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: y = yo sin(ωt), 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 = Eosin(ωt)      and      B = Bosin(ωt)      where      ω = 2πf.

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 the repeated oscillation of an electric charge creates sine waves that propagate throughout the entire space.   Such waves are called the "electromagnetic waves."   The graphical representation of such 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.

Figure 1

Note that E = cB where c is the speed of light (3.00x108 m/s in vacuum).    This figure shows one out of almost countless horizontal propagation directions for the vertical oscillation of Charge q.  See the foot note regarding Figure 1 at the end of this chapter.

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 (visible 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 onto E&M waves and send them at the speed of light, c.  This is the maximum possible speed according to the "Einstein's Theory of Relativity."  Mounting sound waves on electromagnetic waves is called "modulation."

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

Solution:  From c = f λ  λ c/f = {3.00x108/60.0}m = 5,000,000m = 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  = {3.00x108/0.15}s-1 = 2.0x109Hz = 2000MHz.

Example 3:  White light is a mixture of a large number of different electromagnetic waves with wavelengths ranging 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 

Use the values of  εo = 8.85x10-12F/m and  μo =  x10-7Tm/A to verify that  c = 3.00x108 m/s.

 

Energy carried by Electromagnetic Waves:

The energy carried by a mechanical wave is proportional to the square of its amplitude A2.  For  E&M waves, it will be Eo2, or Bo2, or the product 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 Joule per second per meter squared through space.  Since Joule per second is watt, we may say that the energy carried by a wave is expressed in watt/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 εoEo2     = cBo2 /μo   = EoBo/μo

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)cεoEo2   =   cBo2 /o   =   EoBo/o

 

Example 4:  A radio set is tuned on a 10-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 Irms 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 (8000m or 5 miles in radius) that 10,000 watts of energy is to be distributed over its surface.  How much energy will every m2 of it receive?  The simple division below will give us the value of  Irms. 

Irms= 10,000watts / [4(3.14 x80002)m2] = 1.2 x10-5w/m2  = 12μw/m2.

Irms = (1/2)cεoEo2 , solving for Eo  Eo = [2Irms/cεo]1/2  =  9.4 x10-3 V/m.

Verify that V/m is the same as N/C.

Since E = cB, we get:  Bo = Eo/c  = (9.4x10-3/3.00x108T  =  3.1 x10-11T.

 

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) with 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) 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)cμoEo2  (b) (1/2)cεoEo2  (c) neither a nor b.   click here.

9) The speed of E&M waves in vacuum is (a) the same as the speed of light  (b) 3.00x1010cm/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 = 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 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 (c = 300,000km/s in vacuum), they do not accelerate (a = 0), and the equation of motion for them is (a) x = (1/2)at2 +vi 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) 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, as a result a charge that is on Star, Alpha Centauri, 4 light-years away, will start oscillating (a) 3x108s 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 2000MHz.  This frequency is (a) 2.0x109Hz  (b) 2.0x106Hz  (c) 2.0x1012Hz.   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) 4.4x1014 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, (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 (33 years) is practically motionless!  (d) b & c.   click here.

26) The frequency of a certain violet light (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 3.00x108m/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 3.00x108m/s  (b) has a speed of 3.00x105km/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 (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) 3.2x10-7m  (b) 320nm  (c) both a & b.

31) X-rays (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 an X-ray and not visible.  It has a wavelength of (a) 4.6x10-7m  (b) 4.6nm  (c) both a & b.   click here.

32) Gamma rays (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 femto-meter 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.