EM Waves

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18 Οκτ 2013 (πριν από 3 χρόνια και 11 μήνες)

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Chapter 32: EM Waves
-
updated 5/28/08


J. C. Maxwell’s modification to Ampere’s law was a stroke of genius enabled him to develop a complete
theory of electricity & magnetism. The most surprising consequence is that light turns out to be an
electromagneti
c wave; but
not

“the only type” of electromagnetic wave. Now we know that space is full
of different types of EM waves that can be detected and yield information about the cosmos. Maxwell’s
equations revolutionized physics and astronomy, gave rise to a hos
t of useful inventions, and was the
jumping off point for Einstein’s theory of Relativity.


All EM waves travel with the same speed c=3.00 x 10
8

m/s in a vacuum, although they slow down when
traveling inside other substances. Like mechanical waves, EM wav
es reflect, refract, interfere, and
transfer energy. Unlike mechanical waves, EM waves can exert pressure, transfer momentum, and they
do not need a
medium

in which to travel. The direction and intensity of an EM wave is expressed in the
Poynting vector,
S
=
E
x
B
/

o
, which has units of watts/m
2
.


1. Answer the following questions. Always explain your reasoning.

a)

There is a
displacement current

between the plates of a charging capacitor, yet no charge is
moving between the plates. Explain the meaning of “displac
ement current”. In what sense is the
word "current" appropriate here?

b)

List some similarities and differences between electromagnetic waves and sound waves.

c)

The speed of an electromagnetic wave is given by c =

f. How is it that we can say that the speed
of light is independent of frequency or of wavelength?

d)

Define the Poynting vector. Explain its meaning and its usefulness.

e)

The Sun emits most of its electromagnetic wave energy in the visible region of the spect
rum
. Our
eyes are sensitive to the same range
. Is this a coincidence?

f)

Suppose your eyes were sensitive to radio waves rather than light. What things would look
bright?

g)

The intensity of light falls off as the inverse square of the distance from the source. Does this
mean that electromagnetic wave en
ergy is lost? Explain.

h)

Some long
-
distance power transmission lines use DC rather than AC, despite the need to convert
between DC and AC at either end. Why might this be? What energy loss mechanism occurs with
AC but not DC?

i)

Electromagnetic waves do not r
eadily penetrate metals. Why might this be?


2. The following chart summarizes the development of Maxwell’s equations through the course. The
first column lists the static version of Maxwell’s equations. Fill in the other columns with the dynamic
version
of Maxwell’s equations and the hypothetical version if magnetic monopoles were found to exist.
Finally, explain the significance of each law.


Static version of
Maxwell’s equations

Dynamic version of
Maxwell’s equations

Significance of the law

V
ersion
wi
th h
ypothetical
magnetic monopole
“p”


Gauss’ Law:





Gauss’ Law:





Ampere’s Law:


Faraday’s Law:




Ampere’s Law:






a)

Explain why M
axwell thought that the laws of electricity& magnetism were “unfinished” or
“lacked symmetry” before he added his modification.

b)

Why i
s Maxwell's modification of Ampe
re's law essential to the existence of electromagnetic
waves?

c)

What would be the implication
s of the

existence

of magnetic monopoles
?


3
. A parallel
-
plate capacitor of plate area
A

and spaci
ng
l

is charging at the rate
(
dq
/dt)
.
(a)
Show that
the
displacement current

(I
D
=

o
d

E
/dt)
in the capacitor is equal to the conduction current flowing in the
wires feeding the capacitor.
(b) You can also show that the displacement current is equal to
C
(
dV/dt
)
,
where
C

is the capacitance and (
dV/d
)
t

is the rate at which the capacitor v
oltage changes.


4.
A capacitor with circular plates
of radius R spaced a distance

l

apart,
is fed with

long, straight wires along the axis of the plates.
Determine the magnetic field
in a

plane that passes through the interior of the capacitor and is pe
rpendicular to the
axis.

Assume that the electric field inside the capacitor is uniform and confined to the inside.

a)
Show that the magnetic field
outside

the capacitor is given by
.
Here
(
dV/
dt
)

is
t
he rate of change of the capacit
or voltage,

and
r

the distance from the axis.

b)
Show that the magnetic field
inside

the capacitor

is given by
.

c) As a numerical application of the problem above, consider a

parallel
-
plate capacitor
with

circular
plates with radi
us 50 cm and spacing 1.0 m
m. A uniform electric field be
tween the plates is

changing at
the rate dE/dt=1.0 MV/m
.
s. What is the mag
netic field betwee
n the plates (i) on the symmetry axis, (ii)
15 cm from the axis, and (iii
) 150 cm from the axis?


5
. An ele
ctric field points into

the page and occupies a circu
lar region

of
radius 1
.0 m, as shown in
the illustration
. There are no electric charges

in the region, but there is a magnetic field forming closed loops pointing

clockwise, as shown. The magnetic fie
ld strength 50 cm from the center

of the region is 2.0

T. (a) What is the rate of change of the electric field?

(b) Is the e
lectric field increasing or de
creasing?
(c) What is the value of the

magnetic field 1.2 m from the center?


6
.
The following four illustrations represent EM waves. (a) Determine the di
rection of propagation in
each case. (b) Which wave has the longest wavelength?
Use the scale of the drawings.
(c) Which wave
has the least intensity? (d) In each case there is a wire connected to a light bulb lying on the x
-
y plane, in
the lower two examp
les the wire is rotated
on the x
-
y plane

through an angle of 30
o

from the vertical.
Compare the brightness of the bulbs.


A






B






C






D


l


R






I


x


x


dE/dt

B

x x


x x




1.0m

x x
0.5m

x x




x x


y


E





B

z

x


y


B





E

z

x


y


E






B

z

x


y


B




E

z

x

30
o



7. A light
-
year

is the distance light travels in one
year
.
(a)
Show that the Sun is about 8 light
-
minutes
f
rom Earth.

Use the astronomical data appendix in the back of the book to get the information you need.
(b) The nearest star to our sun, Alpha Centauri, is about 4 lys away. How far is this in meters?...in miles?
(c) Our galaxy, the Milky Way, is a flattene
d spiral about 20,000 lys by 100,000 lys, by 5,000 lys in
size. How much larger are these dimensions compared to the distance between our sun and the earth?


8.
A computer can fetch information from its memory in 3.0 ns, a pro
cess that involves sending a s
ig
nal
from the central processing unit (CPU) to memory and awaiting the return of the information. If

signals
in the computer's wir
ing travel at 0.60c, what is the maximum distance between the CPU and the
me
mory? Your answer shows why high
-
speed computers
are necessarily compact.


9.
Show by differentiation and substitution that
the “wave equation”
can be
satisfied by fields of the
form: E(x, t) = E
p
f
(kx
+

t) and B(x, t) = B
p

f
(kx
+

t),

where
f is any function

of the argument
(
kx
+

t
)
.


1
0.
Determine

the wavelengths
or frequency
of
the following:
(a) a 100
-
MHz FM radio wave, (b) a 3.0
-
GH
z radar wave, (c) a 500 nm

light wav
e, (d) a 1.0 x 10
18
-
Hz X
-
ray, and
(e) a 1 cm microwave?


11. Draw an “electromagnetic
-
spectrum” line ranging from the highest to lowest frequencies. Roughly
identify the various portions of the range from radio waves to cosmic rays. Give an example of an
application for each identified por
tion of the spectrum.


12
. The “solar constant” is the intensity of sunlight reaching the earth. This “constant” actually varies
depending on where on earth it is determined. The average value in the upper atmosphere is
1368 W/m
2
.

(a) Use the solar consta
nt on earth to
calculate

the Sun's total power output. The average distance from
the Sun to the Earth is 1.5 x 10
11
m.

(b)
Assume that about two
-
thirds of the solar energy at Earth's orbit
actually
reaches the planet's surface.
At what rate is solar energy

incident on the entire E
arth?

(c)
What is the solar
energy gathered by a
collector
on the surface
measuring 60 cm by 2.5 m if it is
oriented at right angles to the incident l
ight and absorbs all the light?...What if it absorbs only 50% of
the light and
reflects 50%?

(d)
What is the radiation force on a solar collector
on the surface
measuring 60 cm by 2.5 m if it is
oriented at right angles to the incident l
ight and absorbs all the light?...What if it absorbs only 50% of
the light?


1
3. A typical flu
ores
cent lamp is a little over 1

m long and a few
centimeters

in diameter. How do you
expect the light intensity to vary with distance (a) near the lamp but not near either end and (b) far from
the lamp?

(c)
The
intensity is

measured at 8.0 cm
from a light so
urce is 150 W/m
2
, whil
e at 12 cm it measures 100
W/
m
2
. Roughly d
e
scribe the shape of the source. How would your description change if the intensity at
12 cm had measured 67
W/
m
2
?


14
. Serious proposals have been made to "sail" spacecraft to the outer sola
r system using the pressure of
sunlight
. (a) How much sail area must a l0
3
-
k
g spacecraft have if its accel
eration at Earth's orbit is

to be
1 m/s
2
?

Assume the sails are made from
100%
reflecting material.
(b) How far from the earth must the
spacecraft be b
efore this “solar propulsion” can compete with the earth’s gravitational attraction? (c)
Why can you neglect the Sun's gravity? Use the solar constant given in problem 12.


15
. A 65
-
kg astronaut is floatin
g in empty space. If the astro
naut shines a 1.0
-
W f
lashlight in a fixed
d
irection, how long will it tak
e the astronaut
to accelerate to a speed of 10 m
/s?
How does this compare
with the average lifetime of a person (75 years)?


16
.
(a)

A 60
-
W light bulb is 6.0 cm in diameter. What is the radiation pressure

on an opaque object at
the b
ulb's sur
face?

Assume the light bulb is a point source.



(b)
A white dwarf star is approximately the size of Earth but radiates about as much energy as the Sun.
Estimate the radiation pressure on an absorbing object at the whi
te dwarf's surface.


17
. A radar system produces pu
lses consisting of 100 full cy
cles of a sinusoidal 70
-
GHz
electromagnetic wave. The average power while

the transmitter is “on” is 45 MW,

and the waves are
confined to a beam 20 cm in diameter. Find (a)
the peak electri
c field, (b) the wavelength, (c
) the total
energy in a pulse, and (d) the total m
omentum in a

pulse. (e) If the transmitter produces 1000 pulses per
second, what is its
overall
average power output?


18.


The “solar wind” that blows from t
he sun consists of particles pushed by solar radiation
. To see how
small such particles must be, compare the force of sunlight with the force of gravity

from the sun
, and
solve for the particle radius at which the two are equal. Assume the particles are sp
herical and have
density 2 g/cm
3
. Why do you not need to worry about the distance from the Sun?
The mass of the Sun is
1.99 x 10
30

kg, the universal gravitational constant G=6.67 x 10
-
11

N m
2
/kg
2
, and use the solar power
output value determined in problem

12a.


19
. A cylindrical resistor of length
l,

radius
a
, and resistance R carries a current
I
.

Calculate the electric and magnetic fields
at the surface

of the resistor, assuming


the electric field is uniform throughout
, including at the surface. De
termine

the

Poynting vector, and show th
at it points into the resistor.
Calculate the flux
of the

Poynting vector (that is
SA
|

) over the surface of the resistor to get the rate of electromagnetic energy
flow
ing

into the resistor, and show that the resul
t is just
I
2
R. Your result
demonstrates

that the energy
heating the resistor comes from the fields surrounding it. These fields are sustained by the source of
electrical energy that drives the current.




E


B

I