Important points

• Time-varying#»E and

#»B fields propa-

gate through space in waves.• EM waves have both the electric and

magnetic fields perpendicular to thedirection of motion.

• EM wavelengths fall along a largespectrum, including visible light.

Important equations

• EM wave speed (≡ speed of light):

c = 1/√

ε0µ0

i = 3.00× 108 m/s

PHYS 202 Notes, Week 8Greg Christian

March 8 & 10, 2016Last updated: 03/10/2016 at 12:30:44

This week we learn about electromagnetic waves and optics.

Electromagnetic Waves

So far, we’ve learned about magnetic and electric fields as separateentities. However, as it turns out, they are very closely linked; twosides of the same coin, so to speak. We’ve had hints of this already,as Faraday’s law shows that time-varying magnetic fields are a sourceof an electric field (manifested as an emf). As it turns out, the reverseit true too: time varying electric fields are also a source of magneticfield.

Fundamentally, electric and magnetic fields are described by fourequations called Maxwell’s equations. These use highly advanced math-ematics, so we won’t go into detail; we’ll just examine the conse-quences. The main consequence is that time-varying electric and mag-netic fields can travel, or propagate, through space in the form of waves.These are called Electromagnetic Waves (or EM waves for short).

Figure 1: Propagation of an electromag-netic wave.Figure 1 shows how electromagnetic waves propagate through space.

As shown in the top panel, the electric and magnetic field directionsare always perpendicular to the direction of motion. To get the fielddirections, we can use a variation of the right hand rule:

• Point your thumb in the direction of the direction of motion.• Point your fingers in the direction of the

#»E field.

phys 202 notes, week 8 2

• Rotate your fingers by 90 deg . They now point in the direction ofthe

#»B field.

Propagation Speed

Electromagnetic always waves travel through empty space with a con-stant speed,

c =1

√ε0µ0

. (1)

Numerically, this comes out to

c =1√

[8.85× 10−12 C2/ (N ·m2)] (4π × 10−7 N/A2)(2)

= 3.00× 108 m/s. (3)

As you may recognize, this is exactly the speed of light in a vacuum!This is because light is a specific form of electromagnetic wave.

Another property of EM waves is the relationship between the mag-nitudes of

#»E and

#»E . They are always related by the speed of propaga-

tion c,E = cB. (4)

Properties Summarized

The following points summarize the properties of EM waves:

1. Both#»E and

#»B are ⊥ to the direction of propagation and to each

other.2. The magnitudes of

#»E and

#»B are always related by E = cB.

3. The wave travels in a vacuum with unchanging speed

c = 1/√

ε0µ0 = 3.00× 108 m/s. (5)

4. Electromagmetic fields do not need a medium to travel through. Itis the

#»E and

#»B fields themselves that are doing the “waving.”

The Electromagnetic Spectrum

As mentioned, visible light is a particular form of electromagneticwave. More generally, EM waves span a wide range of wavelengthscalled the electromagnetic spectrum. This is summarized in Figure 2.

Electromagnetic waves span many orders of magnitude in wave-length, from ∼ 10 m on the high end to ∼ 10−13 m on the low end.Equivalently, they span a wide range of frequencies given by

f = c/λ. (6)

phys 202 notes, week 8 3

Figure 2: The electromagnetic spectrum.

Important points

• EM wave propagation is described byequations called wave functions.

• The power carried by EM waves canalso be described mathematically.

Important equations

• Wave functions

E = Emax sin (ωt∓ kx)

B = Bmax sin (ωt∓ kx)

• Energy density

u = ε0E2/2 + B2/(2µ0)

= ε0E2

• Average intensity

I = ε0cE2max/2

= Emax Bmax/(2µ0)

1 Note the sign flip!!!

EM waves are roughly broken down into seven (overlapping) cate-gories as shown in Figure 2. You are probably already familiar withall of these from your everyday life. At the low end of the spectrumare long wavelength, high frequency radio waves like the ones youpick up in an FM radio. At the high end are short wavelength, highfrequency gamma rays, which are the result of nuclear de-excitations.Visible light lies roughly in the middle of these two extremes.

EM Wave Properties

Like all waves, EM wave propagation can be described by mathemati-cal functions called wave functions. These describe the amplitude of the#»E and

#»B fields as a function of time t and position x:

E = Emax sin (ωt∓ kx) (7)

B = Bmax sin (ωt∓ kx) , (8)

where

• ω = 2π/T and k = 2π/λ

• T is the period of the wave and λ is the wavelength• Emax = cBmax• The negative sign means a wave traveling in the +x direction; the

positive sign means a wave traveling in the −x direction;1 and• When E is positive, B may be positive or negative; get the sign using

the right hand rule as in Figure 1.

Figure 1 shows what a propagating wave described by Eqns. (7) and (8)and traveling in the +x direction looks like. A picture of an EM wavetraveling in the −x direction is shown in Figure 3.

phys 202 notes, week 8 4

Figure 3: An EM wave traveling in the−x direction.

Energy in EM Waves

As we’ve already learned, energy can be stored in#»E and

#»B fields.

Recall that the equations for the respective energy densities are

uE =12

ε0E2 (9)

and

uB =B2

2µ0. (10)

In EM waves, the total energy density is the sum of these two,

u =12

ε0E2 +B2

2µ0. (11)

Relating E and B by B = E/c =√

ε0µ0E, we can express the energydensity just in terms of the electric field strength:

u =12

ε0E2 +1

2µ0(√

ε0µ0E)2 (12)

= ε0E2. (13)

Note that the energy oscillates with time since E also oscillates. It is nota constant quantity.

We can also talk about the power carried by EM waves. In particular,let’s talk about the power per unit area, S. This is equal to the energydensity times the velocity of the wave c,

S = cε0E2. (14)

Again, this is an instantaneous quantity; it’s constantly changing withtime.

If we want to talk about an average quantity, we can define theintensity I as the average power per unit area. This is found simply by

phys 202 notes, week 8 5

Important points

• Light and other EM waves can bemodeled as rays.

• Light rays undergo optical phenom-ena: refraction and reflection.

Important equations

• Index of refraction

n = c/v

• Snell’s law

na sin θa = nb sin θb

• Total internal reflection

sin θcrit = nb/na

Figure 4: A spherical wave front.

replacing E with Emax/2 in Eq. (14),

I =12

ε0cE2max. (15)

This can also be expressed using Emax and Bmax:

I =12

ε0c2EmaxBmax (16)

=12

ε0

(1

ε0µ0

)EmaxBmax (17)

=1

2µ0EmaxBmax. (18)

Remember: this is an average quantity that does not vary sinusoidally.

Optics

The nature of light has been pondered for hundreds of years. Startingaround the 1600s, there were competing theories: one treating lightas a steady stream of particles (as proposed by Isaac Newton) andthe other treating it as a wave. As it turns out, both of these theoriesare correct. Sometimes light behaves like a wave while other times itbehaves like a particle. Only with the advent of quantum mechanics inthe 20th century can we fully resolve these two competing properties.

For the time being, we’ll focus on the wave nature of light. Recallthat light is simply a specific form of EM wave. As such, it travels inwave fronts as represented in Figure 4.

To model the propagation of light in media, we can represent lightin the form of rays, or straight lines, rather than visible wave fronts.Figure 5 shows two different forms of light rays: the first (left panel)is the result of a spherical wave front resulting in light rays divergingfrom a point source; the second (right panel) is the result of a planarwave front, resulting in parallel light rays.

Figure 5: Spherical (left) and planar(right) light rays.

phys 202 notes, week 8 6

Figure 6: Reflection and refraction.

Figure 7: Total internal reflection.

Reflection and Refraction

Although light travels at a constant speed c in a vacuum, it slows downwhen inside physical media. The amount by which light slows downdefines something called the index of refraction, deonted by n. This is re-lated to the speed of light in a vacuum c and the speed in the mediumv by

n = c/v. (19)

Note that the speed of light in a medium is always less than c. Thismeans that n is always greater than one.

When traveling in a medium, the frequency of light stays the sameas the vacuum frequency. However, the wavelength changes accordingto

λ =λ0

n. (20)

When light encounters a medium boundary, it can either be re-flected or refracted as shown in Figure 6. Let’s talk about these twopossibilities in a bit more detail.

Reflection

In reflection, the light “bounces off” the medium, staying in the sameplane. In this case, the outgoing angle, θr is always equal to the incom-ing angle θa:

θr = θa. (21)

Refraction

In refraction, the light crosses the medium boundary, changing its di-rection to a new angle θb. The relationship between θa and θb is givenby Snell’s Law,

na sin θa = nb sin θb. (22)

Note that these angles are with respect to the normal to the interface.Furthermore, if the incident light is normal to the interface (θa = 0),there is no bending.

Total Internal Reflection

As shown in Figure 7, there is always some critical angle θcrit abovewhich refraction is impossible because above this angle the outgoingangle is greater than 90◦.

The critical angle for total internal reflection is given by

sin θcrit =nbna

. (23)

phys 202 notes, week 8 7

2 The reason we choose the electric andnot the magnetic field is that most EM-wave detectors, such as your eyes, aresensitive to the electric part of the wave.

Figure 8: Illustration of polarizationalong different axes.

Figure 9: Example of a polarizing filter.

When the incident angle θa is greater than this angle, refraction isimpossible and all of the light will be reflected. This phenomena isreferred to as total internal reflection.

Polarization

Recall (Figure 1) that EM waves are transverse waves, i.e. the directionsof the

#»E and

#»B fields are perpendicular to the direction of propagation,

and to each other. However, within these constraints, the displacementdirections of the

#»E and

#»B fields can be in different orientations. We

refer to the displacement direction of the electric field as the polarizationdirection.2

Now let’s pretend an EM wave has#»E field displacements in only

one direction, for example the y direction. We then say that the wave islinearly polarized in the y direction. Figure 8 shows examples of wavespolarized in the y and z directions, as well as how a slit can act as apolarizing device.

To set the polarization of light, we can use a device called a polar-izing filter, as demonstrated in Figure 9. This device serves to onlytransmit light with a specific polarization direction. However, we haveto pay a price for this: the intensity of the light is reduced. Let’s saywe have light striking a polarizing filter at an angle φ to the directionof polarization. The intensity reduction is described by the equation,

I = Imax cos2 φ, (24)

where Imax is the maximum intensity of the light transmitted (at φ = 0,i.e. already polarized in the direction of the filter).

Note that if unpolarized light strikes the filter, φ is randomized andit turns out that I = Imax/2.

phys 202 notes, week 8 8

Mirrors

Images

The first concept we need to introduce is that of an image. This isbasically the apparent source of light rays as seen by an observer. Thisis illustrated in Figure 10 (left side). Although the real source of thelight rays is point P, when they reach the observer’s eyes it looks likethey came from point P′. This is because the rays reflect off the surfaceof the mirror and reach the observer’s eyes at an angle that makesthem look like they came from P′.

Figure 10: An image due to a mirror(left) and refraction (right).

Similarly, images can also form due to refraction, as shown in theright side of Figure 10. Here, the rays change angle when crossing themedium boundary. Thus they reach the observer’s eyes at an anglethat makes them look like they came from P′.

Figure 11 shows what’s going on in more detail and defines a fewterms. First, we have the distance s, the horizontal distance from Pto the mirror plane. This is called the object distance. When theobject is on the same side of the reflecting (or refracting) surface as theincoming light, we say that the object distance is positive. Otherwise,it’s negative.

We can also define the image distance, s′ in the figure. This isthe horizontal distance from the mirror to the image point. When theimage is on the same side of the reflecting (or refracting) surface as theoutgoing light, we say that the image distance is positive. Otherwise,it’s negative.

The image shown in Figure 11 is a virtual image. This is because thelight rays don’t actually come from the image point P′. They only looklike they do. There is also a type of image called a real image where the

phys 202 notes, week 8 9

Figure 11: An image from a mirror, inmore detail.

Figure 12: A reversed image.3 Or colloquialy, people often use theterm “mirror image”.

rays really do pass through the image point.

Magnification

In the Figure 11, the object distance s and image distance s′ both havethe same magnitude. Thus if we were to place an image of height y infront of the mirror, the apparent image would have height y′ ≡ y. Inthis case, there is no change in the size of the image; the magnificationis unity.

In general, the apparent height of an image is not always the sameas the real height. We can define the lateral magnification of an imageas

m =y′

y. (25)

Depending on the geometry of the problem, images can either ap-pear in the same vertical orientation as the original, in which case theyare upright. They can also appear flipped along the y-axis, in whichcase they are inverted.

The orientation of 3d objects can also be changed by mirrors. Thisis represented in Figure 12, where the orientation of the hand changesfrom left-handed to right. We call this type of image reversed.3 Flatplane mirrors always form images that are the same size as the origi-nal, but reversed.

phys 202 notes, week 8 10

Example Problems

15. Jy @ L (£) ~ 7 F .,,

0+?.: '= 0+ &-., ::::---r

-_J

=) G~:: tg.r1 ' ~ .........._ __

8 b -:::. 3f / 31 °

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a..). 11 <K"', I f,, f. 1,v "",r: 5 j -----

=) S~11 67c,+ .. VS c~) ~ ~ : ~~~ 11~ L ~ I

fc,(ari h f:01 -----.....

11! c~

ID 1( / ~ { s-"l"'t q,,,,,e J

~1 T =- :t _, I" c~ "rh yz ___ T (o

~) rh = a{<PJ (! _!.. r 11;'