Modern Physics: Light

Question 1

Waves are:

Answer 1

Periodic oscillations of a medium

Question 2

Important points about waves to remember are:

Answer 2

1) sin(θ) has an amplitude of 1
2) the period of repetition is 2π radians = 360˚

Question 3

In one dimension, we can write our “wavefunction”, which describes the amplitude of the
wave as a function of position, as follows:

Answer 3

𝜓 𝑥 = 𝐴 sin(2𝜋𝑥/𝜆)= 𝐴 sin(𝑘𝑥)

where we have defined the wavevector k as

𝑘 =2𝜋/𝜆

Question 4

What is the wavevector K defined as:

Answer 4

𝑘 =2𝜋/𝜆

Note that the units of k are inverse length.

Question 5

We know that sin 𝜃 = 0 for:

Answer 5

𝜃 = 𝑛𝜋, 𝑛 =1,2,3,…

Question 6

We know that sin 𝜃 = 0 for 𝜃 = 𝑛𝜋, 𝑛 =1,2,3,… This occurs when:

Answer 6

2 𝜋 𝐿/𝜆= 𝑛 𝜋

⇒

𝜆 =2 𝐿/𝑛

⇒

𝜓 𝑥 = 𝐴 sin(𝑛 𝜋 𝑥/𝐿)

Question 7

We can represent our moving wave as a function of both space and time as:

Answer 7

𝜓 𝑥, 𝑡 = 𝐴 sin(2𝜋𝑥/𝜆−2𝜋𝑡/𝑇)= 𝐴 sin(𝑘𝑥 − 𝜔𝑡)

where we have defined the angular frequency ω as:

𝜔 =2𝜋/𝑇= 2𝜋𝜐

Question 8

Define angular frequency

Answer 8

𝜔 =2𝜋/𝑇= 2𝜋𝜐

Question 9

Notice that in one period, the wave moves by one wavelength, so the“phase velocity” is

Answer 9

𝑣 =𝜆𝑇⟹𝜔 = 𝑣𝑘

Question 10

To complete our definition of the wave, we need
to specify its initial state, usually by setting its value at x=t=0; this is done by introducing a
phase factor φ0. What is the full equation:

Answer 10

𝜓(𝑥,𝑡) = 𝐴 sin 𝑘𝑥 − 𝜔𝑡 − 𝜙)

Question 11

Notice that the complete definition of the wave is given by four independent parameters: A, ω,
k, and φ0. What do these represent?

Answer 11

A= amplitude

ω= angular frequency

k= wavevector

φ0= phase factor

Question 12

Equivalently, we could also describe a wave by specifying A, T, λ, and φ0. What do these represent?

Answer 12

A= amplitude

T= time phase

λ= wavelength

φ0= phase factor

Question 13

The power transported by a wave is proportional to..

Answer 13

𝑃 ∝ l𝐴l^2

Question 14

There are different conventions for how the quantities that determine the properties of waves
are defined and named.

Amplitude
This can be specified as the maximum amplitude 𝐴 or..

Answer 14

the peak-to-peak amplitude (𝐴”” = 2𝐴),
or root mean square amplitude A ‘rms’ = (1/2^0.5)A

Question 15

There are different conventions for how the quantities that determine the properties of waves
are defined and named.

Period / Frequency
For a given period of oscillation 𝑇, we can specify either the frequency 𝜈 = 1/𝑇 or..

Answer 15

the angular frequency 𝜔 = 2𝜋/𝑇 = 2𝜋𝜈.

Question 16

There are different conventions for how the quantities that determine the properties of waves
are defined and named.

Wavelength / Wavevector / Wavenumber
For a given wavelength 𝜆, we can specify two corresponding inverse quantities: 1/𝜆 and 2𝜋/𝜆.
Unfortunately, both of these quantities are referred to as wavevector or wavenumber in
different academic fields. In this class, we will use the convention that:

Answer 16

wavenumber refers to
1/𝜆 and wavevector 𝑘 = 2𝜋/𝜆; sometimes the latter quantity is also referred to as the
“angular wavenumber”. Note that the textbook Jewett & Serway uses a different convention
and refers to 𝑘 = 2𝜋/𝜆 as the “wavenumber”.

Question 17

There are different conventions for how the quantities that determine the properties of waves
are defined and named.

Phase
The sign in front of φ0 can be either positive or negative. If a negative sign is used, a positive
value of φ0 shifts the wave to the right (in the positive x direction); on the other hand, if a
positive sign is used, convention is..

Answer 17

a positive value of φ0 shifts the wave to the left (negative x direction). Either convention is fine as long as it is applied consistently.

Question 18

The most important point about waves is that they…

Answer 18

obey the principle of superposition.

Question 19

What does this picture represent?

and what occurs during this interaction?

Answer 19

Constructive interference.

𝜓1 = 𝐴 sin 𝑘𝑥
𝜓2 = 𝐴 sin 𝑘𝑥
𝜓3 = 𝜓1 + 𝜓2 = 2𝐴 sin(𝑘𝑥).

The new wave now has zero amplitude because the two components cancel each other. These
two waves are said to be “completely out of phase” or “out of phase” by 180˚ or π radians and
therefore undergo “destructive interference”.

Question 20

What does this picture represent?

and what occurs during this interaction?

Answer 20

Destructive Interference.

𝜓1 = 𝐴 sin 𝑘𝑥
𝜓2 = 𝐴 sin 𝑘𝑥 + 𝜋 = −𝐴 sin 𝑘𝑥
𝜓3 = 𝜓1 + 𝜓2 = 0

Question 21

What does this picture represent?

and what occurs during this interaction?

Answer 21

General Interference.

𝜓1 = 𝐴 sin 𝑘𝑥
𝜓2 = 𝐴 sin 𝑘𝑥 +(3𝜋/2)
𝜓3 = 𝜓1 + 𝜓2

If the waves are not perfectly in phase or perfectly out of phase, or do not have the same
frequency, then a more general form of interference will occur.

Question 22

What is coherence?

Answer 22

If a source emits waves that all have the same phase relationship (i.e. they all have the same
φ0), then the source emits light that is “coherent”. On the other hand, if the phases are all
random, then the source emits light that is “incoherent”.

Question 23

What is the classical behaviour of light?

Answer 23

Newton originally proposed a “corpuscular” theory of light, in which light is composed of particles. This view was consistent with the development of “geometric
optics”, in which light moves in straight lines, or “rays”.

Question 24

What does this show?

Answer 24

We see here a profile whose intensity varies as a function of position with a series of maxima
and minima. This behaviour is not consistent with that of a simple particle, but rather seems
like interference, so it is natural to conclude that light must be a wave.

Question 25

Explain what is occuring here

Answer 25

• First, the single slit turns the incoherent light source into a coherent one: because the slit is thin (i.e. its width is much smaller than the wavelength of the light) it looks like a point source that sends out spherical waves.
• This coherent light then proceeds to the two slits, each of
  which act as coherent light sources.
• The two spherical waves will cause an interference pattern because at any given angle the two
  wavefronts have travelled different distances from the slits, and are therefore out of phase. If
  we assume the screen is very far away (L >> d), then the angle between each slit and a point
  on the screen is the approximately same.

Question 26

How is the path difference calculated?

Answer 26

Δ𝑙 = 𝑙2 − 𝑙1
sin 𝜃 =Δ𝑙/𝑑⇒ Δ𝑙 = 𝑑 sin 𝜃

Question 27

If the path difference is an integer multiple of the wavelength, then the waves are in phase and
we have constructive interference. The maxima, then, will occur when the following condition
is met:

Answer 27

𝑑 sin 𝜃 = 𝑛𝜆 for {n=0,±1, ±2, ±3, …}

Question 28

At θ =0, there is a central maximum. As sin(θ) increases from 0 to
1, we will get maxima whenever d sin(θ) is an integer multiple of the wavelength. This means
that if we want to get multiple maxima, we need to have:

Answer 28

𝑑 ≥ 𝜆, otherwise this condition will
never be satisfied.

Question 29

if the path difference is half a wavelength more than an integer multiple of the
wavelength, then the waves are out of phase and we have complete destructive interference.
The minima are then found when the following condition is met:

Answer 29

𝑑 sin 𝜃 = (𝑛 + 1/5)𝜆 for {n=0,±1, ±2, ±3, …}

Question 30

The spectrum of visible light covers a range of wavelengths from:

Answer 30

0.4μm – 0.7μm; wavelengths
longer than ~700nm are called “infrared” (IR) while those that are shorter than ~400nm are
“ultraviolet” (UV).