, I have to find Maxwells equations using the variational principle and the electromagnetic action: S=-\intop d^{4}x\frac{1}{4}F_{\mu\nu}F^{\mu\nu} by using \frac{\delta s}{\delta A_{\mu(x)}}=0 therefore \partial_{\mu}F^{\mu\nu}=0 3. I have had a go at the...
15. ### Self consistent maxwells equations

Are there any articles on solutions for simple self consistent systems in EM, as in when the field equations are coupled with the motion of the particles, I would like to explicitly see the energy conservation in those systems.
16. ### Confused about Maxwells Equations in potential formulation

As i understand it, the Maxwell equations in potential form (in the Lorenz gauge) are basically 4 independent wave equations for the 3 components of A and the 1 component of Phi, with J and ρ acting as source terms: Now from the usual formulation of the Maxwell equations and from experience...
17. ### How do Maxwells equations result from the field tensor?

Hi, I've been trying to solve problem 2.1 a in Peskin and schroeder, an introduction to QFT. The problem is to derive Maxwells equations for free space, which I have almost managed to do, using the Euler- lagrange euqation And the definition of the field tensor as F_{μv} = d_μ A_v - d_v A_μ...
18. ### Maxwells Equations being non-invariant with Galilean transformations

I just purchased a book on the introduction of special relativity and I seem to be stuck on a simple mathematical step. For some reason I just can't see this! This is what it says: Gotta love getting stuck on something when the book says its "Easy to see." Confidence -1.
19. ### Symmetries and Maxwells equations

Hi Maxwells Equations for a time-invariant system are separable, hence we can write a solution as E(r, t) = E(r)E(t). They also mention that if the system is radially invariant, then that implies that the solution splits into a product of radial and angular functions (with 2π periodic angular...
20. ### Maxwells Equations and Time Invariance

Hi In my book it says that if the dielectric function ε is time invariant, we can write a solution to Maxwells equations of the form E(r, t) = E(r)exp(jωt). I agree that the ME are separable, but I don't see how they know that the time-dependence is harmonic? What is so special about exp(jωt)...
21. ### Monochromatic waves and Maxwells equations

Hi Are there other reasons why monochromatic solutions to Maxwells equations of the form E(z, t) = E(z)exp(-iωt) are good other than its plane wave solutions forming a complete set?Niles.
22. ### Solution to Maxwells equations

Homework Statement I am trying to solve prob 4.107 in Schaums' Vector analysis book. Show that solution to Maxwells equations - \DeltaxH=1/c dE/dt, \DeltaxE= -1/c dH/dt, \Delta.H=0, \Delta.E= 4pi\rho where \rho is a function of x,y,z and c is the velocity of light, assumed constant...
23. ### Solutions of maxwells equations in Vaccum

Homework Statement Show that the Fields E= Eo exp{i (k.r-ωt)} and B= Bo exp{i(k.r-ωt) are solutions of Maxwell's Equations in source free vaccum.Starting with maxwells equations in vaccum. And there by Derive the relations between the Magnitudes & Phases of Eo, Bo, ω, k...
24. ### Modified Amperes law, maxwells equations

Using the following identity curl(curl(B))=Grad(Div(B)-Grad^2(B) with maxwells equations in differential form in the absences of sources show that magnetic field obeys the wave equation Grad^2(B)- (1/(C^2))(d^2(B)/dt^2) [b]2. The main probelm is I'm not sure what form of B i should use, i...
25. ### Qualitative description of maxwells equations

Homework Statement Give a qualitative description of maxwell's equation s in non polarizable, non magnetizable media. Homework Equations \oint_{S} E \cdot dA = 4\pi\int_{V}\rho d\tau \oint_{S} B \cdot dA = 0 \oint_{P} E \cdot dl = -\frac{1}{c} \frac{d}{dt} \int_{S} B \cdot dA...
26. ### Charged Massless Waves in Maxwells Equations?

Hello. This is a first post for me. Do Maxwell's equations alone allow for propagating waves in charge/current (\phi,s[/B]J)? I was rather struck dumb by this question out of the blue. I've never seen it addressed, denyed or confirmed. Schematically the electric and magnetic fields are...
27. ### Is Malus' Law a law, or does it derive from Maxwells equations?

Well the question is in the title. Does Malus' Law (http://en.wikipedia.org/wiki/Malus%27s_law ) follow automatically from Maxwell's equations, or is it really an extra thing put in by hand? In particular I'm interested if there is a purely classical electromagnetic explanation (i.e. without...
28. ### Calculating Induced Voltage in a Rotating Rectangular Loop

A 40cmX30cm rectangular loop rotates at 130 r/s in a magnetic field of .06 in the direction normal to the axis of rotation. If the loop has 50 turns, determine the induced voltage in the loop. Heres what I did. I used B dot ds. I had my ds being the sides of the rectangle and coswt in the...