# Q to position

Urban and Civil

Nov 16, 2013 (4 years and 7 months ago)

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Q

Q’

e

Move charge
Q to position
Q’ creates
propagating
‘kink’ in the
electric field,
which affects
the motion of
electron e.

The electric field and the magnetic field in an
electromagnetic wave are perpendicular to each other and
to the direction of propagation of the wave

http://www.phy.ntnu.edu.tw/java/emWave/emWave.html

What do the electric field lines of an electromagnetic wave
look like?

We know that an electromagnetic wave carries energy,
because it can causes charges to accelerate as it passes by
them. If it gives kinetic energy to these “absorbers” or
“receivers” then energy must be lost by the wave.

If electromagnetic waves contain energy, it also follows
that the source of the wave, the charged matter which
made the original disturbance in the field, must lose
energy. This usually means it loses kinetic energy, so its
motion decreases, or is “damped.”

What form does the energy carried by the wave take?

Obviously it is partly electric field energy and partly
magnetic field energy.

The density of electric field energy (i.e. the field energy
per unit volume of space) contained in an electric field of
strength E is

U
e

= ½
e
o

E
2
.

Similarly the density of magnetic field energy contained
in a magnetic field of strength B is

U
m

= ½ B
2

/
m
o
.

You can calculate each of these formulae for the specific
cases of a capacitor (electric) and an inductor (magnetic),
as was done in previous chapters, but it is also true for
any electric or magnetic field, including an
electromagnetic wave.

Now we made the claim earlier that you can’t have just an
electric wave or just a magnetic wave. The only thing
keeping the magnetic field in the wave from disappearing is
that the electric field is constantly changing. The only thing
keeping the electric field from disappearing is that the
magnetic field is constantly changing.

Given this, which of the two fields is stronger?

There is a surprisingly strong rule in physics that if energy
has the choice of going into two or more forms, that it will
choose between them equally.

So it is not surprising that the magnetic and electric field
energy densities in an electromagnetic wave are always
equal to each other.

So we have that U
e

= ½
e
o

E
2
= U
m

= ½ B
2

/
m
o
.

What does this tell us about the relative strengths of E, the
electric field of the wave, and B, the magnetic field of
the wave?

a)
E = B

b)
E = B
e
o
/
m
o

c)
E = B/(
e
o
m
o
)
1/2

d)
E = B/
e
o
m
o

So we have the interesting result that E/B = 1/(
e
o
m
o
)
1/2

But what does this mean? The units of
e
o

are C
2
/(N m
2
)

The units of
m
o
are T m/A

Keeping in mind that the force on a moving charged particle
is
F

= q (
v

x
B
), which of the following is correct for the
units of 1/(
e
o
m
o
)
1/2
?

a)
m

b)
m/s

c)
T C/s

d)
N A

So this funny looking quantity, which is equal to the ratio
between the electric and magnetic field strengths of an
electromagnetic field, is a velocity.

Since it is the tugging of each field on the other which keeps
the wave going, it is rather like the tension in water
waves or waves on strings that determines the speed of
the wave.

So if this is the speed of an electromagnetic wave, how fast
is it?

m
o
= 1.26 x 10
-
6

T m/A and
e
o
= 8.85 x 10
-
12

C
2
/(N m
2
)

e
o
m
o
)
1/2
= 3 x 10
8

m/s = c the speed of
light

The person who realised this was James Clerk Maxwell
(1831
-
1879) a Scottish theoretical physicist who was a

Unlike Faraday he came from a very wealthy background and
was very well educated. But he thought in a similar way and
seized on Faraday’s experimental results and on his way of
thinking about the electric and magnetic fields to produce a set
of equations, known as Maxwell’s equations, which described
the behavior of both fields. Because the two fields were mixed
together in the equations people began to speak of the

Electromagnetic field.

Maxwell quickly realized that he could combine his
equations together to create a wave equation, the type of
equation which described waves on strings, in water or in
air. But in this case the wave would exist only in the
electromagnetic field.

All wave equations contain a factor which is the velocity of
the wave and in Maxwell’s calculation this worked out to
be 1/(
e
o
m
o
)
1/2
.

When he calculated this number and found it was the same as
the speed of light he realized he had discovered the true
nature of light itself, it was simply a kind of
electromagnetic wave.

The Intensity of an electromagnetic wave is the amount of
energy it delivers to a unit area per unit time.

So we define it as I =
D
U/(A
D
t) = u
D
V/(A
D
t)

Where I is the intensity and u is the energy density in the wave.

Which of the following is the correct expression for I?

a)
I = u c

b)
I = u/c

c)
I = u A

d)
I = u

A

First of all, it has the right units.

Also I =
D
U/(A
D
t) = u
D
V/(A
D
t) = u c A
D
t /(A
D
t) = u c

So the intensity is simply the energy density in the wave
times the speed with which the wave is carrying that energy
to you.

But how do receivers or absorbers actually absorb the energy?
Usually by the fact that the electric and magnetic field vectors
in the wave impart a force on the particles in the receiver and
make them move, thus giving them kinetic energy.

D

E

B

+e

Suppose you have an proton +e upon which
an electromagnetic wave impinges. The
direction of propagation of the wave
(labelledD), and the directions of the
electric (E) and magnetic fields (B) are
as shown. Which direction will the force
exerted on the proton by the magnetic
field point (assuming the proton is not
moving at all when the wave first
reaches it)?

a)
E

b)
B

c)
Opposite to D

d)
D

D

To begin with the electron isn’t moving, but as the electric
field operates upon it, it will begin to move along the electric
field line in the direction of its arrow, so in the direction E.
Once it is moving the magnetic field can affect it, according to
the equation which gives the magnetic force exerted on the
proton by the wave
F

= q
v

x
B
.

If v is in the direction of E then
v

x
B
, by the right hand rule,
will be in the direction of D.

v

B

F

What this means is that electromagnetic fields exert a
force on charged particles which causes them to
recoil when hit. So it gives them some momentum in
the direction of the wave. This momentum has to
come from somewhere. Where can it have come
from?

a)
Empty space

b)
The particle which was the source of the wave

c)
The previous particle the wave struck

d)
It came into existence from nowhere

B

Just as the receiver recoils when the wave hits it, so the source of
the wave must recoil when it emits it. If the wave goes off in one
direction the source recoils in the other direction, just as a cannon
does when it fires off a cannonball.

So electromagnetic waves carry momentum from the source to the
receiver. The amount of momentum carried by an electromagnetic
wave is a simple formula

p = E/c = U/c where U or E is the amount of energy in the wave.

As we shall see later on, it is from this formula that Einstein
derived his famous equation E = m c
2
.