PHYS102: Introduction to Electromagnetism

6a. State the information contained in each of Maxwell's equations in words

• Can electric field lines form loops?
• Can magnetic field lines emanate from a single point?
• Is an electric current the only way to create a magnetic field?

There are four Maxwell's equations. Each describes possible sources for electric and magnetic fields:

1. Gauss' Law for electricity states that the electric field lines begin at positive charges and terminate at negative charges.
   
   
2. Ampere-Maxwell's Law states that a magnetic field can be created by a current or a moving charge, and by the presence of a changing electric field. To understand how a changing electric field would create a magnetic field, think of dielectric material between the plates of a capacitor. When the capacitor charges, a current flows onto one plate and off of the other. In the gap between the plates, there is a movement of charges, too: it is the individual molecules getting polarized. Although there is no way for a charge to cross all the way from one side of the gap to the other, the charges inside each molecule can move slightly toward or away from the plates. Collectively, this movement of charge within the molecules is enough to create a magnetic field – Maxwell called this the displacement current.
   
   
3. Faraday's Law states that an electric field can be created by the presence of a changing magnetic field. The lines of the electric field created in this way form closed loops, and the strength of the field is proportional to the magnetic flux through those loops.
   
   
4. Gauss' Law for magnetism states that there is no such a thing as a magnetic charge: all sources of magnetic fields, such as bar magnets and currents, contain north and south poles. Separating them and creating a magnetic monopole (another term for a hypothetical magnetic charge) is impossible. Magnetic field lines always form closed loops; they do not have starting or ending points, which would be necessary to have a magnetic charge.

Review Maxwell's Equations: Electromagnetic Waves Predicted and Observed.

6b. Explain how Maxwell's equations resulted in the prediction of electromagnetic waves and the realization that light was an electromagnetic wave

• What mathematical form does the solution of Maxwell's equations take that indicates that electric and magnetic fields can propagate?
• What combination of quantities indicates the speed of propagation? Calculate the value of this combination of quantities to show that it equals the known speed of light.

Maxwell's addition to Ampere's law predicted that a changing electric field can create a magnetic field; on the other hand, Faraday's law states that a changing magnetic field can create a magnetic field. This means that electric and magnetic fields can create each other. This is analogous to how a water wave appears on the surface of a lake when you throw in a pebble: the water level at the impact point changes, and this change creates a slope in the surface. This slope makes the water level at neighboring points change, which in turn changes the slope of the surface next to them, in a repeating chain of events that we call a wave, propagating outward from the center.

Because electric and magnetic fields influence each other in a similar way, they can also create a wave. This is called an electromagnetic wave because neither electric nor magnetic fields alone can create it.

The frequency $f$ and wavelength $\lambda$ depend on the interplay between the inertia of the medium in which the wave is traveling, and the stiffness of the connections between different points in the medium. In all wave phenomena, one finds that the wave speed is related to wavelength and frequency by the same relationship: $v=f \lambda$.

In an electromagnetic wave, the medium is empty space itself, but even this emptiness has two characteristic quantities that are contained in Maxwell's equations: the dielectric permittivity and magnetic permeability, $v=\frac{1}{\sqrt {\varepsilon_{0}\mu_{0}}}$.

The speed of propagation $v$ v from Maxwell's equations is $v=\frac{1}{\varepsilon_{0}\mu_{0}}$. Plugging in the values for $\varepsilon_{0}$ and $\mu _{0}$ results in $v=3\times 10^{B}\ m/s$. One conventionally uses the symbol $c$ instead of $v$ for this special speed.

At the time, when Maxwell and others were developing the theory of electromagnetism, the speed of light was already measured and known to be equal to this value. Therefore, the fact that electric and magnetic fields propagate with this speed served to indicate that light is an electromagnetic wave. Hertz, who showed that light has the properties that electromagnetic waves are expected to have, confirmed this experimentally.

Review how antennas produce and interact with electromagnetic waves in Production of Electromagnetic Waves.

6c. List several examples of electromagnetic phenomena

• What wavelength and frequency do radio waves exist within? How are radio waves used? What are some common phenomena involving radio waves? How are radio waves produced?
• What wavelength and frequency do microwaves exist within? How are microwaves used? How are microwaves produced?
• What wavelength and frequency does visible light exist within? On which side of the visible light spectrum (lower or higher frequency) are infrared and ultraviolet radiation? What are some phenomena that involve infrared and ultraviolet radiation? How is visible light produced?
• What wavelength and frequency do x-rays exist within? What are some applications of x-rays? How are x-rays produced?
• What wavelength and frequency do gamma rays exist within? What are some phenomena involving gamma rays? How are gamma rays produced?

Different types of electromagnetic waves are classified according to their frequency or wavelength. Each can be obtained from the other by using the formula $\lambda = \frac{v}{f}$, where $v$ is the speed of propagation of the wave and $f$ is frequency. Since electromagnetic waves propagate at the speed of light $c=3\times 10 ^{8} \ m/s$, their wavelength and frequency are related as $\lambda = \frac{3\times 10^{8}}{f}$. Together, different kinds of electromagnetic waves form a continuous electromagnetic spectrum.

Radio waves are on the longest wavelength/lowest frequency end of the electromagnetic spectrum. They have wavelengths between several centimeters and thousands of kilometers, which correspond to frequencies from 100 Hz to $10^{10}$ Hz. Devices that operate on radio waves include television and cell phones. The common way to produce, transmit, and receive radio waves is via an antenna: a conductor connected to a circuit with an AC current. Astronomical events also produce radio waves naturally in space.

Microwaves overlap with radio waves but have shorter wavelengths and higher frequencies. Their wavelength is between several millimeters and several centimeters, which correspond to frequencies from $10^{8}$ to $10^{10}$ Hz. This is the highest possible frequency of electromagnetic waves that electronic circuits can produce. Molecules can also produce and absorb microwaves, which results in thermal agitation (rotational and vibrational motion). Thus, microwaves can be used to increase the internal energy (and therefore temperature) of various objects. The microwave ovens in our kitchens also use this property.

Visible light is the most familiar and the narrowest part of the electromagnetic spectrum: it ranges from red light, with a wavelength of about 700 nm ($7\times 10^{-7}$ m) and frequency of about $4\times 10^{14}$ Hz, to violet light, with a wavelength of about 400 nm ($4\times 10^{-7}$ m) and frequency of about $7\times 10^{14}$ Hz. Visible light is produced by atomic vibrations and electronic transitions within atoms and molecules. Infrared radiation has a frequency just below that of red light ($3\times 10^{14}$ Hz, or wavelength of 800 nm), and is produced by the thermal motion of atoms and molecules. It is also known as thermal radiation, which is one of the ways heat transfers.

Ultraviolet (UV) light, as the name implies, has a higher frequency than violet light (it has wavelengths of $10^{-10}$ to $10^{-7}$ m and frequencies of $10^{15}$ to $10^{18}$ Hz). Electronic transitions also produce UV light. It can damage skin and affect vision.

X-rays have a wavelength of $10^{-12}$ to $10^{-10}$ m and a frequency of $10^{18}$ to $10^{20}$ Hz. They overlap with high-frequency UV waves and low-frequency gamma rays. X-rays are produced by high-energy electronic transitions. They are widely known for their applications in medical imaging.

Gamma rays are emitted as a result of nuclear decay and transitions between the energy states of nuclei. Their wavelengths are smaller than $10^{-15}$ m, which corresponds to frequencies higher than $10^{23}$ Hz. Nuclear medicine is based on the properties of gamma rays, and we use them in many of the same applications as x-rays.

Review the wide range of phenomena that involve electromagnetic waves in The Electromagnetic Spectrum.

6d. Solve problems involving properties of electromagnetic waves

• What is the relationship between the wavelength and frequency of electromagnetic waves?
• What does the intensity of electromagnetic radiation depend on?
• What is the relationship between the magnitude of the electric and magnetic fields in an electromagnetic wave?

Typically, problems involving the properties of electromagnetic waves include:

1. Calculating wavelength $\lambda$ and frequency $f$ of the wave. They are related to the speed of light in vacuum, $c=3\times 10^{8}\ m/s$: $\lambda=\frac{c}{f}$
   
   

2. The electric and magnetic field vectors in an electromagnetic wave are always perpendicular to each other, and the direction of the propagation of the wave is perpendicular to both vectors. The $E(t)$ and $B(t)$ functions are in phase (reach their maxima and minima at the same times) and they have a proportional relationship: $E=cB$.
   
   

3. The intensity of radiation is measured using power per unit area: $I=\frac{P}{A}$. For a source of spherical radiation, which spreads out uniformly in all directions, the intensity at distance $R$ away from the source will be $I=\frac{P}{4\pi R^{2}}$; thus, intensity is inversely proportional to the distance away from the source. The average intensity for propagating an electromagnetic wave is proportional to the product of the amplitudes of the electric or magnetic field: $I_{ave}=\frac{E_{max}B_{max}}{2\mu _{0}}$.

This can be expressed in terms of the amplitude of electric or magnetic field alone, by using $E=cB$ and $c=\frac{1}{\sqrt{\varepsilon_{0}\mu_{0}}}$:

$I_{ave}=\frac{cB^{2}_{max}}{2\mu_{0}}=\frac{c\varepsilon_{0}E^{2}_{max}}{2}$.

Review how energy, intensity, and power help us characterize electromagnetic waves in Energy in Electromagnetic Waves.

Unit 6 Vocabulary

You should be familiar with these terms to complete the final exam.

• Ampere-Maxwell's Law
• antenna
• displacement current
• Faraday's Law
• flux
• frequency
• gamma ray
• Gauss' Law
• infrared radiation
• intensity
• light
• magnetic monopole
• Maxwell's Equations
• microwave
• radiation
• radio wave
• ultraviolet radiation
• visible light
• wave
• wavelength
• x-ray