• ### Unit 6: Electromagnetic Waves

Maxwell's four equations describe classical electromagnetism. James Clerk Maxwell, (1831–1879), the Scotish physicist, first published his classical theory of electromagnetism in his textbook, A Treatise on Electricity and Magnetism in 1873. His description of electromagnetism, which demonstrated that electricity and magnetism are different aspects of a unified electromagnetic field, holds true today. In fact, Maxwell's equations are consistent with relativity, which was not theorized until 30 years after he had completed his equations.

Completing this unit should take you approximately 8 hours.

• ### 6.1: Maxwell's Equations

In this unit, we turn from circuits back to a more general setting where it is no longer enough to talk about voltages and currents. We return to electric and magnetic fields to finish constructing our structure of physical laws that govern how electric and magnetic fields are created and how they mutually affect each other. We have already laid the foundation for our construction with Coulomb's and Ampère's Laws (the origins of the fields), and with Faraday's Law.

It is beyond the scope of this course to write these laws out in their modern mathematical form, but we can state them from a conceptual standpoint, in more general terms than we have done so far. In this form they are simply called Maxwell's equations, and there are four of them. Let's state them here before we get started on the relevant section in the text since the formulation in the text may look confusing.

1. Electric charges produce electric field lines that emanate radially from them. This is known as Gauss' Law for Electricity; you can think of Coulomb's Law as a logical consequence of it (valid when there is only a single static point charge).

2. Magnetic field lines are continuous; they have no beginning or end. This second of Maxwell's equations is known as Gauss' Law for Magnetism.

3. A changing magnetic field induces an electromotive force (emf) and, hence, an electric field whose field lines form closed loops. The direction of the emf opposes the change. This third of Maxwell's equations is Faraday's Law of Induction, and includes Lenz's Law.

4. Magnetic fields are generated by moving charges or by changing electric fields. This fourth of Maxwell's equations encompasses Ampère's Law and adds another source of magnetism – changing electric fields.

The statement of Faraday's Law in his third equation mentions loops of electric field lines. To understand this, think about the wire loops we used to experimentally verify the phenomenon of magnetic induction: if a current goes around a loop in the presence of even the tiniest bit of resistance, there must be an electric field that continues in the same direction all around the loop so it can propel the current forward. This means the electric field forms a loop.

The surprising consequence of Faraday's Law is that electric field lines created by changing magnetic fields can, in fact, form closed loops; therefore they do not start and end on electric charges.

So do not be confused by this version of Maxwell's first equation above. Our textbook makes it sound as if all electric field lines start and end on charges. That is true only in electrostatics, where changing fields are not allowed. The electric field lines in the third equation are closed loops, very similar to the magnetic field lines around a current-carrying wire, described by the Law named after Ampère; but closed electric field line loops can only happen if fields change in time.

Maxwell's main contribution (beyond realizing that these four equations provided a complete theory of electromagnetism) was that he postulated an additional source of magnetism mentioned in his fourth equation above. We can call this effect electric induction because it is formally almost identical to magnetic induction, except with the roles of electric and magnetic fields interchanged.

Maxwell postulated this without having any experimental evidence, purely based on a "symmetry argument". He knew magnetic field lines come in loops, and he knew electric field lines can also come in loops – but only if created by changing magnetic fields. This suggested that magnetic and electric fields could be almost mirror images of each other, provided that loops of magnetic field could also be created by changing electric fields.

This purely hypothetical new effect in Maxwell's fourth equation had one dramatic consequence that could be used to test its existence experimentally: that is the existence of electromagnetic waves. They represent the crowning achievement of the theory of electromagnetism, and also illustrate the power of the scientific method. A hypothesis is formed based on concepts that have previously been verified experimentally (e.g., magnetic induction); by using logical reasoning (usually in the language of mathematics) predictions are derived from the new hypothesis; and these predictions lead to new experiments.

• ### 6.2: Electromagnetic Waves

Now, we go into more detail to explain why electric and magnetic fields can form waves – a phenomenon we have only previously encountered in the context of mechanical waves where the medium was something a lot more tangible, such as a water surface.

While electromagnetic waves are a more abstract kind of wave, this designation makes them no less real. What seals their status as a real phenomenon is the fact that electromagnetic waves can transport energy. But unlike all the other waves we encountered earlier, this energy transport can even go through empty space – such as a vacuum, which is the very definition of having no medium present at all!

• ### 6.3: Spectrum of Electromagnetic Radiation

Like all waves, electromagnetic waves are characterized by their speed, frequency, and wavelength. Because the speed is equal to c for all of these waves (as long as they are traveling in air or in empty space), we can actually characterize all electromagnetic waves by a single number: their frequency. The wavelength λ is related to the frequency f by $\lambda =c/f$, so it is known as soon as you specify the frequency.

When you think of electromagnetic waves, you may imagine your radio dial which is labeled in units of frequency (kilohertz or megahertz), or your wifi router which operates at say 2.4 or 5 gigahertz. The prefixes in these numbers (kilo = thousand, mega = million and giga = billion) indicate that there are electromagnetic waves with a huge range of frequencies. But your radio dial does not even begin to scratch the surface as far as the electromagnetic spectrum is concerned.

For example, take a typical musical note of 440 Hz. Sound waves with this frequency are audible to us. But there are also electromagnetic waves of this same frequency! We cannot hear them because our ears can only react to sound waves, not electric or magnetic fields. On the other hand, one of our senses can react to electromagnetic waves: vision. Visible light is an electromagnetic wave, but its frequency is in the range of 400 to 750 terahertz (tera = trillion).

• ### 6.4: Energy and Intensity of Electromagnetic Waves

When a radio station transmits a program on a certain frequency, it uses energy at a certain rate. Likewise, when electromagnetic waves deliver energy to a radio antenna, your eyes, or to a molecule in the microwave oven, that energy arrives at a certain rate. The physical quantity that measures energy transfer per unit time is called power, and it is measured in joules per second, also called watt.

The radio station does not have to send out more power when more people tune in to its program, because the energy put into the antenna is completely decoupled from the antenna as soon as the wave has traveled just a few wavelengths into the surrounding space. So the transmitting antenna neither knows nor cares about how the energy is being used once it has been sent out – just like a lightbulb does not know whether anyone is looking at it or not.

But the receiver does care. To get a good radio reception, we want large currents to be induced in the receiving circuit. Since radio waves spread out from the antenna similar to waves on a pond, the transmitted energy is "diluted" with increasing distance from the source, and you will therefore have better reception if you are close to the transmitter. The same is true for light sources, and any localized source of waves. The amount by which the power carried in a wave is diluted or, conversely, concentrated is quantified by the new concept of intensity.

When you focus sunlight with a burning glass, you concentrate the power entering the glass onto a smaller area (the focal point). That means you deliver the same energy per unit time (power) to a smaller area. This is how sunlight can start a fire: you have increased the intensity of the sunlight. Intensity is defined as power (in watt) delivered per unit area (in square meters). It is the quantity introduced in the following reading.