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 = yo 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 = Eo sin (ω t )      and      B = Bo 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 (3.00x108 m/s in vacuum). The scale on the E-axis is therefore much greater than the scale on the B-axis ( an order of 108 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 3.00x108 m/s in vacuum. This means 300,000 km/s or 186,000 miles /sec.  You may put 186,000 miles on your car in 8 to10 years.  Electromagnetic waves ( Light being one type of it ) travel that distance in one second!  All radio transmitters and cellular phone systems take advantage of E&M waves.  The trick is to mount sound waves (voice) onto the E&M waves (called modulation) and send them at the speed of light ( E&M waves).  This is the maximum possible speed according to the "Einstein's Theory of Relativity."

 

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 = 3.00x108 m/s.  Find the wavelength, λ, for such waves.

Solution:  From c = f λ,   we get    λ  = c /f   =   (3.00x108 / 60.0 ) m   =   5,000,000 m   =  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 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  =  (3.00x108 / 0.15 )s -1  =  2.00x109 Hz = 2000 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  εo  and  μo  (Form previous chapters or the front or back cover of your text) to verify that  c = 3.00x108 m/s.

 

Energy carried by Electromagnetic Waves:

The energy carried by a wave is proportional to the square of its amplitude (A2).  For  E&M waves, it will be (Eo2), or (Bo2), or ( EoBo) where Eo and Bo are the maximum values of the electric and magnetic fields intensities.   Eo= cBo.  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 / m2.   It can be shown that the formula for maximum energy density carried by an electromagnetic wave is either of the three following forms:

Io  =  c εo Eo 2 =  c Bo 2 / μo  =  Eo Bo / μo

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 εo Eo 2 =  c Bo 2 / 2μo  =  Eo Bo / 2μo

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 / m2 at the 8.0-km radius, and (b) the magnitude of the electric and magnetic field strength ( Eo and Bo ) 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 m2 of it receive?  This simple division will give us the value of  Irms. 

Irms = 10,000watts  / [ 4(3.14 x 80002 )m2 ] = 1.2 x 10-5 watts / m2.    (or 12 micro-watts per m2)

Since I rms   = (1/2)cεoEo2 , solving for Eo we get : Eo = [2I rms/cεo]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:  Bo = Eo / c = (9.4x10-3 / 3.00x108)  =  3.1 x 10-11 T.

 

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 =Eosin (ωt)  where ω = 2πf   (b) E = (1/2)Eot2 + ωt   (c) E = (1/2)Eot2.    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) 3.00x1010 cm/s  (c) 3.00x105 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, A2  (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 t2 + vi t   (b) x =  vi 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) 500s 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) 3x108s. 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 2000MHz.  This frequency is (a) 2x109Hz  (b) 2x106Hz  (c)2x1012Hz.    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) 4.4x1014 s-1  (b)4.4x1014/s  (c)  4.4x1014Hz   (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 109 seconds means practically motionless  (d) b & c.   click here.

26) The frequency of a certain violet light (of course, an E&M wave) is 7.3x1014/s.   Its wave length is (a) 4.1x10-9m   (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 3.0x108m/s  (b) has a speed of 3.0x105km/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 3.0x108m/s  (b) has a speed of 1.86x105mi/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 3.0x108m/s  (b) has a speed of 3.0x105km/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 (fv = 7.5x1014/s).   An E&M wave of frequency 9.5x1014/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 (fUV > 7.5x1014/s).  An E&M wave of frequency 6.5x1016/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 (fX > 1016/s).   An E&M wave of frequency 5.0x1021/s is of course of Gamma type, not visible, and very penetrable.  It's wavelength is (a) 6.0x10-14m  (b) 60fm  (c) a & b.  Note: fm means femtometer that is10-15m.  click here.

33) In general, for E&M waves, the speed is constant (3.00x108m/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.