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C3 HL Diffraction Formative WORKED SOLUTIONS

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'Universe without Color' Chapter 1 Theory Section

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'Universe without Color' Chapter 1 Theory Section

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The DP physics theory behind 'What if electromagnetic waves only had one wavelength?'

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Path difference - The difference in distance/wavelengths at which two different waves intersect

Radian - A unit of measurement used to express angles relative to the radius of a circle. 360* is 2π

Sine and Cosine functions - opp/hyp and adj/hyp. Can be used to model waves by rotating a point along the circumference of a circle and graphing its x and y coordinates relative to the center.

Initial condition - The starting conditions of the wave at t = 0, eg. initial angle, velocity, displacement, etc.

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Simple Harmonic Motion - Repetitive motion where the acceleration is always directed toward a fixed equilibrium point, eg. simple pendulums, mass-spring systems.

Wave - A disturbance that transfers energy, often through a medium (mechanical waves) although it is not required (non-mechanical waves) eg. sound waves, light waves

Frequency/angular frequency - The rate at which an oscillation occurs, either in hertz or rad/s

Wavelength λ - The distance between the crests of a wave

Amplitude - The maximum displacement of a wave measured from its equilibrium equation

Phase Angle φ - the angular displacement of a wave from a reference point in time

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Damping - losing energy in oscillations by dissipating energy

Types of Damping

Critical - Immediately to equilibrium

Light - Still continues to oscillate about equilibrium, but amplitude decays over time until it reaches 0

Heavy - Slowly to equilibrium

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1. Conservative forces only? No

2. Dissipative forces? Yes

3. Time-dependent amplitude? Yes

4. Form of potential energy function? ET = 1/2kx^2

5. Total mechanical energy? Decreases over time

6. Energy loss over time? Yes

7. Graph of displacement over time? Sinusoidal but with decreasing amplitudes

8. Equation of motion? Equation of motion? a = -kAsin(ωt+Φ)

9. Real-world examples? Real pendulums and mass-spring system, vehicle shock absorbers, building mass dampers

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1. Conservative forces only? Yes

2. Dissipative forces? No

3. Time-dependent amplitude? No

4. Form of potential energy function? Ep = 1/2kx^2

5. Total mechanical energy? Constant

6. Energy loss over time? No

7. Graph of displacement over time? Sinusoidal

8. Equation of motion? a = -kAsin(ωt+Φ)

9. Real-world examples? Pendulums and mass-spring systems under a vacuum

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Follow up Q: What is the potential energy function for a test mass in a gravitational field?

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Follow up Q: What is the potential energy function for a test mass in a gravitational field?

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1. Write your ideas and research here.

2. Apply your understanding to these systems:

What is the complete total energy function for an oscillating mass on a spring system inside a viscous liquid?

What is the complete total energy function for raindrop falling through the atmosphere of a planet?

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No net energy transfer

Certain nodes where there are points of 0 displacement

Certain antinodes where there are points of max displacement

Oscillates only between two defined boundaries

Constructive interference - two waves in phase combine to create a wave with greater amplitude

Constructive interference - two waves out of phase combine to create a wave with lower amplitude

How do standing waves form?

Reflection at the boundaries, then constructive and destructive interference forms nodes and antinodes for specific frequencies

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How does changing the type of string (mass per unit length) affect the wavelength of a standing wave on a vibrating string system?

How does changing the strobe light frequency affect the apparent motion of a standing wave on a vibrating string?

How does changing the diameter of the measuring cylinder affect the resonance length for a given tuning fork frequency?

What equipment did you find most interesting today? Why?

The tuning fork is the most interesting. Metals vibrate at frequencies, that is common knowledge but its uses and applications in experiments I'm not sure about, which is why it's interesting.

What was one thing you found confusing or surprising?

What standing wave related experiments could be conducted with a laser pointer? I struggle to think of a situation where a laser pointer could relate to standing waves.

Which of your personal research questions would you most like to test? Why?

I want to do the stroboscope research question, but mostly because I just want to see flashing lights and the way you described the visuals made me curious.

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Wavefronts show an array of waves propagating in the same direction. The visible section is the crests of the wavefront.

Rays are straight lines that shows average direction of movement, perpendicular to the ray, showing the peaks/crests of waves.

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Normal - A theoretical line that is perpendicular to the surface at which the light strikes

Angle of Incidence - The angle between the normal and the ray that strikes the object

Angle of reflection - The angle between the reflected ray and the normal

Angle of incidence = Angle of reflection

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Transmission - passes through

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Does the color of light change as it when it passes across a boundary?

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Does the color of light change as it when it passes across a boundary?

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No, it does not.

Energy determines colour, given by the equation E = hf

When an object passes across a boundary, it can be reflected or refracted. For both of these, the frequency does not change but its wavelength does, therefore colour does not change.

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1. Fiberoptics TIR

2. Constructive and destructive double slits

3. Air/water TIR

4. Atmosphere TIR

5. Water refraction

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Where does light in a laser come from?

The light comes from excited electrons in atoms that releases photons when it increases and then eventually drops an energy level.

How can the colour of the laser be changed?

The colour of a laser can change when it gains energy.

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Diffraction is a phenomenon where waves bend as they pass through or around objects.

The surface of diffraction grates contain many tiny grooves. When a ray of white light diffracts on the grooves, the different wavelengths diffract in different angles, separating the light into multiple colours. When a ray of a single wavelength (single colour) diffracts on the grooves, it shows the areas of constructive and destructive interference instead.

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Fringe Width (s) vs Distance along the screen (y)

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Fringe Width (s) vs Distance along the screen (y)

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Single Slit Diffraction (Common Misconceptions)

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Single Slit Diffraction (Common Misconceptions)

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Double Slit Diffraction (Common Misconceptions)

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Double Slit Diffraction (Common Misconceptions)

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Data Analysis Task (Connecting C5 with E5: Fusion & Stars)

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Data Analysis Task (Connecting C5 with E5: Fusion & Stars)

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nλ = dsinθ

maximum when sinθ = 1

nmax = d/λ

maxima = 2nmax + 1 (central maxima)

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With diffraction gratings, there can be smaller secondary maxima

Dark fringes in between two principal maxima = N-1

Bright fringes in between two principal maxima = N-2

so depending on the number of slits, there will be greater number of secondary maxima

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Missing maxima occur when the single slit and double slit interfere with each other given

sintheta = nλ/b = nλ/d

wavelength is same fo rboth

λ/b - λ/d

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When an object emitting any waves moves closer to a stationary observer, the wavelengths will be compressed. When an object moves away, the wavelengths will be stretched.

This means that objects moving away will look red and objects that are moving closer will look blue.

Similarly, the pitch of sound moving away will be low and moving closer will be high

HOWEVER if the object is stationary and the observer is moving, the wavelength does not change, the frequency still does

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Given nλ = dsinθ, as λ increases, so does dsinθ and therefore the angle θ. This means that red light diffracts more strongly than blue, for example as it has a higher wavelength.

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Which component of the relative velocity do we consider?

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Which component of the relative velocity do we consider?

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Relative velocity - difference in velocity between the source and observer

Only consider the component of the relative velocity that is parallel to the line of sight

For an object moving in a circle, the relative velocity will be changing as it goes through a revolution

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Continuous range of values VS discrete values

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Continuous range of values VS discrete values

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For the doppler effect under a constant v, it is a discrete value because as soon as it passes, it jumps from a constant +v to a constant -v

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When damping on an oscillation is reduced, the amplitude increases and frequency increases. This is because the oscillation is allowed to move closer to its natural pattern. As a result, it also moves faster which increases the frequency required to drive the system.

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R = R0A1/3

R0 is a constant 1.2x10^-15, radius of one proton or neutron (fermi radius)

A = Nucleon number

DENSITY ALWAYS REMAINS THE SAME

Density vs nucleon number graph is FLAT

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1ev = 1.6x10^-19J

E = hf

lambda = c/f

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How can energy conservation be used to derive the Bohr energy levels En? (mock prep task)

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How can energy conservation be used to derive the Bohr energy levels En? (mock prep task)

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E = -13.6/n^3 eV

lambda = h/mv

mvr = nh/2pi

r = n^2a0

en = -ke^2/2rn

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ΔQ/Δt = kAΔT/Δx

Q can be energy

k is conductivity

A is area

ΔX is usually length of the segment

When two objects are in contact and insulated ΔQ/Δt are equal for both, so you can solve

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P = eσAT^4

e is emmisivity

σ is Stefan-Boltzmann consant, 5.67E-8

A is area

T is temperature, in K

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(i) estimate a sensible level of uncertainty for measuring devices.

(ii) Use error propagation (in data booklet)

(iii) calculate the uncertainty in repeated measurements.

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Luminosity is just power assuming blackbody

b (brightness) = L (luminosity)/4pid^2 (distance from object)

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E = mcΔT

E is energy absorbed by the object, it can be from a source of power, gravitational potential energy, etc.

Egpe = mgh, so

snmgL = nmcΔT

sgL = cΔT

Isolating heat capacity,

c = sgL/ΔT

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Q = mcΔT

c = Q(J)/m(kg)ΔT(K), units are Jkg⁻¹K⁻¹

Materials have specific latent heat, the amount of energy required to change state (Lvapourisation, Lfusion)

Calculate power from change in temperature over time, then calculate Q during only freezing from power, then use Q = mL

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Solar constant - the Intensity of radiation received by the Earth from the Sun

Albedo (reflectiveness) - scattered power/incident power

solar constant x cross sectional area of earth x (1-albedo) = eσAT^4

where eσAT^4 is with or without the atmosphere

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60W of thermal radiation from Earth is given

Incoming Intensity is 100W, with an albedo of 0.3, giving 30W of reflected energy.

The remainder is absorbed and then reflected again, 70W + 60W = 130W

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Gases absorb energy, causes oscillation and resonate. This resonance must release some energy, emitting it in all directions (some of which returns to Earth)

Planets need atmospheres to retain heat otherwise they end up extremely cold, like Mars, or if they have thick atmospheres they get extremely hot like Venus

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Gray bodies have emissivity 0 < e < 1

As it is insulated, both bodies will eventually reach the same temperature

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https://docs.google.com/presentation/d/142AcgOwzuwM0Hl_s3i8N3eEhbILrZLko2AYlCdtys_w/edit?usp=sharing

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Remember: I = I₁ + I2 for parallel

Just trace the charge, it splits at the node

Current will be the same through the circuit, so it must be the same before and after

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Ideal voltmeter has infinite resistance

Ideal ammeter has 0 resistance

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Be VERY VERY careful about outliers as they will mess up your uncertainty and give you faulty data.

Remove outliers and do extra trials (and don't forget to justify)

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To determine if something has a relationship, you should test whether:

1. when linearized, the graph creates a straight trendline

2. if there is no constant, then the graph should pass through the origin

3. calculating the slope should give you the value of a constant (and be the same at all points)

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1. When adding, add uncertainties

2. When multiplying, add FRACTIONAL uncertainties

3. Exponents multiply the fractional uncertainties by the power

4. Always give absolute uncertainty to 1 sf, and place should match the value

5. To reduce uncertainties, if the absolute remains unchanged you should just use a bigger range to lower fractional uncertainty

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Never forget to always transform given values back into SI form (eg. m, s, kg) before calculating so that you don't get confused

And also use powers in calculator, because it's easy and costs you nothing to be 100% sure

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Trendlines are NOT the same as regression

Regression lines are ALWAYS linear

Trendlines can be exponential, logarithmic, etc etc

IF THE TRENDLINE IS NONLINEAR AND YOU ADD A REGLINE YOU ARE MAKING A BIG MISTAKE

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E = mc^2

E in MeV

m in u

m = E/c^2

m = MeVc^-2

REMEMBER THAT IF YOU WANT TO START WORKING WITH KG AND J YOU NEED TO CONVERT

u = 1.66E-27kg

eV = 1.60E-19J

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Shot alpha particles (2 protons 2 neutrons) at gold foil

Some of the particles deflected at angles > 90*

As energy increases, the number of particles deflected drops exponentially, then linearly

Shows that a repulsive force in atoms exist (strong nuclear force)

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BE = Δmc^2

Δm is mass defect

Δm = (Z*mp + (A-Z)mn) - matom

INFO WILL ALWAYS BE PROVIDED

BE eats some of the mass, which is hwy when BE is broken, atoms get more mass

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Nuclear Stability Concepts and Explanations

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Nuclear Stability Concepts and Explanations

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Why do heavier nuclei need more neutrons than protons to be stable?

D. Neutrons add to the strong nuclear binding without increasing repulsive forces, since they are neutral. In heavier nuclei, more neutrons are needed to overcome the growing proton-proton repulsion and to help bind the nucleus together.

Why does proton repulsion become a bigger problem in larger nuclei?

B. As the number of protons increases, the repulsive electrostatic (Coulomb) force between them grows stronger. Since this force acts over long ranges and the protons are all positively charged, the total repulsion becomes more significant in larger nuclei.

Why are nuclei with too many or too few neutrons unstable?

C. If the neutron-to-proton ratio is too high or too low, the balance of nuclear forces is disrupted. The nucleus becomes energetically unstable and may undergo beta decay to adjust the ratio and reach a more stable configuration.

Why can’t the strong nuclear force always prevent decay?

A. The strong nuclear force only acts over very short distances (on the order of femtometers). In large nuclei, many protons are too far apart for the strong force to hold them all together effectively, which limits the ability of the force to prevent decay.

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Alpha decay

When mass A is too large (heavy elements)

Emits a 2p/2n particle

Beta decay

When ratio of neutrons to protons is too large

Neutron changes into a proton

Neutrino released

B- is high energy electron

B+ is a positron (anti electron)

Gamma decay

When need to release a little bit more energy

V with a line is an anti neutrino

NEVER have two antiparticles

only ever have 1 real and 1 anti

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A = A0e^-lambdat

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Work done by the gas W = PdeltaV

Work done on the gas W = -PdeltaV

deltaU = W + Q

where W depends on whether it is done by or on the gas.

for change in pressure AND volume,

deltaU = delta(PV) + Q

(PV has to be non zero)

for change in volume only,

deltaU = PdeltaV + Q

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For one cycle of the pressure/volume diagram, total work done is 0 (but work is still being done)

Isothermal vs Adiabatic

Isothermal is when no change in T (P proportional 1/V, or deltaPV=0)

Adiabatic is really fast with no change Q (with a smaller change in V)

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A latte coffee has both milk and coffee particles. When separate, arrangement is highly ordered with low entropy. When mixed, the combinations of possible particle microstates  is  disordered, thus high entropy.

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It is impossible for thermal energy to flow from a cold to a hot object without performing work

(Clausius form)

It is impossible in a cyclical process to completely convert heat into mechanical work

(Kevin Planck)