Symmetry in Maxwell’s Equations

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Symmetry in Maxwell’s Equations

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The four fundamental Maxwell's equations exhibit symmetry between electric and magnetic fields.

Consider a space without a charge and consequently with no conduction current.

In this case, Maxwell's equations can be reduced to these four equations.

The first two equations are found to be analogous to each other, with the only difference being the electric and magnetic fields.

The third and fourth equations are also perceived to be similar to each other, implying that a time-varying magnetic field creates an electric field, and symmetrically, a time-varying electric field also creates a magnetic field.

Thus, the symmetry in these four equations indicates the existence of electromagnetic waves, which include time-varying electric and magnetic fields.

Further, due to the simultaneous presence of electric and magnetic fields, a combinational force called Lorentz force acts on any point charge moving under an electromagnetic field.

The Lorentz force equation consists of electric and magnetic fields, and together with Maxwell's equations, comprises all the fundamental laws of electricity and magnetism.

Symmetry in Maxwell’s Equations

Once the fields have been calculated using Maxwell's four equations, the Lorentz force equation gives the force that the fields exert on a charged particle moving with a certain velocity. The Lorentz force equation combines the force of the electric field and of the magnetic field on the moving charge. Maxwell's equations and the Lorentz force law together encompass all the laws of electricity and magnetism. The symmetry that Maxwell introduced into his mathematical framework may not be immediately apparent. Faraday's law describes how changing magnetic fields produce electric fields. The displacement current introduced by Maxwell results instead from a changing electric field and accounts for the fact that changing electric fields produce magnetic fields. The equations for the effects of both changing electric fields and changing magnetic fields differ in form only when the absence of magnetic monopoles leads to missing terms. This symmetry between the effects of changing magnetic and electric fields is essential in explaining the nature of electromagnetic waves.

The later application of Einstein's theory of relativity to Maxwell's complete and symmetric theory showed that electric and magnetic forces are not separate but are different manifestations of the same thing—electromagnetic force. The electromagnetic force and weak nuclear force are similarly unified as the electroweak force. This unification of forces has been one motivation for attempts to unify all of the four basic forces in nature—gravitational, electrical, strong, and weak nuclear forces.