16. Maxwell’s equations 16.1. Gauss’ law for the magnetic field Electric charges can be separated into positive and negative. If we cut the magnet to separate the north and south poles, we obtain again two smaller magnets with both poles, two magnetic dipoles. The cutting process can be continued until single electrons and nuclei are obtained and still we have north and south poles. The conclusion is that: magnetic monopoles do not exist and the simplest magnetic structure that can exist is a magnetic dipole. In other words it can be said that: the magnetic flux through any closed Gaussian surface is zero Gauss’ law for magnetic fields (16.1) Cutting of the bar magnet gives two separate magnets From HRW 3

16.2. Induced magnetic fields In a previous chapter we saw that Faraday’s law of induction can be written in a form (16.2) what means that the changing magnetic flux ΦB induces an electric field E. From the Amper’s law we know that the flowing current generates the magnetic field (16.3) Equations (16.2) and (16.3) are not symmetrical. J.C. Maxwell (1873) was the first who postulated extension of the Amper’s law (16.3) to the form (16.4) that is both flowing current I and a changing electric flux ΦE induce a magnetic field. This idea was next verified experimentally. The magnetic field B between the plates of a charging capacitor is induced by the changing electric field E, the flowing current I generates the magnetic field around a wire.

Induced magnetic fields, cont. Sample problem Calculate the induced magnetic field between the plates of a charged capacitor. The circular plates are of radius R. r ≤ R We start using the Maxwell’s law of induction (16.5) ΦE is a flux of electric field through the surface circumferenced by loop L. Due to the circular symmetry we assume that B has the same magnitude around the loop and also assume that electric field is uniform and perpendicular to the plates, then from (16.5) one gets For r = R = 5 cm and dE/dt = 1012 V/m·s Such a small magnitude of B (hard to measure) results from the low rate of change of E. For higher frequencies the induced magnetic field increases. The induced electric fields (emf) are higher because in experiments the coils of many turns are used.

Maxwell’s equations The last equation together with previously discussed Gauss’ equations for electric and magnetic fields and the Faraday’s law, form the four fundamental equations of electromagnetism, called Maxwell’s equations (16.6) (16.7) (16.8) (16.9) With the help of these equations Maxwell introduced the hypothesis of light as an electromagnetic wave and obtained the magnitude of its speed. The existence of EM waves was verified experimentally by H. Hertz after the Maxwell’s death. Electric charge is a source of an electric field Magnetic monopoles do not exist Changing magnetic flux induces an electric field Both flowing current and a changing electric flux induce a magnetic field