 # Maxwell’s Equations and Electromagnetic Waves

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Maxwell’s Equations and Electromagnetic Waves
Chapter 31 opener. These circular disk antennas, each 25 m in diameter, are pointed to receive radio waves from out in space. Radio waves are electromagnetic (EM) waves that have frequencies from a few hundred Hz to about 100 MHz. These antennas are connected together electronically to achieve better detail; they are a part of the Very Large Array in New Mexico searching the heavens for information about the Cosmos. We will see in this Chapter that Maxwell predicted the existence of EM waves from his famous equations. Maxwell’s equations themselves are a magnificent summary of electromagnetism. We will also examine how EM waves carry energy and momentum.

Content Changing Electric Fields Produce Magnetic Fields; Ampère’s Law and Displacement Current Gauss’s Law for Magnetism Maxwell’s Equations Production of Electromagnetic Waves Electromagnetic Waves, and Their Speed, Derived from Maxwell’s Equations Light as an Electromagnetic Wave and the Electromagnetic Spectrum

Content Measuring the Speed of Light
Energy in EM Waves; the Poynting Vector Radiation Pressure Radio and Television; Wireless Communication

Displacement Current Ampère’s law relates the magnetic field around a current to the current through a surface. Current I passes through both surface 1 and 2. Figure Ampère’s law applied to two different surfaces bounded by the same closed path.

Displacement Current In order for Ampère’s law to hold, it can’t matter which surface we choose. But look at a discharging capacitor; there is a current through surface 1 but none through surface 2: Figure A capacitor discharging. A conduction current passes through surface 1, but no conduction current passes through surface 2. An extra term is needed in Ampère’s law.

Displacement Current Maxwell proposed a new type of current, called the displacement current, ID. Therefore, Ampère’s law is modified accordingly as

Displacement Current Maxwell realized that the changing electric flux must be associated with a magnetic field. For example, for a parallel-plate capacitor:

Displacement Current With Maxwell’s modification, Ampere’s law now becomes Where the second term is called the displacement current

Displacement Current Charging capacitor.
A 30-pF air-gap capacitor has circular plates of area A = 100 cm2. It is charged by a 70-V battery through a 2.0-Ω resistor. At the instant the battery is connected, the electric field between the plates is changing most rapidly. At this instant, calculate (a) the current into the plates, and (b) the rate of change of electric field between the plates. (c) Determine the magnetic field induced between the plates. Assume E is uniform between the plates at any instant and is zero at all points beyond the edges of the plates. Solution: a. At t = 0, all the voltage is across the resistor, so the current is V/R = 35 A. b. The field at any instant is (Q/A)/ε0. So dE/dt = (dQ/dt)/(Aε0) = (I/A)/ε0 = 4.0 x 1014 V/m·s. c. Due to symmetry, the lines of B are circular and perpendicular to E. If E is constant over the area of the plates, Ampere’s law gives B = μ0ε0r0/2 dE/dt = 1.2 x 10-4 T.

Gauss’s Law for Magnetism
Gauss’s law relates the electric field on a closed surface to the net charge enclosed by that surface. The analogous law for magnetic fields is different, as there are no single magnetic point charges (monopoles):

Maxwell’s Equations The complete set of equations describing electric and magnetic fields is called Maxwell’s equations. In the absence of dielectric or magnetic materials, they are:

Maxwell’s Equations In the absence of currents and charges, they are:

Electromagnetic Waves
According to Maxwell’s Equations in the absence of currents and charges, the E and B fields also satisfy Maxwell’s wave equations:

Electromagnetic Waves
A wave traveling along the x-axis with a speed v satisfies the wave equation Therefore, we see that the wave speed is which is exactly the speed of light.

Production of Electromagnetic Waves
Since a changing electric field produces a magnetic field, and a changing magnetic field produces an electric field, once sinusoidal fields are created they can propagate on their own. These propagating fields are called electromagnetic waves.

Production of Electromagnetic Waves
Oscillating charges will produce electromagnetic waves: Figure Fields produced by charge flowing into conductors. It takes time for the and fields to travel outward to distant points. The fields are shown to the right of the antenna, but they move out in all directions, symmetrically about the (vertical) antenna.

Production of Electromagnetic Waves
Close to the antenna, the fields are complicated, and are called the near field: Figure Sequence showing electric and magnetic fields that spread outward from oscillating charges on two conductors (the antenna) connected to an ac source (see the text).

Production of Electromagnetic Waves
Far from the source, the waves are plane waves: Figure (a) The radiation fields (far from the antenna) produced by a sinusoidal signal on the antenna. The red closed loops represent electric field lines. The magnetic field lines, perpendicular to the page, also form closed loops. (b) Very far from the antenna the wave fronts (field lines) are essentially flat over a fairly large area, and are referred to as plane waves.

Production of Electromagnetic Waves
The electric and magnetic waves are perpendicular to each other, and to the direction of propagation. Figure Electric and magnetic field strengths in an electromagnetic wave. E and B are at right angles to each other. The entire pattern moves in a direction perpendicular to both E and B.

Electromagnetic Waves
This figure shows an electromagnetic wave of wavelength λ and frequency f. The electric and magnetic fields are given by Figure Applying Faraday’s law to the rectangle (Δy)(dx).

Electromagnetic Waves
The electric and magnetic fields are related by

Electromagnetic Waves
Determining E and B in EM waves. Assume a 60-Hz EM wave is a sinusoidal wave propagating in the z direction with E pointing in the x direction, and E0 = 2.0 V/m. Write vector expressions for E and B as functions of position and time. Solution: The wavelength is c/f = 5.0 x 106m. The wave number is 2π/λ = 1.26 x 10-6 m-1. The angular frequency is 2πf = 377 rad/s. Finally, B0 = E0/c = 6.7 x 10-9 T. B must be in the y direction, as E, v, and B are mutually perpendicular. Now substitute in equations 31-7.

Energy and Momentum Energy is stored in both electric and magnetic fields, giving the total energy density of an electromagnetic wave: Since E=cB, each field contributes half the total energy density:

Energy and Momentum This energy is transported by the wave.
Figure Electromagnetic wave carrying energy through area A.

Energy and Momentum The energy transported through a unit area per unit time is called the intensity: The energy floe is perpendicular to both E and B. The Poynting vector is defined as

Energy and Momentum Typically we are interested in the average value of Sav:

Energy and Momentum E and B from the Sun.
Radiation from the Sun reaches the Earth (above the atmosphere) at a rate of about 1350 J/s·m2 (= 1350 W/m2). Assume that this is a single EM wave, and calculate the maximum values of E and B. Solution: The rate at which radiation arrives is the average value of S; we can use equation 31-19a to find E0 and B0. E0 = 1.01 x 103 V/m. B0 = 3.37 x 10-6 T.

Energy and Momentum In addition to carrying energy, electromagnetic waves also carry momentum: If the wave is completely absorbed, the radiation pressure exerted on the surface is

Radiation Pressure If the wave is perfectly reflected, the momentum change is doubled and the radiation pressure exerted on the surface is

Radiation Pressure Solar pressure.
Radiation from the Sun that reaches the Earth’s surface (after passing through the atmosphere) transports energy at a rate of about 1000 W/m2. Estimate the pressure and force exerted by the Sun on your outstretched hand. Solution: Estimate P = S/c = 3 x 10-6 N/m2. If your hand is about 10 cm by 20 cm, or 0.02 m2, this translates to a force of 6 x 10-8 N.

Radiation Pressure A solar sail.
Proposals have been made to use the radiation pressure from the Sun to help propel spacecraft around the solar system. (a) About how much force would be applied on a 1 km x 1 km highly reflective sail, and (b) by how much would this increase the speed of a 5000-kg spacecraft in one year? (c) If the spacecraft started from rest, about how far would it travel in a year? Solution: a. Since it is highly reflective, the pressure is 2S/c, so the total force is 6 N. (Clearly, you would want to start your journey far from the Earth so its gravity doesn’t interfere!) b. The acceleration is about 1.2 x 10-3 m/s2. Using standard kinematics with constant acceleration (assuming the sail doesn’t get far enough away from the Sun that the radiation pressure decreases noticeably) gives the final speed as 4 x 104 m/s. c. Again using standard kinematics with constant velocity, the distance is 6 x 1011 m. This is about four times the Sun-Earth distance.

Light as an Electromagnetic Wave and the Electromagnetic Spectrum
The frequency of an electromagnetic wave is related to its wavelength and to the speed of light:

Light as an Electromagnetic Wave and the Electromagnetic Spectrum
Electromagnetic waves can have any wavelength; we have given different names to different parts of the wavelength spectrum. Figure Electromagnetic spectrum.

Light as an Electromagnetic Wave and the Electromagnetic Spectrum
Wavelengths of EM waves. Calculate the wavelength (a) of a 60-Hz EM wave, (b) of a 93.3-MHz FM radio wave, and (c) of a beam of visible red light from a laser at frequency 4.74 x 1014 Hz. Solution: The wavelength is the speed of light multiplied by the frequency. a. 5.0 x 106 m b m c x 10-7 m

Light as an Electromagnetic Wave and the Electromagnetic Spectrum
Cell phone antenna. The antenna of a cell phone is often ¼ wavelength long. A particular cell phone has an 8.5-cm-long straight rod for its antenna. Estimate the operating frequency of this phone. Solution: The frequency is the speed of light divided by the wavelength (which is 4 times the antenna length): f = 880 MHz.

Light as an Electromagnetic Wave and the Electromagnetic Spectrum
Phone call time lag. You make a telephone call from New York to a friend in London. Estimate how long it will take the electrical signal generated by your voice to reach London, assuming the signal is (a) carried on a telephone cable under the Atlantic Ocean, and (b) sent via satellite 36,000 km above the ocean. Would this cause a noticeable delay in either case? Solution: The time is the distance divided by the speed; assume the speed of light in both cases. a. t = s. b. Remember the signal must make a round trip: t = 0.24 s, which is noticeable.

Measuring the Speed of Light
The speed of light was known to be very large, although careful studies of the orbits of Jupiter’s moons showed that it is finite. One important measurement, by Michelson, used a rotating mirror: Figure Michelson’s speed-of-light apparatus (not to scale).

Measuring the Speed of Light
Over the years, measurements have become more and more precise; now the speed of light is defined to be c = × 108 m/s. This is then used to define the meter.

Radio and Television; Wireless Communication
This figure illustrates the process by which a radio station transmits information. The audio signal is combined with a carrier wave. Figure Block diagram of a radio transmitter.

Radio and Television; Wireless Communication
The mixing of signal and carrier can be done two ways. First, by using the signal to modify the amplitude of the carrier (AM): Figure In amplitude modulation (AM), the amplitude of the carrier signal is made to vary in proportion to the audio signal’s amplitude.

Radio and Television; Wireless Communication
Second, by using the signal to modify the frequency of the carrier (FM): Figure In frequency modulation (FM), the frequency of the carrier signal is made to change in proportion to the audio signal’s amplitude. This method is used by FM radio and television.

Radio and Television; Wireless Communication
At the receiving end, the wave is received, demodulated, amplified, and sent to a loudspeaker. Figure Block diagram of a simple radio receiver.

Radio and Television; Wireless Communication
The receiving antenna is bathed in waves of many frequencies; a tuner is used to select the desired one. Figure Simple tuning stage of a radio.

Radio and Television; Wireless Communication
A straight antenna will have a current induced in it by the varying electric fields of a radio wave; a circular antenna will have a current induced by the changing magnetic flux. Figure Antennas. (a) Electric field of EM wave produces a current in an antenna consisting of straight wire or rods. (b) Changing magnetic field induces an emf and current in a loop antenna.

Radio and Television; Wireless Communication
Tuning a station. Calculate the transmitting wavelength of an FM radio station that transmits at 100 MHz. Solution: The wavelength is c/f = 3.0 m.

Summary Maxwell’s equations are the basic equations of electromagnetism:

Summary Electromagnetic waves are produced by accelerating charges; the propagation speed is given by The fields are perpendicular to each other and to the direction of propagation.

Summary The wavelength and frequency of EM waves are related:
The electromagnetic spectrum includes all wavelengths, from radio waves through visible light to gamma rays. The Poynting vector describes the energy carried by EM waves:

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