Maxwell's equations

Examples of Maxwell's equations in the following topics:

• Maxwell's Equations

  • Maxwell's equations help form the foundation of classical electrodynamics, optics, and electric circuits.
  • Maxwell's equations are a set of four partial differential equations that, along with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits.
  • Maxwell's equations can be divided into two major subsets.
  • The other two, Faraday's law and Ampere's law with Maxwell's correction, describe how induced electric and magnetic fields circulate around their respective sources.
  • Each of Maxwell's equations can be looked at from the "microscopic" perspective, which deals with total charge and total current, and the "macroscopic" set, which defines two new auxiliary fields that allow one to perform calculations without knowing microscopic data like atomic-level charges.

• Gallilean-Newtonian Relativity

  • Assuming that mass is invariant in all inertial frames, the above equation shows that Newton's laws of mechanics, if valid in one frame, must hold for all frames.
  • Both Newtonian mechanics and the Maxwell's equations were well established by the end of the 19th century.
  • The puzzle lied in the fact that the Galilean invariance didn't work in Maxwell's equations.
  • That is, unlike Newtonian mechanics, Maxwell's equations are not invariant under a Galilean transformation.

• Further Reading

  • The general development of Maxwell's equations and the polarization of radiation are examined in Chapter 6 of

• Maxwell's Predictions and Hertz' Confirmation

  • Maxwell's prediction of the electromagnetic force was confirmed by Hertz who generated and detected electromagnetic waves.
  • Maxwell's equations predict that regardless of wavelength and frequency, every light wave has the same structure.
  • This means Maxwell's equations predicted that radio and x-ray waves existed, even though they hadn't actually been discovered yet.
  • Simple and brilliant in their insight, Maxwell's famous equations would still be hard to prove.
  • Explain how Maxwell's prediction of the electromagnetic force was confirmed by Hertz

• Back to Maxwell's Equations

  • and a similar equation for the magnetic field.
  • Maxwell's and his contemporaries spoke of light traveling through some medium known as the aether.

• Maxwell's Equations

  • The force on the particle is given by the Lorentz force equation,
  • The equations that describe the dynamics of the fields are Maxwell's equations,
  • The equations were discovered by various people.
  • It might be more appropriate to call the penultimate, Maxwell's equation, because Ampere's law as it was originally formulated was
  • where we have used the first of Maxwell's equations to simply the result and find that the Maxwell's addition makes the full set of equations consistent with charge conservation.

• Waves

  • Let's take the curl of the third equation and combine it with the fourth to get
  • The first term on the right-hand side vanishes so we get the final wave equation,
  • We write a general solution to the wave equation as a sum of harmonically varying waves such as

• Electromagnetic Potentials

  • Looking at the structure of Maxwell's equations, we can see that we can express the magnetic field as the curl of another field, the vector potential,
  • The expression of the fields in terms of the vector and scalar potential guarantees that two out of four of Maxwell's equations are satisfied.
  • Let's substitute our results into the remaining Maxwell's equations,
  • Let's look at the last of the charge density equations equations some more,
  • Although it looks like that the equation is a bit more complicated than before, it now has the precise form of the equation with the current.

• Antennae

  • Maxwell's equations predicted that all light waves have the same structure, regardless of wavelength and frequency.
  • Maxwell's 1865 prediction passed an important test in 1888, when Heinrich Hertz published the results of experiments in which he showed that radio waves could be manipulated in the same ways as visible light waves.

• Gauss's Law

  • It is one of the four Maxwell's equations which form the basis of classical electrodynamics, the other three being Gauss's law for magnetism, Faraday's law of induction, and Ampère's law with Maxwell's correction.