maxwell’s equations and electromagnetic waves

Maxwell’s Equations and Electromagnetic Waves

• 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

Content

• Ch. 33

•Eexr. 34, 38

• Ch. 34

•Exer. 1, 2, 5, 6, 11, 12,

•Exer. 13, 14, 15,

•Exer. 19, 20, 23, 24, 27, 28

Homework

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.

Displacement Current

IdBdB 02 surface1 surface

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:


?2 surface1 surface0

dBdBI

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


).( D0 IIdB

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

dt

dI

dt

d

dt

dQEA

A

Q

ddAQ

Ed

AC

C

QV

d

VE

E0D

E0E

00

0

Therefore,

,

and , ,


With Maxwell’s modification, Ampere’s law now becomes

Where the second term is called the displacement current

dt

dIdB E

00

.E0D dt

dI

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.


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:

dt

dIdB

dt

ddE

AdB

QAdE

E00

B

0

0

Maxwell’s Equations

In the absence of currents and charges, they are:

dt

ddB

dt

ddE

AdB

AdE

E0

B

0

0

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:

2

2

002

2

2

2

002

2

t

B

x

B

t

E

x

E


A wave traveling along the x-axis with a speed v satisfies the wave equation

2

2

22

2 1

t

f

vx

f

Therefore, we see that the wave speed is

m/s1000.31 8

00

c

which is exactly the speed of light.

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:



Close to the antenna, the fields are complicated, and are called the near field:

Far from the source, the waves are plane waves:


The electric and magnetic waves are perpendicular to each other, and to the direction of propagation.



This figure shows an electromagnetic wave of wavelength λ and frequency f. The electric and magnetic fields are given by

tkxBBB

tkxEEE

z

y

sin

sin

0

0


light of speed the,

frequencyangular the,2

r wavenumbe the,2

ck

f

k

The electric and magnetic fields are related by

.cBE


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.

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

2

0

20BE 2

1

2

1BEuuu

EBB

EEEu

0

0

0

2

20

2

0

0020 22

1

This energy is transported by the wave.

Energy and Momentum

uAcdtuAdxdU

The energy transported through a unit area per unit time is called the intensity:

Energy and Momentum

The energy floe is perpendicular to both E and B. The Poynting vector is defined as

0

11

EB

ucdt

uAdx

Adt

dU

AS

0BE

S

Energy and Momentum

Typically we are interested in the average value of Sav:

0

rmsrms

0

00

0

2

0

00

00

0av

2

sin

BEBE

dttBE

dtEB

SdtSTTT

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.

Energy and Momentum

In addition to carrying energy, electromagnetic waves also carry momentum:

.c

Up

If the wave is completely absorbed, the radiation pressure exerted on the surface is

uc

S

cdt

dU

Adt

dp

AA

FP

11

Radiation Pressure

If the wave is perfectly reflected, the momentum change is doubled

.2c

Up

and the radiation pressure exerted on the surface is

uc

S

A

FP 22

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.

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?

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.



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.


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.


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?

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:

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 × 108 m/s.

This is then used to define the meter.

Measuring the Speed of Light

This figure illustrates the process by which a radio station transmits information. The audio signal is combined with a carrier wave.

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):


Second, by using the signal to modify the frequency of the carrier (FM):


At the receiving end, the wave is received, demodulated, amplified, and sent to a loudspeaker.


The receiving antenna is bathed in waves of many frequencies; a tuner is used to select the desired one.



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.


Tuning a station.

Calculate the transmitting wavelength of an FM radio station that transmits at 100 MHz.

• 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:

Summary

maxwell’s equations and electromagnetic waves

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