TOPIC 01
Waves and Particle Nature of Light
Unit 2 begins with the behaviour and properties of waves, followed by refraction, diffraction, interference and stationary waves. The unit then develops the idea that electromagnetic radiation can behave both as a wave and as a stream of photons.
A wave transfers energy and information without requiring the overall transfer of matter from one place to another.
Key Wave Quantities
- Amplitude: maximum displacement from the equilibrium position.
- Frequency: number of complete oscillations per second.
- Period: time taken for one complete oscillation.
- Wavelength: distance between successive points in phase.
- Wave speed: speed at which the wave pattern travels.
TOPIC 02
Transverse and Longitudinal Waves
Transverse Waves
In a transverse wave, the oscillations are perpendicular to the direction in which the wave transfers energy.
Examples include electromagnetic waves and waves travelling along a stretched string.
Longitudinal Waves
In a longitudinal wave, the oscillations are parallel to the direction of energy transfer.
Sound in air is a longitudinal wave. It consists of compressions and rarefactions caused by variations in pressure and molecular displacement.
Wave Graphs
Transverse waves can be represented by displacement against distance or time. Longitudinal waves can be represented using pressure or molecular displacement.
TOPIC 03
Wavefronts, Phase and Superposition
Wavefront
A wavefront is a line or surface joining points that are at the same phase of oscillation.
Phase
Two points are in phase when they have the same displacement and are moving in the same direction at the same time.
Path Difference
Path difference is the difference between the distances travelled by two waves to reach a particular point.
Superposition
When two or more waves overlap, their resultant displacement is the algebraic sum of their individual displacements.
Constructive interference occurs when waves reinforce each other, while destructive interference occurs when they oppose each other.
TOPIC 04
Stationary Waves
A stationary wave is produced when two waves of the same frequency, wavelength and amplitude travel in opposite directions and superpose.
Nodes
A node is a point where the displacement is always zero.
Antinodes
An antinode is a point where the displacement has its maximum amplitude.
Adjacent nodes are separated by λ/2, while a node and its nearest antinode are separated by λ/4.
Wave on a Stretched String
Here T is the tension in the string and μ is the mass per unit length.
WORKED EXAMPLE
A string has a tension of 80 N and a mass per unit length of 0.020 kg m⁻¹. Find the wave speed.
v = √(80 / 0.020)
v = √4000
v ≈ 63.2 m s⁻¹
TOPIC 05
Intensity of Radiation
Intensity is the power transferred per unit area perpendicular to the direction of energy transfer.
where I is intensity, P is power and A is area.
Inverse Square Relationship
For radiation spreading uniformly from a point source, intensity decreases as the square of the distance from the source increases.
TOPIC 06
Refraction
Refraction occurs when a wave changes direction because its speed changes when it enters a different medium.
Refractive Index
Here c is the speed of light in vacuum and v is the speed of light in the medium.
Snell's Law
The angles are measured from the normal to the boundary.
Critical Angle
For light travelling from a material of refractive index n into air, the critical angle C is the angle of incidence for which the refracted ray travels along the boundary.
Total Internal Reflection
Total internal reflection occurs when light travels from a more optically dense medium to a less optically dense medium and the angle of incidence is greater than the critical angle.
CORE PRACTICAL 04
Measuring Refractive Index
The refractive index of a transparent solid or liquid can be determined experimentally by measuring angles of incidence and refraction.
- Place the transparent material on paper.
- Draw a normal at the point where the light enters.
- Use a ray box or laser to produce an incident ray.
- Measure the angle of incidence and angle of refraction.
- Repeat for several different angles.
- Use Snell's law or a suitable graph to determine n.
Take care to avoid parallax when measuring angles and repeat measurements to improve reliability.
TOPIC 07
Polarisation
Plane polarisation occurs when the oscillations of a transverse wave are restricted to one direction.
Polarisation is evidence that a wave is transverse, because longitudinal waves cannot be plane-polarised.
Electromagnetic waves are transverse and can therefore be polarised.
TOPIC 08
Diffraction
Diffraction is the spreading of waves when they pass through a gap or around an obstacle.
Huygens' Construction
Huygens' construction treats every point on a wavefront as a source of secondary wavelets. The new wavefront is obtained by drawing the envelope of these wavelets.
Diffraction becomes more significant when the size of the gap is comparable to the wavelength.
Diffraction Grating
Here n is the order of the maximum, λ is wavelength, d is the grating spacing and θ is the diffraction angle.
WORKED EXAMPLE
A diffraction grating has 600 lines per millimetre. Light of wavelength 500 nm produces a first-order maximum. Find the diffraction angle.
Number of lines per metre = 600 × 1000 = 6.00 × 10⁵ m⁻¹
d = 1 / (6.00 × 10⁵)
d = 1.67 × 10⁻⁶ m
sin θ = λ / d
sin θ = (5.00 × 10⁻⁷) / (1.67 × 10⁻⁶) ≈ 0.300
θ ≈ 17.5°
CORE PRACTICAL 05
Finding the Wavelength of Light
A diffraction grating can be used to determine the wavelength of monochromatic light such as laser light.
- Place the diffraction grating perpendicular to the incident beam.
- Observe the central maximum and higher-order maxima.
- Measure the angle θ for a known diffraction order.
- Use the number of lines per unit length to calculate d.
- Use nλ = d sin θ to determine λ.
- Repeat measurements for greater reliability.
TOPIC 09
Wave Nature of Electrons
Electrons can behave as waves as well as particles. Electron diffraction experiments provide evidence for the wave nature of matter.
de Broglie Wavelength
Since momentum p = mv for a non-relativistic particle:
The wave nature becomes more significant for particles with small momentum.
TOPIC 10
Pulse-Echo Techniques
A pulse-echo system sends a short pulse of radiation or ultrasound towards an object and detects the reflected echo.
Finding Distance
The factor of 2 is required because the pulse travels to the object and then back to the detector.
Resolution
The amount of information obtained can be limited by the wavelength of the radiation and the duration of the pulse. Shorter wavelengths and shorter pulses can generally provide better resolution.
TOPIC 11
Wave-Particle Duality of Light
Electromagnetic radiation can be described using both a wave model and a photon model.
Wave Model
The wave model explains phenomena such as interference, diffraction, refraction and polarisation.
Photon Model
The photon model treats electromagnetic radiation as packets of energy called photons.
Higher-frequency photons have greater energy.
TOPIC 12
The Photoelectric Effect
The photoelectric effect is the emission of electrons from a metal surface when electromagnetic radiation of sufficiently high frequency is incident on it.
Threshold Frequency
The threshold frequency is the minimum frequency required for photoemission from a particular metal.
Work Function
The work function is the minimum energy required to remove an electron from the surface of a metal.
Electronvolt
The electronvolt is a convenient unit for small amounts of energy associated with particles.
The instantaneous emission of electrons above a threshold frequency and the dependence of maximum kinetic energy on frequency provide evidence for the particle nature of electromagnetic radiation.
WORKED EXAMPLE
Light of frequency 8.0 × 10¹⁴ Hz falls on a metal with a work function of 2.0 eV. Find the maximum kinetic energy of an emitted electron.
Photon energy: E = hf
E = (6.63 × 10⁻³⁴)(8.0 × 10¹⁴)
E ≈ 5.30 × 10⁻¹⁹ J
Work function: φ = 2.0 × 1.60 × 10⁻¹⁹ = 3.20 × 10⁻¹⁹ J
Kinetic energy = 5.30 × 10⁻¹⁹ − 3.20 × 10⁻¹⁹
KEmax ≈ 2.10 × 10⁻¹⁹ J
TOPIC 13
Atomic Line Spectra
Electrons in atoms can occupy only certain discrete energy levels. When an electron moves between energy levels, electromagnetic radiation can be emitted or absorbed.
Emission
When an electron moves from a higher energy level to a lower energy level, a photon is emitted.
Absorption
When an electron absorbs a photon with the correct energy, it can move to a higher energy level.
Therefore:
The discrete nature of atomic energy levels produces discrete spectral lines rather than a continuous spectrum.
TOPIC 14
Electric Current, Charge and Potential Difference
Electric Current
Electric current is the rate of flow of electric charge.
Potential Difference
Potential difference is the energy transferred per unit charge when charge moves between two points.
Resistance
Ohm's law is the special case where current is directly proportional to potential difference at constant temperature.
TOPIC 15
Charge and Energy Conservation in Circuits
Current at a Junction
Charge cannot accumulate indefinitely at a junction. Therefore the total current entering a junction equals the total current leaving it.
Potential Differences
Energy is conserved around a complete circuit. The total potential difference supplied is equal to the total potential difference across the components.
TOPIC 16
Resistors in Series and Parallel
Series
In series, the same current passes through each resistor.
Parallel
In parallel, the potential difference across each branch is the same.
WORKED EXAMPLE
Two resistors of 6 Ω and 3 Ω are connected in parallel. Find the combined resistance.
1/R = 1/6 + 1/3
1/R = 1/6 + 2/6 = 3/6 = 1/2
R = 2 Ω
TOPIC 17
Electrical Power and Energy
Using V = IR, further useful forms are:
WORKED EXAMPLE
A 12 V supply drives a current of 2.5 A through a device. Calculate its power.
P = VI
P = 12 × 2.5
P = 30 W
TOPIC 18
Current-Potential Difference Characteristics
Ohmic Conductor
At constant temperature, an ohmic conductor has current directly proportional to potential difference. Its I-V graph is a straight line through the origin.
Filament Bulb
As current increases, the filament becomes hotter. Its resistance increases, so the I-V relationship is non-linear.
Thermistor
A negative temperature coefficient thermistor has decreasing resistance as temperature increases.
Diode
A diode allows current to flow much more easily in one direction than the other and therefore has a strongly non-linear I-V characteristic.
TOPIC 19
Resistivity
Resistivity is a property of a material that describes how strongly it opposes the flow of electric current.
Therefore resistance increases with length and decreases with cross-sectional area, for a given material and temperature.
WORKED EXAMPLE
A wire has resistivity 1.7 × 10⁻⁸ Ω m, length 2.0 m and cross-sectional area 1.0 × 10⁻⁶ m². Find its resistance.
R = ρl / A
R = (1.7 × 10⁻⁸ × 2.0) / (1.0 × 10⁻⁶)
R = 0.034 Ω
CORE PRACTICAL 06
Determining Electrical Resistivity
- Measure the length of the wire accurately.
- Measure the diameter at several points using a micrometer.
- Calculate the cross-sectional area.
- Connect the wire in a suitable circuit.
- Measure potential difference and current.
- Calculate resistance using R = V/I.
- Repeat for different lengths.
- Use R = ρl/A to determine the resistivity.
Multiple diameter measurements are useful because the cross-sectional area depends on the square of the radius.
TOPIC 20
Current and Charge Carriers
In a metal, current is produced by the movement of conduction electrons through the material.
Here n is the number density of charge carriers, q is the charge of each carrier, v is the drift velocity and A is the cross-sectional area.
The drift speed of electrons in a metal is much slower than the speed at which electrical effects propagate through a circuit.
TOPIC 21
Potential Along a Uniform Wire
For a uniform current-carrying wire, the potential changes steadily with distance along the wire when its resistance per unit length is constant.
Equal lengths of a uniform wire have equal resistance and therefore equal potential drops when the same current flows through them.
TOPIC 22
Potential Dividers
A potential divider uses resistors in series to produce a chosen fraction of the supply potential difference.
The resistor connected to the output determines the fraction of the supply voltage appearing across it.
Variable Resistance
A variable resistor, thermistor or light-dependent resistor can be used to make the output voltage depend on an external condition.
Thermistor
For a negative temperature coefficient thermistor, resistance decreases as temperature increases.
LDR
The resistance of a light-dependent resistor decreases as light intensity increases.
WORKED EXAMPLE
A 12 V supply is connected across a 2 kΩ resistor and a 4 kΩ resistor in series. Find the potential difference across the 4 kΩ resistor.
Vout = 12 × 4/(2 + 4)
Vout = 12 × 4/6
Vout = 8.0 V
TOPIC 23
e.m.f. and Internal Resistance
Electromotive Force
The electromotive force of a source is the energy supplied by the source per unit charge passing through it.
Internal Resistance
Real cells have internal resistance. Some of the energy supplied by the cell is dissipated inside the cell itself.
Equivalently, the terminal potential difference is:
Here ε is the e.m.f., I is current, R is external resistance and r is internal resistance.
e.m.f. refers to the energy supplied per unit charge by the source, while terminal p.d. is the energy transferred per unit charge in the external circuit.
CORE PRACTICAL 07
Determining e.m.f. and Internal Resistance
- Connect the cell to a variable external resistance.
- Measure the current I and terminal potential difference V.
- Repeat for several different values of external resistance.
- Plot V against I.
- Use the relationship V = ε − Ir.
- The y-intercept gives ε.
- The magnitude of the gradient gives r.
TOPIC 24
Resistance and Temperature
Metallic Conductors
As temperature increases, the lattice ions vibrate more strongly. Electrons experience more frequent collisions, so the resistance of a metal generally increases.
Negative Temperature Coefficient Thermistor
In an NTC thermistor, resistance decreases as temperature increases. This can be explained by an increase in the number of charge carriers available for conduction.
TOPIC 25
Light-Dependent Resistors
An LDR changes its resistance depending on the intensity of incident light.
Increasing illumination increases the number of charge carriers available for conduction, reducing resistance.
LDRs can be incorporated into potential-divider circuits to produce automatic responses to changes in light level.
PRACTICE
Unit 2 Mixed Practice
Show answer
v = fλ = 500 × 0.68
v = 340 m s⁻¹
Show answer
A stationary wave is formed by superposition of two identical waves travelling in opposite directions. Destructive interference produces nodes and constructive interference produces antinodes.
Show answer
The wave must travel from a medium of higher refractive index to a medium of lower refractive index, and the angle of incidence must exceed the critical angle.
Show answer
500 lines/mm = 5.00 × 10⁵ lines/m.
d = 1/(5.00 × 10⁵)
d = 2.00 × 10⁻⁶ m
Show answer
E = hf
E = (6.63 × 10⁻³⁴)(6.0 × 10¹⁴)
E ≈ 3.98 × 10⁻¹⁹ J
Show answer
Q = It = 3 × 20
Q = 60 C
Show answer
R = 4 + 8
R = 12 Ω
Show answer
P = I²R
P = 2² × 6
P = 24 W
Show answer
Increasing temperature increases the number of charge carriers available for conduction, causing the resistance to decrease.
Show answer
V = ε − Ir
V = 1.50 − (2.0 × 0.20)
V = 1.10 V
QUICK REVIEW