Maxwell’s Equations and Electromagnetic Waves

Maxwell’s Equations and Electromagnetic Waves

Introduction

In 1864, James Clerk Maxwell (1831-1879) took all of the then known equations of electricity and magnetism, and with the addition of a new term to one of the equations, combined them into only four equations that could be used to derive all the results of electromagnetic theory. These four equations came to be known as Maxwell’s equations. The four Maxwell’s equations are (1) Gauss’s law for electricity, (2) Gauss’s law for magnetism, (3) Ampere’s law with the addition of a new term called the displacement current, and (4) Faraday’s law of electromagnetic induction. With these four equations, Maxwell predicted that waves should exist in the electromagnetic field. Thirteen years later, in 1887, Heinrich Hertz (1857-1894) produced and detected these electromagnetic waves. Maxwell also predicted that the speed of these electromagnetic waves should be 3  108 m/s. Observing that this is also the speed of light, Maxwell declared that light itself is an electromagnetic wave. In fact it eventually became known that there was an entire spectrum of these electromagnetic waves. They differed only in frequency and wavelength. Finally, it was found that these electromagnetic waves are capable of transmitting energy from one place to another, even through the vacuum of space.

The Displacement Current and Ampere’s Law

In the study of a capacitor  (where we assumed that the current was conventional current, that is a flow of positive charges) we saw that when the switch in the circuit is closed, charge flows from the positive terminal of the battery to one plate of the capacitor, called the positive plate, and charge also flows from the negative plate of the capacitor back to the negative terminal of the battery. This is  Until the plates are completely charged, there is a current into the positive plate, and a current out of the negative plate, yet there seems to be no current between the plates. There is thus a discontinuity in the current in the circuit because of the capacitor.
As charge is placed on the plates of the capacitor an electric field is set up between the plates. The electric field between the plates of a capacitor was found by Gauss’s law as

E=qoA