Maxwell’s Equations Maxwell Summarizes all of Physics using Fields

Gauss’ Law The net electric field exiting a closed surface is proportional to the charge enclosed by the surface. All field lines must end on charges. The experimental source of Gauss’ Law is Coulomb’s Law. ©2008 by W.H. Freeman and Company

Gauss’ Law for Magnetism The net magnetic field exiting a closed surface is 0. Magnetic field lines do not end. The experimental “confirmation” of Gauss’ law is the fact that magnetic point charges, called magnetic monopoles, have never been observed experimentally.

Faraday’s Law in Field Form In our form: The emf in Faraday’s Law is the induced emf caused by the changing field. We could write it in terms of a path integral of the electric field, since by definition of emf, We could actually take this integral with all electric fields over a closed loop, since by Kirchhoff’s Law, all the other emf’s add to zero over a closed loop. The loop doesn’t even have to be a real loop. It can be any closed curve.

Faraday’s Law The line integral of the electric field around any closed curve is proportional to the change in flux in the area bounded by the curve. Changing magnetic fields cause electric fields. ©2008 by W.H. Freeman and Company

Ampere’s Law The line integral of the magnetic field around a closed curve will be proportional to the current flowing through the area bounded by the curve. Currents produce magnetic fields. ©2008 by W.H. Freeman and Company

What’s Wrong with This Picture? The left hand side of the equations are symmetric in pairs. However, the right hand sides are not. The asymmetry in the first two equations arises because there are no magnetic monopoles, so a hypothetical Qmagnetic= 0 .

What’s Wrong with This Picture? Similarly, there is a electric current in the third equation, but no magnetic current in the fourth, again because as far as we know, there are no magnetic “charges” to form a current. These asymmetries have driven the search for magnetic monopoles.

What’s Wrong with This Picture? These equations exhibit another asymmetry: Changing magnetic flux can produce an electric field, but not vice versa. Physics equations and nature are rarely asymmetry. Maxwell sensed something was missing from the 3rd equation.

Maxwell’s Displacement Current Consider two surfaces bounded by the same closed curve while a capacitor is charging According to Ampere’s Law, the current flowing through both surfaces should be the same. However, a current flows through S1 but not S2. ©2008 by W.H. Freeman and Company

Maxwell’s Displacement Current A current flowing through S1 implies the charge on the capacitor is changing. Hence the electric field in the capacitor is changing. Thus the electric flux through S2 changes. This changing flux has the same effect as a current. ©2008 by W.H. Freeman and Company

Maxwell’s Displacement Current Maxwell’s prediction is generally applicable, not just in capacitors. A changing electric flux will produce a magnetic field. Ampere’s Law, as generalized by Maxwell, is ©2008 by W.H. Freeman and Company

Maxwell’s Equations completed Electric field lines start and end on charges. Magnetic field lines don’t end. Electric currents or changing electric fields produce magnetic fields. Changing magnetic fields produce electric fields. (As would a “magnetic monopole current”, if one were to exist.) As far as we know. Which it does not.

Maxwell’s Equations Maxwell showed that these equations explain the existence of the waves that Hertz discovered. The speed of these waves would be Using µ0=4π x 10-7 N/A2 and ε0= 8.85 x 10-12 C2/(Nm2), find the speed of these waves.

Electromagnetic Waves Maxwell showed that the waves that are predicted by these equations have the same speed as the speed of light. He concluded, correctly, that light is an electromagnetic phenomenon. An electromagnetic wave consists of perpendicular electric and magnetic fields. When the electric field changes, a magnetic field is produced. When the magnetic field changes, an electric field is produced.