maxwell’s equations and - people.virginia.edupeople.virginia.edu/~ben/2415131/lecture_20.pdf ·...

Copyright © 2009 Pearson Education, Inc.

Chapter 31

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

Units of Chapter 31


• Measuring the Speed of Light

• Energy in EM Waves; the Poynting Vector

• Radiation Pressure

• Radio and Television; Wireless

Communication

Units of Chapter 31


E&M Equations to date

0

0

enc

B

QE dA

dE d

dt

B d I


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

0

0

0

mag

e

enc

B

nc

QE dA

B dA

dE d

dt

B d I

Q

No effect since RHS

identically zero

These two not pretty,

i.e., not symmetric

Now, I suggest 0s mag

magd dQI

t

Q

dt d


E&M Equations to date – more

updated

0

0

0

0

???

mag

enc

mag

enc

B

QE dA

B dA

dE d

t

d I

Q

I

d

B

Wouldn’t it be nice if we could replace ??? with something?


Ampère’s law

relates the

magnetic field

around a current

to the current

through a

surface.

31-1 Changing Electric Fields

Produce Magnetic Fields; Ampère’s

Law and Displacement Current

0???B d I


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:

31-1 Changing Electric Fields Produce

Magnetic Fields; Ampère’s Law and

Displacement Current

0???B d I


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:








The second term in Ampere’s law has the

dimensions of a current (after factoring out

the μ0), and is sometimes called the

displacement current:

where


Example 31-1: 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.

31-1 Ampère’s Law and

Displacement Current

E


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:

0

0

0

0 0 0

mag

enc

enc

B

mag

E

QE dA

B dA

dE d

dt

d

Q

Id

I

B dt


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.

31-4 Production of Electromagnetic

Waves

ConcepTest 31.1a EM Waves I

Plastic

Copper

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?

1) the plastic loop

2) the copper loop

3) voltage is same in both

Faraday’s law says nothing about

the material:

The change in flux is the same (and

N is the same), so the induced emf

is the same.

ConcepTest 31.1a EM Waves I

Plastic

Copper

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?

1) the plastic loop

2) the copper loop

3) voltage is same in both

BdN

dt

%


Oscillating charges

will produce

electromagnetic

waves:


Waves



Waves

Close to the antenna,

the fields are

complicated, and are

called the near field:


Far from the source, the waves

are plane waves:


Waves


The electric and magnetic waves are

perpendicular to each other, and to the

direction of propagation.


Waves

ConcepTest 31.2 Oscillations

The electric field in an EM

wave traveling northeast

oscillates up and down. In

what plane does the

magnetic field oscillate?

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.

ConcepTest 31.2 Oscillations

The electric field in an EM

wave traveling northeast

oscillates up and down. In

what plane does the

magnetic field oscillate?

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


31-5 Electromagnetic Waves, and

Their Speed, Derived from Maxwell’s

Equations

In the absence of currents and charges,

Maxwell’s equations become:




Equations

This figure shows an electromagnetic wave of

wavelength λ and frequency f. The electric and

magnetic fields are given by

where

.




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:

.




Equations

Similarly, we apply

Maxwell’s fourth

equation to the

rectangle of length Δz

and width dx, which

gives

.




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,

.




Equations

The magnitude of this speed is

3.0 x 108 m/s – precisely equal

to the measured speed of light.




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 E0 = 2.0 V/m.

Write vector expressions for E and B as

functions of position and time.

E

E

E

B

B


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.






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 1014 Hz.




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.




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?


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

108 m/s.

This is then used to define the meter.

31-7 Measuring the Speed of Light


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.


Poynting Vector


The energy transported through a unit area

per unit time is called the intensity:


Poynting Vector

Its vector form is the Poynting vector:



Poynting Vector

Typically we are interested in the average

value of S: S

.



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·m2 (= 1350 W/m2). Assume that this is

a single EM wave, and calculate the maximum

values of E and B.


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/m2.

Estimate the pressure and force

exerted by the Sun on your

outstretched hand.



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?


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):


Communication


Second, by using the signal to modify the

frequency of the carrier (FM):


Communication


At the receiving end, the wave is received,

demodulated, amplified, and sent to a

loudspeaker.


Communication


The receiving

antenna is

bathed in

waves of many

frequencies; a

tuner is used to

select the

desired one.


Communication



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.



Communication

Example 31-9: Tuning a station.

Calculate the transmitting wavelength

of an FM radio station that transmits

at 100 MHz.

ConcepTest 31.3 TV Antennas

Before the days of cable,

televisions often had two

antennae on them, one straight

and one circular. Which antenna

picked up the magnetic

oscillations?

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 31.3 TV Antennas

Before the days of cable,

televisions often had two

antennae on them, one straight

and one circular. Which antenna

picked up the magnetic

oscillations?

1) the circular one

2) the straight one

3) both equally; they were

straight and circular for

different reasons

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.

ConcepTest 31.4 Radio Antennas

E field of wave

E field of wave

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


• 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.



• 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:


maxwell’s equations and - people.virginia.edupeople.virginia.edu/~ben/2415131/lecture_20.pdf ·...

Documents