Chapter 31 Maxwell s Equations and Electromagnetic Waves
Units of Chapter 31 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
Units of Chapter 31 Measuring the Speed of Light Energy in EM Waves; the Poynting Vector Radiation Pressure Radio and Television; Wireless Communication
E&M Equations to date E da E d B d Q 0 enc 0 d dt I B
31-2 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):
E&M Equations to date - updated E da Q enc 0 B da E d B d Q 0 0 d dt I mag enc B No effect since RHS identically zero These two not pretty, i.e., not symmetric mag dq mag dq Now, I suggests I 0 dt dt
E&M Equations to date more updated Qenc E da 0 B da Q E d 0 d dt mag enc B I mag B d??? I 0 0 Wouldn t it be nice if we could replace??? with something?
31-1 Changing Electric Fields Produce Magnetic Fields; Ampère s Law and Displacement Current Ampère s law relates the magnetic field around a current to the current through a surface. B d??? I 0
31-1 Changing Electric Fields Produce Magnetic Fields; Ampère s Law and 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: B d??? I 0
31-1 Changing Electric Fields Produce Magnetic Fields; Ampère s Law and Displacement Current Therefore, Ampère s law is modified to include the creation of a magnetic field by a changing electric field the field between the plates of the capacitor in this example:
31-1 Changing Electric Fields Produce Magnetic Fields; Ampère s Law and Displacement Current The second term in Ampere s law has the dimensions of a current (after factoring out the μ 0 ), and is sometimes called the displacement current: where
31-1 Ampère s Law and Displacement Current Example 31-1: Charging capacitor. A 30-pF air-gap capacitor has circular plates of area A = 100 cm 2. 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.
31-3 Maxwell s Equations We now have a complete set of equations that describe electric and magnetic fields, called Maxwell s equations. In the absence of dielectric or magnetic materials, they are: E da Q enc mag B da 0Qenc d Ed dt 0 B mag de B d 0 0 0I dt I 0
31-4 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.
ConcepTest 31.1a A loop with an AC current produces a changing magnetic field. Two loops have the same area, but one is made of plastic and the other copper. In which of the loops is the induced voltage greater? EM Waves I 1) the plastic loop 2) the copper loop 3) voltage is same in both Plastic Copper
ConcepTest 31.1a A loop with an AC current produces a changing magnetic field. Two loops have the same area, but one is made of plastic and the other copper. In which of the loops is the induced voltage greater? EM Waves I 1) the plastic loop 2) the copper loop 3) voltage is same in both Faraday s law says nothing about the material: % N d dt The change in flux is the same (and N is the same), so the induced emf is the same. B Plastic Copper
31-4 Production of Electromagnetic Waves Oscillating charges will produce electromagnetic waves:
31-4 Production of Electromagnetic Close to the antenna, the fields are complicated, and are called the near field: Waves
31-4 Production of Electromagnetic Waves Far from the source, the waves are plane waves:
31-4 Production of Electromagnetic Waves The electric and magnetic waves are perpendicular to each other, and to the direction of propagation.
ConcepTest 31.2 The electric field in an EM wave traveling northeast oscillates up and down. In what plane does the magnetic field oscillate? Oscillations 1) in the north-south plane 2) in the up-down plane 3) in the NE-SW plane 4) in the NW-SE plane 5) in the east-west plane
ConcepTest 31.2 The electric field in an EM wave traveling northeast oscillates up and down. In what plane does the magnetic field oscillate? Oscillations 1) in the north-south plane 2) in the up-down plane 3) in the NE-SW plane 4) in the NW-SE plane 5) in the east-west plane The magnetic field oscillates perpendicular to BOTH the electric field and the direction of the wave. Therefore the magnetic field must oscillate in the NW-SE plane.
31-5 Electromagnetic Waves, and Their Speed, Derived from Maxwell s Equations In the absence of currents and charges, Maxwell s equations become:
31-5 Electromagnetic Waves, and Their Speed, Derived from Maxwell s Equations This figure shows an electromagnetic wave of wavelength λ and frequency f. The electric and magnetic fields are given by. where
31-5 Electromagnetic Waves, and Their Speed, Derived from Maxwell s Equations Applying Faraday s law to the rectangle of height Δy and width dx in the previous figure gives a relationship between E and B:.
31-5 Electromagnetic Waves, and Their Speed, Derived from Maxwell s Equations Similarly, we apply Maxwell s fourth equation to the rectangle of length Δz and width dx, which gives.
31-5 Electromagnetic Waves, and Their Speed, Derived from Maxwell s Equations Using these two equations and the equations for B and E as a function of time gives. Here, v is the velocity of the wave. Substituting,
31-5 Electromagnetic Waves, and Their Speed, Derived from Maxwell s Equations The magnitude of this speed is 3.0 x 10 8 m/s precisely equal to the measured speed of light.
31-5 Electromagnetic Waves, and Their Speed, Derived from Maxwell s Equations Example 31-2: 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 E 0 = 2.0 V/m. Write vector expressions for E and B as functions of position and time.
31-6 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:
31-6 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.
31-6 Light as an Electromagnetic Wave and the Electromagnetic Spectrum Example 31-3: 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 10 14 Hz.
31-6 Light as an Electromagnetic Wave and the Electromagnetic Spectrum Example 31-4: 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.
31-6 Light as an Electromagnetic Wave and the Electromagnetic Spectrum Example 31-5: 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?
31-7 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:
31-7 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 = 2.99792458 10 8 m/s. This is then used to define the meter.
31-8 Energy in EM Waves; the Poynting Vector Energy is stored in both electric and magnetic fields, giving the total energy density of an electromagnetic wave: Each field contributes half the total energy density:
31-8 Energy in EM Waves; the Poynting Vector This energy is transported by the wave.
31-8 Energy in EM Waves; the Poynting Vector The energy transported through a unit area per unit time is called the intensity: Its vector form is the Poynting vector:
31-8 Energy in EM Waves; the Poynting Vector Typically we are interested in the average value of S: S.
31-8 Energy in EM Waves; the Poynting Vector Example 31-6: E and B from the Sun. Radiation from the Sun reaches the Earth (above the atmosphere) at a rate of about 1350 J/s m 2 (= 1350 W/m 2 ). Assume that this is a single EM wave, and calculate the maximum values of E and B.
31-9 Radiation Pressure In addition to carrying energy, electromagnetic waves also carry momentum. This means that a force will be exerted by the wave. The radiation pressure is related to the average intensity. It is a minimum if the wave is fully absorbed: and a maximum if it is fully reflected:
31-9 Radiation Pressure Example 31-7: 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/m 2. Estimate the pressure and force exerted by the Sun on your outstretched hand.
31-9 Radiation Pressure Example 31-8: 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?
31-10 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.
31-10 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):
31-10 Radio and Television; Wireless Communication Second, by using the signal to modify the frequency of the carrier (FM):
31-10 Radio and Television; Wireless Communication At the receiving end, the wave is received, demodulated, amplified, and sent to a loudspeaker.
31-10 Radio and Television; Wireless Communication The receiving antenna is bathed in waves of many frequencies; a tuner is used to select the desired one.
31-10 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.
31-10 Radio and Television; Wireless Communication Example 31-9: Tuning a station. Calculate the transmitting wavelength of an FM radio station that transmits at 100 MHz.
ConcepTest 31.3 Before the days of cable, televisions often had two antennae on them, one straight and one circular. Which antenna picked up the magnetic oscillations? TV Antennas 1) the circular one 2) the straight one 3) both equally; they were straight and circular for different reasons
ConcepTest 31.3 Before the days of cable, televisions often had two antennae on them, one straight and one circular. Which antenna picked up the magnetic oscillations? TV Antennas 1) the circular one 2) the straight one 3) both equally; they were straight and circular for different reasons The varying B field in the loop means the flux is changing and therefore an emf is induced.
ConcepTest 31.4 Radio Antennas If a radio transmitter has a vertical antenna, should a receiver s antenna be vertical or horizontal to obtain the best reception? 1) vertical 2) horizontal 3) doesn t matter
ConcepTest 31.4 Radio Antennas If a radio transmitter has a vertical antenna, should a receiver s antenna be vertical or horizontal to obtain the best reception? 1) vertical 2) horizontal 3) doesn t matter If a wave is sent out from a vertical antenna, the electric field oscillates up and down. Thus, the receiver s antenna should also be vertical so that the arriving electric field can set the charges in motion. E field of wave E field of wave
Summary of Chapter 31 Maxwell s equations are the basic equations of electromagnetism:
Summary of Chapter 31 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 of Chapter 31 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: