__Chapter 24__

__Electromagnetic Waves__

Anywhere there is **an
electric charge**, there exists an **electric
field**** around it** throughout
the entire space**.** If the charge
moves, its electric
field (its effect of
attraction or repulsion on other charges) moves or changes as
well**.** The motion of the charge causes change in its distance from
any other charge**.** Consequently, the electric field intensity **E** of
the charge at any point in space changes 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 coexist.**

Now, **if** * the
motion of the charge is of oscillatory nature* [with
a sinusoidal equation

**E = E _{o}sin(ωt)**
and

Since such oscillations cause any other charge at some distance away (no matter
how far) to oscillate * accordingly
at a later time*, we believe that

**Figure 1**

**Note** that **E
= cB** where **c** is
the **speed of light (****3.00x10 ^{8} m/s
in vacuum).**

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 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 this 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 ****; ** **λ = ****c/****f =** {3**.**00x10^{8}**/**60**.**0**}**m =
5,0**0**0,000m = **3100 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 15cm or 6**.**0
inches**.** Calculate the frequency of such waves and express the result in
MHz**.** Note that **MHz** stands for **Mega
Hertz** or **Million
Hertz.**

Solution: c =
f λ **;** **f = ****c****/****λ**
= {3**.**00x10^{8}**/**0**.**15}**s**** ^{-1}** = 2

**
Example 3: ** White
light is a mixture of a large number of different electromagnetic waves with
wavelengths ranging from about **400nm (violet****)** to
about 7**00nm**** (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

Use the values of **ε _{o}**

__Energy carried by Electromagnetic Waves:__

The energy carried by a mechanical wave is proportional to the square of its
amplitude **A ^{2}.** For
E&M waves, it will be

**I _{o}= c **

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

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

Example 4: A
radio set is tuned on a 1**0**-kilowatt AM radio station that is 5**.**0
miles (8**.**0 km) away**.** ** 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

**Solution:** Visualize
a huge sphere (8000m or 5 miles in radius) that 1**0**,000 watts of energy is
to be distributed over its surface**.** How much energy will every m^{2} of
it receive? The simple division below will give us the value of I_{rms}**.**

**I _{rms}= **10,000watts

**I**_{rms }**=** **(1/2)**cε_{o}E_{o}^{2 },
solving for E_{o}**, ****E**_{o}** =**** [2I _{rms}**

* Verify* that V

Since **E = **cB,
we get**:** **B _{o} =
E_{o}/c = **(9

__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) with 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) both a & 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 **rms** energy density of an E&M wave is (a) **
(1/2)cμ _{o}E_{o}^{2}** (b)

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
= f **/**λ (a) v
= λ**/**t.

12) Frequency **f** is
defined as (a) the number of meters per second (b) the number of ω's 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) the square of
amplitude **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 (c = 30**0**,000km/s
in vacuum), they do not accelerate (a = 0), and the equation of motion for them
is (a) x = (1/2)at^{2} +v_{i} t
(b) x = **ct**
(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 minutes 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, as a result a charge that is on Star, Alpha Centauri, 4 light-years away,
will start oscillating (a) 3x10^{8}s later (b) 2yrs 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
for cellular phones is about 2**0**00MHz**.** This frequency is (a) 2**.**0x10^{9}Hz
(b) 2**.**0x10^{6}Hz (c) 2**.**0x10^{12}Hz**.** click
here**.**

23) The** λ** of
E&M waves used for cellular phones (Question 22) is (a) 25cm (b) 25 inches
(c) 0**.**25m (d) 15cm**.**

24) The wavelength of a certain red
light (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,
(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
(33 years)
is practically motionless**!** (d) b & c**.** click
here**.**

26) The frequency of a certain violet
light (E&M wave) is 7**.**3x10^{14}**/**s**.**
Its wave length is (a) 4**.**1x10** ^{-9}**m (b) 41

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 15**0**0m
in vacuum and it occurs 20**0**,000 times per second**.** The wave (a)
has a speed of 3**.**00x10^{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 30**0**0m
in vacuum and it occurs 10**0**,000 times per second**.** The wave (a)
has a speed of 3**.**00x10^{8}m**/**s (b) has a speed of 3**.**00x10^{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 (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**.**2x10^{-7}m
(b) 320nm (c) both a & b**.**

31) X-rays (E&M waves)
have frequencies more than that of ultraviolet (**f _{UV}** >
7

32) Gamma rays (E&M waves)
have frequencies more than that of X-Rays (**f _{X}** >
10

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**.**