__Chapter 34__

__Electromagnetic Waves__

Recall that anywhere there is an electric charge, there exists
an
electric field around it throughout space**.** If the
charge moves, the electric
field (or any field-line) around it moves as well**.** The motion of the charge causes change in
the electric field intensity (E) at any point in space that varies with distance and time**.**
Also,
recall that the motion of a charge creates a magnetic field (B) that is
perpendicular to the direction of motion of the charge.
The important aspect is
that the two effects (E) and consequently (B) turn out to be perpendicular to
each other and they coexist.

Now, if the motion of the charge is of oscillatory nature [with a
sinusoidal equation**:** y = y_{o }sin( ω t)
],
an up-and-down motion for example, the variations of
E and B will also be sinusoidal
as well with equations of the type

E = E_{o} sin (ω
t ) and B = B_{o }sin (ω
t ) where
ω = 2πf.

Since such oscillations cause other charges at some distance away ( no
matter how far ) to oscillate accordingly, we believe that **oscillations of an
electric charge creates waves that propagate through space.** The graphical
representation of the propagation of such waves called the "electromagnetic
waves" through space may be pictured as shown below**.** **This is the way
fields variations are sensed by a distant charge as the waves pass by it.**

Note that E is much greater than B. In fact
E
= cB where c is the speed of light
(3**.**00x10^{8} m/s in vacuum). The scale on the E-axis is
therefore much greater than the
scale on the B-axis ( an order of 10^{8} times greater)**.** This means that the
magnetic effect is much weaker than the electric effect**.**
Also, this
figure shows
one propagation direction only**.**

Note that the E and
B that reach a **distant charge** as a result of
E&M waves propagation cause that distant charge
to oscillate accordingly**.**
**The transmission of this effect is very fast but not instant.** The speed
of propagation of charge oscillations, **E&M Waves**, is measured to be **
3.00x10 ^{8} m/s in vacuum.** This means 30

__Wave Speed:__

Recall that wave speed ( v ) is related to wavelength ( λ ) pronounced " lambda" and frequency ( f ) by

v = f λ

For electromagnetic
E&M waves, letter (
c ) is commonly used for the
wave speed**.** c = f
λ

Since c is a **constant** for any given medium, if
f increases, then
λ has to decrease**
**in that medium**.**

Example 1:
An AC source is running at a
frequency of 60**.**0Hz**.** This causes the current ( moving charges ) in wires
connected to this generator to flow back-and-forth at that frequency**.** The
charge oscillations in such wires produce E&M waves that as we know propagate at
the speed of light
c =
3**.**00x10^{8} m/s**.** Find the wavelength,
λ, for
such waves**.**

Solution: From
c = f λ, we get
λ = c /f
= (3**.**00x10^{8} / 60**.**0 ) m = 5,0**0**0,000 m =
31**0**0** miles**

This
is a very**
**long wavelength and therefore
very weak! Only 60**.**0 of such waves
pass by a given point in space every second**.** This means that if there is a
charge at one point in space, it oscillates only 60**.**0 times per second as
such waves keep coming to it**.**
The
shorter the wavelength, the more energetic the wave is or the more energy it
carries**.** Shorter wavelengths are associated with higher frequencies (
c = f λ )**.** Higher frequencies make a distant charge
to oscillate faster**;** thus, imparting more energy
to that charge**.**

Example 2: Waves transmitted or received by
cell-phones have wavelengths of about 15 cm or 6**.**0 inches**.** Calculate the
frequency of such waves and express it in MHz**.** Note that
MHz means
Mega Hertz or Million Hertz.

Solution:
c = f λ
**;** f = c / λ = (3**.**00x10^{8} / 0**.**15
)s^{
-1} = 2**.**00x10^{9} Hz = 20**0**0 MHz

Example 3:
White light is a mixture of a large number of different electromagnetic waves
whose wavelengths range from about
400nm
( violet
) to about
700nm
( red
)**.** Find the corresponding frequency for each of these two limiting values
in the visible range**.**

Solution: To be solved by students**.**

Speed of E&M Waves by Calculation:

By solving the wave equation ( a differential equation not shown here ), it is possible to show that speed of E&M waves is given by

Solution:**
**Use the values of
ε_{o}
and
μ_{o} (Form
previous chapters or the front or back cover of your text) to verify that
c
= 3**.**00x10^{8
}m/s.

__Energy carried by
Electromagnetic Waves: __

The energy carried by a wave is proportional to the square of
its amplitude (A^{2})**.** For E&M waves, it will be (E_{o}^{2}), or (B_{o}^{2}), or ( E_{o}B_{o})
where E_{o} and
B_{o} are the maximum values of
the electric and
magnetic fields** **intensities**.** E_{o}=
cB_{o}. This energy transfer is expressed in units of
**Joules per
second per meter squared** through space**.** Since **Joules per second** is **watt**,
we may say that the energy carried by a wave is expressed in **watts / m ^{2}.**
It can be shown that the formula for maximum energy
density carried by an electromagnetic wave is either of the
three following forms

I_{o}
= c ε_{o} E_{o} ^{
2 }= c B_{o} ^{2 }/ μ_{o} = E_{o }B_{o}
/ μ_{o}.

For a
continuous sinusoidal wave, we may calculate an RMS ( Root Mean Square) value**.**
Since (rms) power is 1/2 of the max.
power, we may write**:** [subscript (o) denotes maximum value]

I _{rms
} = (1/2)
c ε_{o} E_{o} ^{2
}= c B_{o} ^{2 }/ 2μ_{o}
= E_{o }B_{o} / 2μ_{o}.

Example 4:
A 10-kilowatt AM
radio station is tuned on by a radio set at a distance of 5**.**0 miles
= 8**.**0 km from the transmitter antenna**.** ** Isotropic wave propagation **
in space means that waves are sent out by the transmitter uniformly in all
directions**.** This is not really the case with actual antennas, but for
simplicity here, we suppose isotropic propagation of wave energy in space**.**
Assuming isotropic, calculate **(a)** the wave intensity (I _{rms}) in
watts / m^{2} at the 8**.**0-km radius, and (b) the magnitude of the
electric and magnetic field strength ( E_{o} and B_{o} ) at that
radius**.**

Solution: Visualize a huge
sphere ( of 8000m radius or 5 miles ) that 10,000 watts of energy is to be
distributed over its surface**.** How much energy will every m^{2} of it
receive? This simple division will give us the value of I_{rms}**.**

I_{rms}
= 10,000watts **/ **[ 4(3**.**14 x 8000^{2} )m^{2} ]
= 1**.**2 x 10^{-5} watts / m^{2}**.**
(or 12 micro-watts per m^{2})

Since I _{
rms } = (1/2)cε_{o}E_{o}^{2
}, solving for E_{o} we get **:**
**E**_{o}
= [2I _{rms}_{/}cε_{o}]^{1/2}
=
9**.**4 x 10^{-3} V/m**.**

(Can you verify that Volt/meter is the same as N/Coul.?)

Since E = c B, we get: **B _{o}**
= E

__Test Yourself 1:__ click here**.**

1) An electromagnetic wave is a result of the oscillation of (a)
an electron (b) a proton (c) a neutron (d) a & b**.**

2) When a charged particle moves, its electric field (a) moves
accordingly (b) remains unchanged (c) varies as sensed by other
charges elsewhere (d) a & c**.** click here**.**

3) If a charged particle oscillates, its equation of motion is
(a) quadratic in time (b) sinusoidal in time (c) neither a nor b**.**

4) A particle oscillating in the y-direction at a frequency of (f)
Hz and an amplitude of (A) meters follows
(a) y=Asin(2πft) (b) y=Asint (c) y=Asin(ft)**.**

5) If a charged
particle oscillates, the ripples
generated in its electric field follow (a) E =E_{o}sin (ωt)
where ω =
2πf (b) E = (1/2)E_{o}t^{2}
+ ωt
(c) E = (1/2)E_{o}t^{2}**.** click here**.**

6) Anywhere a charged particle moves, it generates a magnetic
effect that is (a) parallel to its direction of motion (b) perpendicular
to its direction of motion (c) neither a nor b**.**

7) The magnetic effect of an electromagnetic wave is (a)
separable from its electric effect (b) is not separable from its electric
effect (c) can only be separated at high frequency oscillations of a
charged particle**.**

8) The magnetic effect of an E&M wave is (a) much stronger than
its electric effect (b) much weaker than its electric effect (c)
has the same strength as its electric effect**.** click here**.**

9) The speed of E&M waves in vacuum is (a) the same as the speed of light
(b) 3**.**00x10^{10} cm/s (c) 3**.**00x10^{5} km/s
(d) 186,000mi/s (e) a, b, & c**.**

10) Light is (a) an E&M wave (b) a mechanical wave and
cannot travel in vacuum (c) is a longitudinal wave (d) is a
transverse wave (e) a & d**.** click here**.**

11) The formula for wave speed is (a) v = f λ (b) v = ω λ (a) v = f ω.

12) Frequency is
defined as (a) the number of meters per second (b) the number of
ωs
that occur per second (c) the number of wavelengths (full cycles)
that occur per second**.**

13) Wavelength is (a)
the distance between any two crests on a wave (b) the distance between a
crest to the next one on a wave (c) the distance between a trough to the
next one on a wave (d) b & c**.** click here**.**

14) The energy
carried by a wave is proportional to (a) its amplitude, A (b) its
amplitude squared, A^{2} (c) neither a nor b**.**

15) When a charged
particle oscillates at one point in space, other charges in space (a) oscillate
instantly as a result (b) will oscillate accordingly at some later time
depending on their relative distances (c) both a & b**.** click here**.**

16) Since E&M waves
move at a **constant velocity **in a medium with fixed properties (v
= 300,000km/s in vacuum), they do not accelerate (a = 0), and the
equation of motion for them is (a) x = (1/2)a
t^{2} + v_{i}
t (b) x = v_{i}
t (c) x = Rθ**.**

17) If a charge
starts oscillating now here on the Earth, a charge that is on the Moon, an
average distance of 384,000km away, will start oscillating (a)1**.**2 min.
later (b) 1**.**28s later (c) one month later**.** click here**.**

18) If a charge
starts oscillating now here on the Earth, a charge that is on the Sun, an
average distance of 150,000,000km away, will start oscillating (a)8**.**3
min. later (b) 5**0**0s later (c) both a & b**.**

19) A light year is
the distance light travels in 1yr**.** If a charge starts oscillating
now here on the Earth, a charge that is on the star Alpha Centauri, 4
light-years away, will start oscillating (a) 3x10^{8}s. later
(b) 2 light-years later (c) neither a nor b**.**

20) From the Earth,
it takes a radio signal (An E&M wave) 5**.**0s to reach a space station an
back**.** The space station is (a) 1,500,000km away (b) 3,000,000km
away (c) 750,000km away**.** click here**.**

21) From the Earth,
it takes a radio signal (An E&M wave) 5**.**0min. to reach a space station an
back**.** The space station is (a) 45,000,000km away (b) 90,000,000km
away (c) 75,000,000km away**.**

22) The frequency of
E&M waves used foe cellular phones is about 2**0**00MHz**.** This frequency
is (a) 2x10^{9}Hz (b) 2x10^{6}Hz (c)2x10^{12}Hz**.** click here**.**

23) The wavelength of
the E&M waves used for cellular phones is (a) 15cm (b) 6**.**0in
(c) 0**.**15m (d) a, b, & c**.**

24) The wavelength of
a certain red light (of course, an E&M wave) is 680nm**.** Its frequency is
(a) 4**.**4x10^{14} s^{-1}
(b)4**.**4x10^{14}/s (c) 4**.**4x10^{14}Hz
(d) a, b, & c**.**

25) If you are
solving for the frequency of an E&M wave, and you come up with f = 2**.**7x10^{-9}/s,
for example, (a) you accept the answer and think it must be correct (b)
you doubt the answer thinking that order of 10^{-9} is extremely small
to be the frequency of an E&M wave (c) you may think that a charge
oscillating once every 10^{9} seconds means practically motionless
(d) b & c**.** click here**.**

26) The frequency of
a certain violet light (of course, an E&M wave) is 7**.**3x10^{14}/s**.**
Its wave length is (a) 4**.**1x10^{-9}m (b)41**.**0nm
(c) 410nm (d) 160nm. click here**.**

27) The wavelength of
a wave is 750m in vacuum and it occurs 400,000 times per second**.** The wave (a) has a
speed of 3**.**0x10^{8}m/s (b) has a speed of 3**.**0x10^{5}km/s
(c) is electromagnetic because only E&M waves can travel at that speed in vacuum
(d) a, b, & c**.**

28) The wavelength of
a wave is 1500m in vacuum and it occurs 200,000 times per second**.** The wave (a) has a
speed of 3**.**0x10^{8}m/s (b) has a speed of 1**.**86x10^{5}mi/s
(c) is electromagnetic because only E&M waves can travel at that speed in
vacuum
(d) a, b, & c**.**

29) The wavelength of
a wave is 3000m in vacuum and it occurs 100,000 times per second**.** The wave (a) has a
speed of 3**.**0x10^{8}m/s (b) has a speed of 3**.**0x10^{5}km/s
(c) is electromagnetic because only E&M waves can travel at that speed in
vacuum
(d) a, b, & c**.** click here**.**

30) Ultraviolet rays
(of course, E&M waves) have frequencies more than that of violet (f_{v}
= 7**.**5x10^{14}/s)**.** An E&M wave of
frequency 9**.**5x10^{14}/s is of course UV and not visible**.** It has a wavelength of (a) 3**.**2E-7m (b)
320nm (c) both a & b**.**

31) X- rays (of
course, E&M waves) have frequencies more than that of ultraviolet (f_{UV}
> 7**.**5x10^{14}/s)**.** An E&M wave of
frequency 6**.**5x10^{16}/s is of course of
X-rays type, and not visible**.** It has a wavelength of (a) 4**.**6E-9m
(b) 4**.**6nm (c) both a & b**.** click here**.**

32) Gamma rays (of
course, E&M waves) have frequencies more than that of X-Rays (f_{X}
> 10^{16}/s)**.** An E&M wave of frequency 5**.**0x10^{21}/s
is of course of Gamma type, not visible, and very penetrable**.** It's wavelength
is (a)
6**.**0x10^{-14}m (b) 6**0**fm (c) a & b**.**
Note: fm means femtometer that is10^{-15}m**.** click here**.**

33) In general, for
E&M waves, the speed is constant (3**.**00x10^{8}m/s in vacuum)**.**
An E&M wave of (a) lower frequency has of course a greater wavelength
(b) higher frequency has of course a smaller wavelength (c) both a & b (d)
neither a nor b**.**
click here**.**