PHYS102 Study Guide

Unit 6: Maxwell's Equations

6a. state Maxwell's equations and identify each mathematical operator and physical quantity in the equations

  1. What is the mathematical expression for the flux of a field? What does the flux of a field represent in the real world?
  2. Write the mathematical expression for the circulation of a field. What does the circulation of an electric or magnetic field represent in the real world?
  3. According to Maxwell's equations, what are the flux and circulation of an electric field equal to? What are the flux and circulation of a magnetic field equal to?

The flux of a field through a surface is the surface integral of the field vector. For example, the flux of an electric field through the surface  S  is  \Phi_E=\int_S \vec{E}\ \mathrm{d}\vec{A} . The flux through a surface indicates the number of field lines passing through the surface. For a closed surface, positive flux indicates the number of the field lines leaving (or coming out of) the surface, while negative flux indicates the number of lines entering the surface.

Maxwell's equations that involve flux are also Gauss' Laws for electricity and magnetism. Gauss' Law for electricity states that the flux of the electric field through a closed surface equals the charge enclosed by the surface divided by electric permittivity of vacuum:  \Phi_E=\oint \vec{E}\ \mathrm{d}\vec{A}=\frac{Q_\mathrm{enc}}{\varepsilon_0} , where  \varepsilon_0=8.85\times 10^{-12}\ \mathrm{\frac{C^2}{Nm^2}} .

Gauss' Law for magnetism states that the flux of a magnetic field through any closed surface is always zero:  \Phi_B=\oint \vec{B}\ \mathrm{d}\vec{A}=0 .

The circulation of a field is the line integral of the field vector. For an electric field, it is expressed as  \int_l \vec{E}\ \mathrm{d}\vec{l} . This is also the expression for the work that the electric field performs on a unit charge. For an electrostatic field, which is conservative, the circulation of the field around a closed loop is zero:  \oint \vec{E}\ \mathrm{d}\vec{l}=0 . However, this changes in the presence of changing magnetic flux. One of Maxwell's equations, Faraday's Law, states that the circulation of electric flux around a closed loop equals the negative rate of change of the magnetic flux through the surface bounded by the loop:  \oint \vec{E}\ \mathrm{d}\vec{l}=\frac{\mathrm{d}\Phi_B}{\mathrm{d}t} .

For magnetic fields, circulation is expressed as  \int_l \vec{B}\ \mathrm{d}\vec{l} . This is not equivalent to work, however, because the magnetic force is perpendicular to the field. Ampere-Maxwell's law states that the circulation of a magnetic field around a closed loop equals the magnetic permittivity of vacuum times the sum of the current and the displacement current through the surface bounded by the  \oint \vec{B}\ \mathrm{d}\vec{l}=\mu_0\left ( I+I_d \right ) . Here,  \mu_0=4\pi\times 10^{-7}\ \mathrm{\frac{Tm}{A}} is the magnetic permeability of vacuum. The displacement current is not an actual current, but a quantity with the same dimensions as current, defined as the rate of change of electric flux times the electric permittivity of vacuum:  I_d=\varepsilon_0\frac{\mathrm{d}\Phi_E}{\mathrm{d}t} .

Note that each of these statements about Maxwell's equations is for fields in a vacuum, as opposed to media that have different electric permittivity and magnetic permeability. Read about Maxwell's equations in the introduction to A Dash of Maxwell's, which summarizes how all that is known about electric and magnetic fields is expressed in these four equations.

 

6b. state in words the information contained in each of Maxwell's equations

  1. According to Gauss' Law for electric fields and Faraday's Law, what are the possible sources of electric fields? How do electric fields differ when created by different kinds of sources?
  2. According to Ampere-Maxwell's Law and Gauss' Law for magnetic fields, what are the possible sources of magnetic fields? What features are common to all magnetic fields?

There are four Maxwell's equations, each of which describes possible sources for electric and magnetic fields:

Gauss' Law for electricity states that the electric flux through a closed surface is proportional to the net charge enclosed by the surface. This means that an electric field is created by the presence of an electric charge. Electric field lines begin at positive charges and terminate at negative charges.

Ampere-Maxwell's Law states that the circulation of a magnetic field around a closed loop is proportional to the sum of currents through the surface bounded by the loop plus the displacement current through that surface. The displacement current is a quantity proportional to the rate of change of the electric flux through the surface. This means that a magnetic field can be created by a current or a moving charge, and by the presence of changing electric flux.

Faraday's Law states that the circulation of an electric field around a closed loop equals the negative rate of change of the magnetic flux through that loop. This means that an electric field can be created by the presence of changing magnetic flux. The lines of the electric field created in this way form closed loops. Unlike the field created by stationary electric charges, this field not conservative.

Gauss' Law for magnetism is analogous to Gauss' Law for electricity. It states that the magnetic flux through any closed surface is zero. This means that there is no such a thing as a magnetic charge: if there was a magnetic charge similar to an electric charge, it would be possible to surround it with a surface, and the magnetic flux through that surface would be non-zero. This is not the case, however: all sources of magnetic fields, such as bar magnets and currents, contain north and south poles. Separating them and creating a magnetic monopole, or a 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.

Watch Maxwell's Equations and Electromagnetic Waves I to review each of Maxwell's equations. You can skip the first twenty minutes, which are devoted to the derivation of the mechanical wave equation.

 

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

  1. What mathematical form does the solution of Maxwell's equations take that indicates that electric and magnetic fields can propagate?
  2. 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.

The solution of Maxwell's equations, which show that electric and magnetic field are a function of time and space, take the form of  \vec{E}(x,t)=\vec{E_0}\cos{(kx-\omega t)} and  \vec{B}(x,t)=\vec{B_0}\cos{(kx-\omega t)} . Alternatively, these can be written as  \vec{E}(x,t)=\vec{E_0}\cos[{k(x-vt)}] and  \vec{B}(x,t)=\vec{B_0}\cos[{k(x-vt)}] . This kind of function of the argument  x-vt describes a wave propagating with a velocity  v . Here, the angular velocity  \omega , wave number  k , and speed of propagation  v are related as  v=\frac{\omega}{k} . The speed of propagation  v  from Maxwell's equations is  v=\frac{1}{\sqrt{\varepsilon_0 \mu_0}} . Plugging in the values for \varepsilon_0 and  \mu_0 results in  v=3\times 10^{8}\ \mathrm{m/s} .

At the time when the theory of electromagnetism was being developed by Maxwell and others, 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. This was confirmed experimentally by Hertz, who showed that light has the properties that electromagnetic waves are expected to have.

For a detailed explanation of how Maxwell's equations result in the prediction of the existence of electromagnetic waves, watch Maxwell's Equations and Electromagnetic Waves II and read Why Things Radiate.

 

6d. list several examples of electromagnetic phenomena

  1. 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?
  2. What wavelength and frequency do microwaves exist within? How are microwaves used? How are microwaves produced?
  3. 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?
  4. What wavelength and frequency do x-rays exist within? What are some applications of x-rays? How are x-rays produced?
  5. 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 with the speed of light  c=3\times 10^8\ \mathrm{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 1010 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. Radio waves are also naturally produced in space by astronomical events.

Microwaves overlap with radio waves but have shorter wavelengths and higher frequencies. Their wavelength is between several millimeters to several centimeters, which correspond to frequencies from 108 to 1010 Hz. This is the highest possible frequency of electromagnetic waves that can be produced by electronic circuits. Microwaves can also be produced and absorbed by molecules, 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. This property is widely used in the household appliance of the same name.

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 × 10-7 m) and frequency of about 4 × 1014 Hz, to violet light, with a wavelength of about 400 nm (4 × 10-7 m) and frequency of about 7 × 1014 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 × 1014 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 1015 to 1018 Hz). UV light is also produced by electronic transitions, and it is known to have an effect on vision and can damage the skin.

X-rays have a wavelength of 10-12 to 10-10 m and a frequency of 1018 to 1020 Hz. They overlap with high-frequency UV waves and low-frequency gamma rays. X-rays are produced by high-energy electronic transitions and 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 1023 Hz. Nuclear medicine is based on the properties of gamma rays, and they are used in many of the same applications as x-rays.

See the detailed description of various electromagnetic phenomena and their applications in section 24.3 of College Physics.

 

6e. solve problems involving properties of electromagnetic waves

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

Typically, problems involving the properties of electromagnetic wave include:

  • Calculating wavelength  \lambda , frequency  f , angular frequency  \omega , or wave number  k of the wave. The appropriate formulas are:

     \lambda=\frac{c}{f} ( c =3\times10^8 \mathrm{m/s} is the speed of light in vacuum)

     \omega=2\pi f

     k=\frac{2\pi}{\lambda}=\frac{\omega}{c}
  • 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 and is the same as the direction of the cross-product of  \vec{E} and  \vec{B} . 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 .
  • The intensity of a radiation is measured is power per unit area:  I=\frac{P}{A} . For a source of spherical radiation, spreading 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 of propagating electromagnetic wave is proportional to the product of the amplitudes of the electric or magnetic field:  I_\mathrm{avg}=\frac{E_\mathrm{max} B_\mathrm{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_\mathrm{avg}=\frac{cB^2_\mathrm{max}}{2\mu_0}=\frac{c\varepsilon_0E^2_\mathrm{max}}{2} .

Refer to sections 24.2 and 24.4 of College Physics to see applications of these formulas.

 

Unit 6 Vocabulary

This vocabulary list includes terms that might help you with the review items above and some terms you should be familiar with to be successful in completing the final exam for the course.

Try to think of the reason why each term is included.

  • Ampere-Maxwell's Law
  • Circulation
  • 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
  • Wave number
  • Wavelength
  • X-ray