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gives the force that the fields exert on a particle with charge q moving with velocity . The Lorentz force equation combines the force of the electric field and of the magnetic field on the moving charge. The magnetic and electric forces have been examined in earlier modules. These four Maxwell’s equations are, respectively,
1. Gauss’s law
The electric flux through any closed surface is equal to the electric charge enclosed by the surface. Gauss’s law [ [link] ] describes the relation between an electric charge and the electric field it produces. This is often pictured in terms of electric field lines originating from positive charges and terminating on negative charges, and indicating the direction of the electric field at each point in space.
2. Gauss’s law for magnetism
The magnetic field flux through any closed surface is zero [ [link] ]. This is equivalent to the statement that magnetic field lines are continuous, having no beginning or end. Any magnetic field line entering the region enclosed by the surface must also leave it. No magnetic monopoles, where magnetic field lines would terminate, are known to exist (see Magnetic Fields and Lines ).
3. Faraday’s law
A changing magnetic field induces an electromotive force (emf) and, hence, an electric field. The direction of the emf opposes the change. This third of Maxwell’s equations, [link] , is Faraday’s law of induction and includes Lenz’s law. The electric field from a changing magnetic field has field lines that form closed loops, without any beginning or end.
4. Ampère-Maxwell law
Magnetic fields are generated by moving charges or by changing electric fields. This fourth of Maxwell’s equations, [link] , encompasses Ampère’s law and adds another source of magnetic fields, namely changing electric fields.
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 a changing electric field producing a magnetic field. The equations for the effects of both changing electric fields and changing magnetic fields differ in form only where 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.
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—the 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—the gravitational, electrical, strong, and weak nuclear forces (see Particle Physics and Cosmology ).
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