Light

Question 1

What are waves?

Answer 1

Spatial patterns which fill available space and which oscillate as a function of time.

Question 2

What are light waves made up of?

Answer 2

Oscillating patterns of electric and magnetic fields as functions of r and t.

Question 3

What is the the equation for the wavenumber k?

Answer 3

k =2*pi/λ

Question 4

What are the two equations for wave speed?

Answer 4

v = ω/k = λ/T

Question 5

What is the equation for a harmonic wave?

Answer 5

u(x, t) = Acos(kx -/+ ωt)

Question 6

In thermodynamics, how much energy does each degree of freedom contribute?

Answer 6

1/2 * kb * T

Question 7

What is an “ideal” black body?

Answer 7

An ideal absorber, absorbing all of the radiation that strikes it. It is also an ideal radiator/emitter at the same time.

Question 8

What is the Stefan-Boltzmann law?

Answer 8

I = σT^4

Question 9

How do you get the intensity of a black body from a graph?

Answer 9

Graph of I vs λ, the area under the graph (integral from 0 to infinity of I(λ) dλ

Question 10

How do the I-λ graphs differ for black bodies at different temperatures?

Answer 10

The higher the temperature, the more intense the black body is, and the lower the wavelength.

Question 11

What is Wien’s displacement law?

Answer 11

λmax T = const = 2.898*10^-3 mK

Question 12

What is the value of the Stefan-Boltzmann constant?

Answer 12

5.67*10^-8 W m^-2 K^-4

Question 13

What is the equation for intensity of a black body in terms of temperature a d wavelength? (classical approach)

Answer 13

I(λ) = (2pickb*T)/(λ^4)

Question 14

Why can’t the classical approach to intensity of a black body be used experimentally?

Answer 14

It leads to the Ultraviolet catastrophe, where the intensity of UV rays tends to infinity.

Question 15

What did Planck say about energy of BB radiation?

Answer 15

Energy is quantised, allowed energies are En = nhf, where h is the constant of proportionality.

Question 16

What is Planck’s model for BB radiation?

Answer 16

E = (hf)/(exp(hf/kbT)-1)

Question 17

What was Planck’s equation for the intensity of black body radiation?

Answer 17

I(λ) = (2pic^2h)/(λ^5) * 1/(exp(hc/λkbT) - 1)

Question 18

What did Albert Einstein find to do with the photoelectric effect?

Answer 18

E = hf, where h = 6.626 * 10^-34 Js

Question 19

Describe apparatus for the photoelectric experiment.

Answer 19

• Dot anode and curved cathode opposite eachother

- Anode connected to a Galvanometer and both connected to a power supply

Question 20

Describe how the photoelectric experiment apparatus was used.

Answer 20

Monochromatic light shone onto cathode which emits electrons. Electric field pushes electrons to anode. Electrons return to cathode via circuit and Galvanometer measures current - reverse electric field, and increase the field strength until electrons no longer reach anode and no current flows - stopping potential

Question 21

What was the intensity equation proposed for black body spectral emittance?

Answer 21

I(λ, t) = aexp(-β/λt)/(λ^5)

Question 22

How many degrees of freedom and therefore energy does each mode bring to a wave?

Answer 22

Each mode brings 2 degrees of freedom which is equal to kbT energy per mode.

Question 23

What do we need to describe I(λ)?

Answer 23

Need to find dn, the number of modes in small wavelength interval λ to λ - dλ (dn * kbT -> I(λ))

Question 24

What is the equation for energy due to only some wavelengths being allowed on a black body curve of E-c
x, from x=0 to x=L.

Answer 24

En = sin(npix/L)*sin(ωnt), and λn = 2L/n - allowed standing waves.

Question 25

How do you calculate dn?

Answer 25

Assume a continuum of permitted standing waves, with n >> 1 and λ << 2L, so dn = dn/dλ * (-dλ) = 2L/λ^2 dλ

Question 26

What is the equation for energy in the modes with wavelengths between λ and λ-dλ?

Answer 26

E(λ, t) = 2LkbT/λ^2 dλ

Question 27

How do you find the equation for energy of a 3D black body? What is the equation for intensity?

Answer 27

assume a cubic box of size L: E(λ, t) = (8pi *L^3 kbT)/(λ^4) dλ I(λ) = 2pi*c*kbT/λ^4

Question 28

What is the equation of the probability of a certain energy (from Boltzmann distribution)? What is, therefore the probability of the energy from plancks model?

Answer 28

p(E) ~ exp(-βE), where β = 1/kbT p(nhf) = exp(-βnhf)/sum from n=0 to infinity of exp(-βnhf) = exp(-βnhf)*(1-exp(-βhf))

Question 29

How do you find the average energy, knowing the equation for the probability of a certain energy? (first step)

Answer 29

Average energy = = sum from n=0 to infinity of p(E) * E = sum of p(nhf) * nhf = sum of nhf *exp(-βnhf)*(1-exp(-βhf)

Question 30

How do you find the average energy, knowing the equation for the probability of a certain energy? (second step)

Answer 30

Now differentiate exp(-βnhf) to get -nhf * exp(-βnhf) = d/df * 1/(1-exp(-βhf)) = (-hf*exp(-βhf))/((1-exp(-βhf))^2)

Question 31

How do you find the average energy, knowing the equation for the probability of a certain energy? (third step)

Answer 31

= (exp(-βhf) * hf)/(1-exp(-βhf)) = hf/(exp(βhf)-1)

Question 32

What does this final equation for average energy tell us?

Answer 32

That for high T, the model is the same as classical result because kbT >> hf, so βhf << 1, giving kbT. For low T, kbT << hf, so βhf >>1, so = hf*exp(-βhf) -> is small compared to kbT.

Question 33

How do you find intensity using the equation for average energy?

Answer 33

Substitute in for kbT.

Question 34

How do you find λmax?

Answer 34

Differentiate Planck's formula to get λmax*T = 2.898*10^-3.

Question 35

How do you get the total power radiated?

Answer 35

Integrate I(λ) from 0 to infinity.

Question 36

What equation do you get for the total power radiated?

Answer 36

I = (2pi^5 * kb^4)/(15h^3 * c^2) * T^4