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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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Use the instructions from today’s slides here (What? So What? Now what?)

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Oscillate 震荡 （描述SHM的行为）

ep:The mass on the spring reciprocates up and down.

displacement 位移（从平衡位置的距离）

ep:If the equilibrium position of a spring is x=0, and at a certain moment the block is at x=+0.10m, then the displacement is +0.10 m.

frequency 频率（每秒的震动此事）

If an object vibrates up and down 2 times per second, its frequency is f=2Hz.

period 周期（完成一次震荡所用的时间）

If a pendulum clock takes 2 seconds to swing from left to right and back to left, then its period is T=2s.

total energy 总能量（SHM中恒定）

A block with a mass of 0.2 kg and an amplitude of 0.05 m has an angular frequency of w=10rad/s

ET=1/2*0.2*10^2*0.05^2=0.25J

This is the total mechanical energy that remains constant throughout its simple harmonic motion.

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kiretic energy 动能

potential energy 势能

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a. Conservative forces only? （保守力）

Simple Harmonic Oscillators：✅

Damped Harmonic Oscillators：❌

(not only the consrvative forces.There is friction摩擦力)

b. Dissipative forces?(耗散力)

Simple Harmonic Oscillators：❌

Damped Harmonic Oscillators：✅

(There are dissipative forces (such as air resistance and friction))

c. Time-dependent amplitude?（幅度随着时间变化）

Simple Harmonic Oscillators：❌

Damped Harmonic Oscillators：❌

(Floating constant and decaying over time)

d. Form of potential energy function.

Simple Harmonic Oscillators：U=1/2 kx^2

Damped Harmonic Oscillators：the total energy will be less

f. Energy loss over time

Simple Harmonic Oscillators：❌

Damped Harmonic Oscillators：✅

(the reason that damped harmonic oscillators energy loss over time,conect with b)

g. Graph of displacement vs time.

Simple Harmonic Oscillators：cos/sin

Damped Harmonic Oscillators：get more lower sin/cos

h. Equation of motion.

Simple Harmonic Oscillators：ma + kx = 0

Damped Harmonic Oscillators：ma + bv + kx = 0, where b is the damping coefficient

i real world example

Simple Harmonic Oscillators：Pendulum clock

Damped Harmonic Oscillators：Shock absorber

vocabulary:

damping减震

criticall damping 临界减震

light sampling 轻减震

heavy sampling 重减震

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Guiding questions :how do standing waves form?

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Guiding questions :how do standing waves form?

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Constructive in phase

Destructive out phase

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

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Your questions should follow this format:  'How does changing (IV) affect (DV) for (system name)?'

Be testable by changing one variable and measuring another (like a real experiment)

Be focused on physics concepts (waves, resonance, standing waves, frequency, amplitude, damping, etc.)

Show curiosity! It’s okay if you’re not sure how you’d do it yet — we’ll develop the ideas in class.

How does changing the frequency of the vibrator affect the number of standing wave loops formed on the string?

Task 2 - Add a reflection called 'Standing Wave Equipment Exploration July 31st' to the IA Skills part of your PADLET.

What equipment did you find most interesting today? Why?

I found the vibrator and string system most interesting. Watching the string form clear standing wave patterns as the frequency changed was really engaging.

What was one thing you found confusing or surprising?

I was surprised by how specific the frequencies had to be for standing waves to appear. If the vibrator was even slightly off, the wave pattern would collapse.

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

How does the height of water in a bottle affect the frequency of sound produced when the bottle is tapped?(ia)

Example Research Questions

These are just examples to inspire you — don’t copy them!

How does the tension in a string affect the frequency of its first harmonic?

What is the effect of changing the liquid in a resonance tube on the resonant frequency?

How does damping affect the clarity of standing wave patterns on a string?

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wavefronts and rays

Rays =drection of the wave

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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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Wavelength, Colour, Frequency, Energy, Electromagnetic Spectrum, Refraction

🙅Frequency is set by the source and stays constant across boundaries.

🙆(change)Since speed changes and frequency stays the same, wavelength must change: v=fλ

🙅Colour depends on frequency, so if frequency doesn’t change, the perceived colour stays the same.

🙅Energy of a photon is E=hf, so if frequency stays the same, energy stays the same.

🙅Still the same type of EM wave (e.g., visible light stays visible).

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Single and Double Slit WHAT IF QUESTIONS

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Single and Double Slit WHAT IF QUESTIONS

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fringe spacing (s) vs distance along the screen (y)

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fringe spacing (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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Diffraction LEARNING CHECK CALCULATION QUESTIONS

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Diffraction LEARNING CHECK CALCULATION QUESTIONS

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Diffraction central maximum width 衍射中央极大宽度 W=2L*入/a

First dark pattern condition第一暗纹条件

asin=入

proportional priority method

W1/W2=L1/L2

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用定角和波长取d

Get d using a fixed angle and wavelength

另一个角度对应的波长，把角带 入=dsinθ

The wavelength corresponding to another angle, substituting the angle into = dsinθ

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2*nmax+1

*2:its means both side of the graph

+1 its means the central maximum, which n=0

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Optical path difference (Δ) refers to the difference in distance traveled by two beams of light when they arrive at the same point from different paths. This optical path difference determines the phase difference between the two beams upon arrival, and thus determines whether they interfere constructively (reinforce) or destructively (cancel).

光程差 ΔΔ 指两束光从不同路径到达同一点时，它们实际传播的距离之差。光程差决定两束光到达时的相位差，从而决定它们是相长干涉（增强）还是相消干涉（抵消）。

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Double slit/grating:

Integer multiples of λ bright

Half multiples of λ dark

Single slit: Integer multiples of λ across the slit dark

Bright fringes appear between dark fringes

Remember to check at the end sinθ≤1

Otherwise, that level does not exist.

Double slit/grating:整倍数λ 就是亮

半倍数λ 就是暗

Single slit:整倍数  暗

亮纹在暗纹之间出现

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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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adjacent=in between /next to

slide 3:diffraction grating N slits

principal vs. secondary maxima

N-1 minima

N-2 secondary maxima

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How would an ambulance siren sound if:

(i) you are stationary AND the ambulance is stationary.

The sound heard /the same as the original siren, no change.

(ii) you are stationary AND the ambulance is moving away from you.

Sound you hear/The pitch is lower (the frequency is lower), and the sound sounds "deeper"

(iri) you are stationary AND the ambulance is moving towards you.

Sound you hear / The pitch of the sound increases (the frequency becomes higher), and the sound sounds "sharper".

(iv) you moving towards a stationary ambulance.

The sound you hear / the pitch is rising because you are actively "moving towards the source of the wave".

(v) you are moving away from a stationary ambulance.

The sound you hear /the pitch is lower because you are “moving further” away from the source of the wave.

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The shift in the wavelength of light emitted by a galaxy is measured on Earth. The shift allows the determination of

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The shift in the wavelength of light emitted by a galaxy is measured on Earth. The shift allows the determination of

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relative velocity between the source and observer

component of relve

paralle (to the ine sight)

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use the equation :T=2pai square root m/k

after this we get T1=2pai square root m/k (T)*1/2, also T1=T*1/2

so the answer is 0.5

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use the equation :T=2pai square root m/k

after this we get T1=2pai square root m/k (T)*1/2, also T1=T*1/2

so the answer is 0.5

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ABC no external force, additional force

but D have additional force,the bounce of the basketball when it hits the floor

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Ek=The kinetic energy itself

Ep=Extra energy（such as high）

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The wavelength becomes longer and the light moves away

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1. The Doppler effect can cause a shift in some properties of light waves. Which quantities on the right are affected?

the doppler effect will included

frequency

wavelength

(color&energy)

BUT the wave speed will not change(cause the speed of light in vacuum is c = 3.0 × 10^8 m/s)

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so if a ambulance move towards to me the frequency will be more higher (increase) and when it moves away for me the frequency will be more lower (decreases).

and for the wavelength will decrease when it moves towards to me ,and when it moves away from me it will increase.

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heard when an observer is standing still, and a fire engine is approaching at high speed?

who's moving

so the question said that the observer is STANDING STILL, and a fire engine is APPROACHING at high speed.

so the fire engine is moving, so used the moving source equation.

answer is D

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If a galaxy moves away from us:

The light waves are stretched → their wavelength increases → and they shift toward the red end (red shift).

If a galaxy moves toward us:

The light waves are compressed → their wavelength decreases → and they shift toward the blue end (blue shift).

so the answer is A

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kinetic energy is the energy of motion

so when the speed is increase the kinetic energy increase

potential energy is the energy which connect with the displacement when displacement increase the potential energy will increase

so in SHM

at balance point x=0

⬆️speed /kinetic energy

⬇️potential energy

amplitude position振幅

speed=0

kinetic energy=0

⬆️potential energy

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basic wave

longitudinal wave&transverse wave

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what is electromagnetic wave means?

"Perpendicular to each other

Perpendicular to the direction of wave travel → making EM waves transverse."

so we can know that electromagnetic wave is a transverse wave

No medium needed

Travel speed: In a vacuum, they move at the speed of light, c ≈ 3.0 × 10⁸ m/s.(this one is on the data booklet)

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Visible light wavelength: about 400–700 nm!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

40,000nm=4.0×10−5m

70,000nm=7.0×10−5m

LOOK AT YOUR DATA BOOKLET

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it means that it can't be a transverse waves, cause in the question it says that is "parallel" to the motion of the wave

so its longitudinal wave choose b

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I is not correct because this graph could not prove that is a transverse wave or a longitudinal wave.

II is not correct cause the graph shows that the period is 3

III is not correct because if you use II data then calculate it you will find out that is wrong.

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Strong diffraction occurs when the gap width is about the same size as the wavelength of the wave.

The barrier thickness does not control diffraction, it just blocks the wave.

The “length of the wavefronts” is irrelevant — it’s the wavelength that matters.

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B. Wrong → refracted ray parallel to the normal would mean refracted straight through (0°).

C. Wrong → critical angle is defined by incident, not refracted, angle.

D. Wrong for the same reason.

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the observer is not moving so is moving source ,use the right equation

because the moving source is away from the observer, so it should be -.

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A: ΔL = 0.062 m, λ = 0.031 m, d = 0.10 m→ ΔL/λ=2, so it’s a second-order maximum. Possible, but d = 0.10 m doesn’t quite fit well.

B: ΔL = 0.062 m, λ = 0.062 m, d = 0.05 m→ ΔL/λ=1, so it’s a first-order maximum. Fits perfectly. ✅

C: ΔL = 2.424 m, λ = 1.212 m, d = 0.10 m → Completely wrong.

D: ΔL = 0.031 m, λ = 0.031 m, d = 0.10 m → Path difference is calculated incorrectly.

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Use the double-slit fringe spacing formula!!!!!!!!!!!!!

w=入D/S

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(a)explain why the laser beam stays inside the curved stream of water

Marks cheme

﻿﻿[1] States total internal reflection (TIR) at the water-air boundary.

﻿﻿[1] Explains condition: water has higher n than air and incidence angle exceeds

1. my answer for this queastion

because the laser shoot in the water, and the laser beam stays inside carred of water ,cause light have in caecum is 3.00*10^8 ms^-1 but in water ,we have different density to affect the light speed to affect the light speed ,so if i need to control it in ceacum system, laser beam need to stays inside the curved stream of water.

2. what should i write

TIR/critical angle

this question is about TIR and different refractive indices cause changes in the speed of light

3. rewrite the answer

The laser beam stays inside the curved stream of water because total internal reflection occurs at the water–air boundary. Since water has a higher refractive index than air, and the incidence angle is greater than the critical angle, the light is totally internally reflected and follows the curve of the water stream.

(b) calculate the speed that the laser beam travels inside the curve steam of water.

v=c/n

=3.00*10^-8/1.33

=2.26*10^-8

(c) determine whether the laser beam can or cannot travel inside the curved stream of water when the angle of incidence is 55(angle)

Marks cheme

Decide if a 55° ray is guided (2 marks)

• [1] Computes critical angle: sine_c = n_air /n_water = 1.00/1.33 → 0_c =48.8°

• [1] Since 55° > 0- -c, TIR occurs → the beam remains inside the water.

1.what should i write

Calculate critical angle

Since 55∘>48.8∘55∘>48.8∘, total internal reflection occurs.
Therefore, the laser beam remains inside the water stream (guided).

⚠️ please be clear that what the question about

(ii) the laser is changed from green to red

marks cheme

[1] Change colour green→red: n slightly smaller for longer ^ → bends less (refracted angle in water slightly larger).

what should i write for this question

Water has a slightly lower refractive index for red light than for green light

Because water’s refractive index is slightly lower for red light, the critical angle is slightly larger, but 55° is still above the critical angle, so total internal reflection still occurs and the red laser stays inside the water.

(iii)the stream flows inside a glass tube instead of air

marks cheme

﻿﻿[1] Water inside a glass tube (n_glass > n_water): no TIR at water-glass → lig refracts into glass; confinement in water lost.

what should i write

Because water’s refractive index is slightly lower for red light, the critical angle is slightly larger, but 55° is still above the critical angle, so total internal reflection still occurs and the red laser stays inside the water.

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mark cheme

(a)(i) Radial speed from Ha shift (2 marks)

﻿﻿[1] Uses v = с(▶️入/入) with ▶️入 = (656.0 - 656.2) nm → ▶️入/入 = -3.05×10-4.

﻿﻿[1] v ~ -9.2×10* m s-4; negative sign (blueshift) indicates motion towards Earth.

⚠️ read the question and understand the question

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rewrite this question

Δλ/λlab=v/c

so we can get v=Δλ/λlab

|Δλ|/λlab=0.2/656.2=3.049*10^-4

so

|v|=3.00*10^-8*3.049*10^-4

=9.15*10^-4

Estimate the angular position of the third minimum for this interference pattem.

0-=2.5*10^-2rad from the centre

Calculate the grating spacing d.

1.31*10^-5

Determine the number of slits N inside the grating.

36

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use the equation for simple harmonic motion

L=入/2

入=2L*0.45=0.90m

speed of wave

v=f入

160=f*0.90

so we can know the frequency is 160/0.90

f=178hz

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rewrite the answer

The small metal spheres move inside the empty spaces when the frame vibrates.
They collide with the walls of the cavities, and energy is transferred to heat and sound.
This dissipates the mechanical energy of the frame, so the oscillations are damped.
As a result, the amplitude of vibration decreases with time, as shown in the graph.

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入=v/f=340/1700=0.2m

one common physical set up here is that d receives sound by two routes the direct ray L to D(length=0.70) and the ray that goes L to P then reflects at the water surface and travels P to D(total lengthLP+PD)

Δs=(LP+PD)−LD=1.00−0.70=0.30 m.

Δs=0.30 m=1.5λ=(1+1/2​)

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Nuclear radius increases with the cube root of the nucleon number A; heavier nuclei are only slightly bigger.

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nuclei are incredibly dense and all have almost the same density regardless of size.

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inding energy per nucleon: shows stability of nucleus.

Mass defect: mass difference between separate nucleons and bound nucleus.

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How can energy conservation be used to derive the Bohr energy levels En?

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How can energy conservation be used to derive the Bohr energy levels En?

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Table 1 shows that when ZZ increases from 6 (C-12) to 82 (Pb-208), the closest distance of approach increases from 1.73 fm to 23.62 fm. For example, for Z=20 (Ca-40) the distance is 5.76 fm, while for Z=79 (Au-197) it is 22.75 fm, demonstrating a clear upward trend with ZZ.

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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 (L) & Apparent Brightness (b)

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Luminosity (L) & Apparent Brightness (b)

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Luminosity (L) — the total power output of a star (energy per second), measured in watts (W).

Apparent brightness (b) — the power per unit area received by an observer at distance d from the star, measured in W m⁻².