Polarisation is a fundamental concept in physics, particularly within the study of waves. Malus's Law, represented by the equation $I = I₀ \cos²θ$, plays a crucial role in understanding how the intensity of a plane-polarised wave changes when it passes through a polarising filter. This article delves into the intricacies of Malus's Law, providing AS & A Level Physics students with a comprehensive guide to its application and significance in the broader context of wave dynamics.

Key Concepts

1. Polarisation of Waves

Polarisation refers to the orientation of the oscillations of a transverse wave, such as light, in a particular direction. While natural light consists of waves vibrating in multiple directions, polarised light vibrates predominantly in one plane. Understanding polarisation is essential in various applications, from reducing glare in sunglasses to enhancing the visibility of displays.

2. Introduction to Malus's Law

Malus's Law quantifies the change in intensity of a plane-polarised light wave as it passes through a second polarising filter. Named after Étienne-Louis Malus, who discovered it in 1809, the law is mathematically expressed as: $$I = I₀ \cos²θ$$ where:

I is the transmitted intensity after passing through the polariser.
I₀ is the initial intensity of the polarised wave before passing through the polariser.
θ is the angle between the transmission axis of the polariser and the direction of polarisation of the incoming light.

3. Understanding the Components of Malus's Law

To fully grasp Malus's Law, it is essential to dissect its components:

Initial Intensity ($I₀$): This represents the intensity of the incoming plane-polarised light before it interacts with the polariser. The intensity is proportional to the square of the amplitude of the electric field of the wave.
Transmission Axis: Every polariser has a transmission axis, a direction in which it allows the passage of the electric field vector of the light wave. The alignment between the incoming light's polarisation direction and the polariser's transmission axis determines the resulting intensity.
Angle of Interaction ($θ$): The angle between the incoming light's polarisation direction and the polariser's transmission axis critically affects the transmitted intensity. This angular dependency is at the heart of Malus's Law.

4. Derivation of Malus's Law

Malus's Law can be derived from the principles of vector projection and the nature of electromagnetic waves. Consider a plane-polarised light wave incident on a polariser:

Represent the electric field vector of the incoming light as $E₀$.
The component of $E₀$ along the transmission axis of the polariser is $E₀ \cosθ$.
The intensity of light is proportional to the square of the electric field amplitude. Therefore, the transmitted intensity $I$ is: $$I = I₀ \cos²θ$$

This derivation assumes an ideal polariser with no absorption losses except those dictated by the alignment angle.

5. Applications of Malus's Law

Malus's Law is not just a theoretical construct but has practical applications in various fields:

Optical Instrumentation: Polarises filters in cameras and microscopes to enhance image contrast and reduce glare.
Telecommunications: Used in the design of optical fibers and understanding signal propagation.
Liquid Crystal Displays (LCDs): Helps in controlling light passage through layers to create images on screens.
Stress Analysis: Polarised light techniques help in analyzing stresses in transparent materials.

6. Experimental Verification of Malus's Law

To validate Malus's Law experimentally, one can perform a series of measurements using a polariser and a light source:

Initialize a plane-polarised light source with known intensity $I₀$.
Place a polarising filter in the path of the light.
Vary the angle $θ$ between the polariser's transmission axis and the incoming polarisation direction.
Measure the transmitted intensity $I$ for each angle.
Plot $I$ against $\cos²θ$ to verify the linear relationship as predicted by Malus's Law.

Such experiments typically confirm the quadratic dependence of intensity on the cosine of the angle, thereby validating Malus's theoretical predictions.

7. Polarisation by Reflection and Its Relation to Malus's Law

When unpolarised light reflects off a surface, the reflected light can become partially polarised. The degree of polarisation depends on the angle of incidence following Brewster's angle. Malus's Law applies when this reflected light passes through an additional polariser, allowing for the calculation of the resulting intensity based on the alignment angle.

8. Mathematical Implications of Malus's Law

The quadratic dependence in Malus's Law implies that the intensity drops to zero as the angle approaches 90 degrees, meaning that no light passes through the polariser when the transmission axis is perpendicular to the light's polarisation direction. Mathematically, this relationship emphasizes the sinusoidal nature of wave interactions in polarised systems.

9. Practical Considerations and Limitations

While Malus's Law provides a straightforward relationship under ideal conditions, real-world applications may encounter deviations due to:

Non-ideal Polarisation: Light may not be perfectly plane-polarised, introducing errors in intensity calculations.
Polariser Imperfections: Real polarisers absorb more light than predicted, affecting the accuracy of $I₀$.
Angular Precision: Small errors in measuring the angle $θ$ can lead to significant intensity discrepancies.

Acknowledging these factors is essential for accurate application and interpretation of Malus's Law in experimental setups.

Advanced Concepts

1. Derivation and Vector Analysis of Malus's Law

Delving deeper into the derivation, consider the electric field vector of a plane-polarised wave represented in a coordinate system aligned with the polariser's transmission axis: $$\vec{E} = E₀ \cosθ \hat{x}$$ where $\hat{x}$ is the unit vector along the transmission axis. The intensity $I$ is proportional to the square of the electric field amplitude: $$I = \frac{1}{2} \epsilon₀ c E²$$ Substituting the projected electric field: $$I = \frac{1}{2} \epsilon₀ c (E₀ \cosθ)^2 = I₀ \cos²θ$$ This vector approach underscores the role of projection in determining transmitted intensity, reinforcing the fundamental principles of vector components in wave interactions.

2. Polarisation States and Malus's Law

While Malus's Law primarily addresses plane-polarised light, understanding its extension to different polarisation states enriches its applicability:

Unpolarised Light: Can be viewed as a combination of plane-polarised waves in all possible directions. Passing unpolarised light through a polariser reduces its intensity by half since only the component aligned with the transmission axis passes through.
Partially Polarised Light: Exhibits varying degrees of polarisation. Malus's Law can be adapted by considering the degree of polarisation in intensity calculations.

These extensions demonstrate the versatility of Malus's Law in handling diverse light states beyond perfect plane polarisation.

3. Quantum Mechanical Perspective

From a quantum viewpoint, light consists of photons, each carrying a quantum of electromagnetic energy. Polarisation corresponds to the photon's spin orientation. Malus's Law can be reinterpreted in terms of the probability of a photon's polarisation state aligning with the polariser's axis, affecting the likelihood of transmission and thus the observed intensity.

4. Interference and Malus's Law

When multiple polarised waves interfere, Malus's Law remains applicable in determining the resultant intensity. Constructive and destructive interference patterns can be analyzed by considering the relative polarisation angles and employing Malus's equation to calculate individual intensities before superimposition.

5. Malus's Law in Optical Coherence

Optical coherence deals with the phase relationship between waves. In coherent light sources, maintaining a fixed phase relationship while applying Malus's Law allows for precise control over intensity modulation. This principle is harnessed in devices like interferometers and quantum optics experiments.

6. Advanced Applications: Optical Sensors and Polarimeters

Malus's Law is integral to the functioning of optical sensors and polarimeters, instruments designed to measure polarisation states and intensities. By adjusting the polariser angles and applying Malus's equation, these devices can accurately determine the degree and angle of polarisation, essential in fields like astronomy, material science, and telecommunications.

7. Nonlinear Optics and Malus's Law

In nonlinear optical materials, the interaction between light and matter leads to phenomena like frequency doubling and self-focusing. Malus's Law, when applied to such environments, requires consideration of intensity-dependent polarisation effects, introducing complexities beyond the linear dependence described in the classical law.

8. Polarisation Entanglement and Quantum Information

In quantum information science, entangled photons with correlated polarisation states are fundamental. Malus's Law aids in predicting measurement outcomes when entangled particles pass through polarising filters, influencing protocols in quantum communication and cryptography.

9. Environmental Factors Affecting Malus's Law

Real-world factors such as temperature fluctuations, material stress on polarisers, and wavelength dependence can influence the applicability of Malus's Law. Understanding these environmental impacts is crucial for precision experiments and the development of robust optical systems.

10. Integration with Other Optical Laws

Malus's Law often interacts with other optical principles, such as Fresnel's equations and Snell's law, especially in complex systems involving refraction and reflection. Integrating these laws allows for comprehensive analysis of light behaviour in multifaceted optical setups.

Comparison Table

Aspect	Malus's Law	Beer's Law
Definition	Describes the intensity of polarised light after passing through a polariser based on angle.	Describes the attenuation of light as it passes through an absorbing medium.
Equation	$I = I₀ \cos²θ$	$I = I₀ e^{-αx}$
Primary Application	Polarisation analysis and intensity calculation.	Determining concentration of absorbing species in a medium.
Dependence	Angle between polarisation direction and polariser axis.	Concentration of absorber and path length.
Relevant Fields	Optics, Photonics, Telecommunications.	Chemistry, Environmental Science, Spectroscopy.
Nature of Law	Geometric dependence based on wave orientation.	Exponential decay based on material properties.

Summary and Key Takeaways

Malus's Law quantitatively describes the intensity reduction of plane-polarised light through a polariser as $I = I₀ \cos²θ$.
Understanding the angle between polarisation direction and polariser axis is crucial for intensity calculations.
The law has diverse applications in optical instrumentation, telecommunications, and display technologies.
Advanced studies reveal connections to quantum mechanics, nonlinear optics, and quantum information science.
Real-world factors and integration with other optical laws are essential for practical applications.

Examiner Tip

Tips

To master Malus's Law, remember the mnemonic "I Cos Squared" to recall the equation $I = I₀ \cos²θ$. When studying, always draw a clear diagram showing the polariser axes and the angle $θ$ to visualize the problem. Practice by solving diverse problems involving multiple polarising filters to strengthen your understanding. During exams, double-check whether the light is initially polarised and adjust your application of Malus's Law accordingly. Lastly, relate real-world applications, like sunglasses or LCD screens, to theoretical concepts to enhance retention and comprehension.

Did You Know

Did you know that Malus's Law not only applies to visible light but also to other electromagnetic waves like X-rays and microwaves? Additionally, the principle behind Malus's Law is utilized in 3D movie technology, where two polarised images are projected simultaneously, allowing each eye to perceive a different image and creating a depth illusion. Another fascinating fact is that Malus's original experiments with polarisation were conducted using sunlight, which he demonstrated to be naturally partially polarised.

Common Mistakes

Students often confuse the angle $θ$ in Malus's Law with its complementary angle, leading to incorrect intensity calculations. For example, using $θ = 90° - θ'$ instead of the actual angle between the polariser axes results in $I \neq I₀ \cos²θ'$. Another common error is neglecting that Malus's Law applies strictly to plane-polarised light; applying it to unpolarised light without proper adjustments can lead to inaccuracies. Additionally, forgetting to square the cosine term when calculating the transmitted intensity often leads to underestimation of the light intensity.

FAQ

What is Malus's Law?

Malus's Law describes how the intensity of a plane-polarised light wave changes as it passes through a polarising filter, following the equation $I = I₀ \cos²θ$.

Who discovered Malus's Law?

Étienne-Louis Malus, a French physicist, discovered Malus's Law in 1809 through his experiments with polarised light.

How does the angle $θ$ affect the intensity of transmitted light?

The intensity of transmitted light decreases with the square of the cosine of the angle $θ$ between the light's polarisation direction and the polariser's transmission axis. At $θ = 90°$, no light is transmitted.

Can Malus's Law be applied to unpolarised light?

Yes, but with modification. When unpolarised light passes through a polariser, its intensity is reduced by half, and subsequent applications of Malus's Law can be used if additional polarising filters are involved.

What are practical applications of Malus's Law?

Malus's Law is applied in areas such as optical instruments, telecommunications, LCD technology, and stress analysis in materials, among others.

What assumptions are made in Malus's Law?

Malus's Law assumes ideal polarising filters with no absorption losses other than those caused by the alignment angle, and that the incoming light is perfectly plane-polarised.

1. Capacitance

1.1 Energy Stored in a Capacitor

1.1.1 Determine electric potential energy stored in a capacitor from the area under the potential–charge g

1.1.2 Recall and use W = ½QV = ½CV²

1.2 Discharging a Capacitor

1.2.1 Analyze graphs of potential difference, charge, and current during capacitor discharge

1.2.2 Recall and use τ = RC for the time constant during discharge

1.2.3 Use equations of the form x = x₀e^(-t / RC) for current, charge, or potential during discharge

1.3 Capacitors and Capacitance

1.3.1 Define capacitance for isolated spherical conductors and parallel plate capacitors

1.3.2 Recall and use C = Q / V

1.3.3 Derive formulae for combined capacitance of capacitors in series and parallel

1.3.4 Use capacitance formulae for capacitors in series and parallel

2. Nuclear Physics

2.1 Mass Defect and Nuclear Binding Energy

2.1.1 Sketch the variation of binding energy per nucleon with nucleon number

2.1.2 Explain nuclear fusion and nuclear fission

2.1.3 Calculate the energy released in nuclear reactions using E = c²∆m

2.1.4 Understand the equivalence between energy and mass (E = mc²)

2.1.5 Define and use the terms mass defect and binding energy

2.2 Radioactive Decay

2.2.1 Understand that fluctuations in count rate provide evidence for random decay

2.2.2 Understand that radioactive decay is spontaneous and random

2.2.3 Define activity and decay constant, and recall A = λN

2.2.4 Define half-life

2.2.5 Use λ = 0.693 / t₁/₂ for the half-life relationship

2.2.6 Understand exponential decay and use the relationship x = x₀e^(-λt)

3. Kinematics

3.1 Equations of Motion

3.1.1 Define and use distance, displacement, speed, velocity, acceleration

3.1.2 Use graphical methods to represent motion

3.1.3 Determine displacement from velocity–time graph

3.1.4 Determine velocity using gradient of displacement–time graph

3.1.5 Determine acceleration using gradient of velocity–time graph

3.1.6 Derive equations for uniformly accelerated motion

3.1.7 Solve problems of uniformly accelerated motion

3.1.8 Describe experiment to determine acceleration due to gravity

3.1.9 Explain motion with uniform velocity and perpendicular acceleration

4. Deformation of Solids

4.1 Stress and Strain

4.1.1 Understand and use the terms load, extension, compression, and limit of proportionality

4.1.2 Recall and use Hooke’s law

4.1.3 Recall and use the formula for spring constant k = F / x

4.1.4 Define and use terms stress, strain, and Young's modulus

4.1.5 Describe an experiment to determine Young’s modulus

4.1.6 Understand deformation caused by tensile or compressive forces

4.2 Elastic and Plastic Behaviour

4.2.1 Understand elastic and plastic deformation and elastic limit

4.2.2 Understand area under the force–extension graph as work done

4.2.3 Calculate elastic potential energy using EP = ½kx²

5. Gravitational Fields

5.1 Gravitational Field of a Point Mass

5.1.1 Understand that g is approximately constant for small height changes near Earth's surface

5.1.2 Recall and use g = GM / r²

5.1.3 Derive and use g = GM / r² for gravitational field strength

5.2 Gravitational Potential

5.2.1 Define gravitational potential as work done per unit mass in bringing a test mass from infinity

5.2.2 Use ϕ = -GM / r for gravitational potential due to a point mass

5.2.3 Use EP = -GMm / r for gravitational potential energy between two point masses

5.3 Gravitational Field

5.3.1 Define gravitational field as force per unit mass

5.3.2 Represent a gravitational field using field lines

5.4 Gravitational Force Between Point Masses

5.4.1 For point outside uniform sphere, treat mass as a point mass at its center

5.4.2 Analyze circular orbits in gravitational fields

5.4.3 Recall and use Newton's law of gravitation F = Gm₁m₂ / r²

6. Electricity

6.1 Resistance and Resistivity

6.1.1 Understand that the resistance of a thermistor decreases as temperature increases

6.1.2 Define resistance and recall and use V = IR

6.1.3 Sketch I–V characteristics for metallic conductor, semiconductor diode, and filament lamp

6.1.4 Explain that the resistance of a filament lamp increases as current increases

6.1.5 State Ohm’s law and recall and use R = ρL / A

6.1.6 Understand that the resistance of an LDR decreases as light intensity increases

6.2 Electric Current

6.2.1 Understand that an electric current is a flow of charge carriers

6.2.2 Understand that the charge on charge carriers is quantised

6.2.3 Recall and use Q = It

6.2.4 Use I = Anvq for current-carrying conductors

6.3 Potential Difference and Power

6.3.1 Define potential difference as energy transferred per unit charge

6.3.2 Recall and use V = W / Q

6.3.3 Recall and use P = VI, P = I²R, P = V² / R

7. D.C. Circuits

7.1 Practical Circuits

7.1.1 Recall and use the circuit symbols in the syllabus

7.1.2 Draw and interpret circuit diagrams using circuit symbols

7.1.3 Define electromotive force (e.m.f.) of a source as energy transferred per unit charge

7.1.4 Distinguish between e.m.f. and potential difference (p.d.)

7.1.5 Understand the effects of internal resistance on terminal potential difference

7.2 Kirchhoff’s Laws

7.2.1 Recall Kirchhoff’s first law: conservation of charge

7.2.2 Recall Kirchhoff’s second law: conservation of energy

7.2.3 Derive the formula for combined resistance of resistors in series using Kirchhoff’s laws

7.2.4 Use the formula for combined resistance of resistors in series

7.2.5 Derive the formula for combined resistance of resistors in parallel using Kirchhoff’s laws

7.2.6 Use the formula for combined resistance of resistors in parallel

7.2.7 Use Kirchhoff’s laws to solve simple circuit problems

7.3 Potential Dividers

7.3.1 Understand the principle of a potential divider circuit

7.3.2 Recall and use the principle of the potentiometer for comparing potential differences

7.3.3 Understand the use of a galvanometer in null methods

7.3.4 Explain the use of thermistors and LDRs in potential dividers to provide a potential dependent on te

8. Thermal Equilibrium

8.1 Thermal Energy Transfer

8.1.1 Understand that thermal energy transfers from higher to lower temperature

8.1.2 Understand that regions of equal temperature are in thermal equilibrium

9. Temperature Scales

9.1 Temperature Measurement

9.1.1 Understand physical properties that vary with temperature (e.g., density, gas volume, resistance)

9.1.2 Convert temperatures between Kelvin and Celsius, and recall T(K) = θ(°C) + 273.15

10. Magnetic Fields

10.1 Concept of a Magnetic Field

10.1.1 Understand that a magnetic field is a field of force produced by moving charges or permanent magnets

10.1.2 Represent a magnetic field using field lines

10.2 Force on a Current-Carrying Conductor

10.2.1 Understand that a force acts on a current-carrying conductor in a magnetic field

10.2.2 Recall and use F = BIL sin θ for the force on a current-carrying conductor in a magnetic field

10.2.3 Define magnetic flux density as the force per unit current per unit length on a wire at right angles

10.3 Force on a Moving Charge

10.3.1 Determine the direction of the force on a charge moving in a magnetic field

10.3.2 Recall and use F = BQv sin θ for the force on a moving charge in a magnetic field

10.3.3 Understand the origin of the Hall voltage and use the expression VH = BI / (ntq) for the Hall voltag

10.3.4 Understand the use of a Hall probe to measure magnetic flux density

10.4 Magnetic Fields Due to Currents

10.4.1 Sketch magnetic field patterns due to currents in a straight wire, flat coil, and solenoid

10.4.2 Understand that magnetic field in a solenoid is increased by a ferrous core

10.4.3 Explain the origin of forces between current-carrying conductors and determine the direction of the

10.5 Electromagnetic Induction

10.5.1 Define magnetic flux as the product of magnetic flux density and area perpendicular to flux directio

10.5.2 Recall and use Φ = BA for magnetic flux

10.5.3 Understand and use the concept of magnetic flux linkage

10.5.4 Explain experiments that show changing magnetic flux induces an e.m.f.

10.5.5 Recall and use Faraday’s and Lenz’s laws of electromagnetic induction

11. Specific Heat Capacity and Latent Heat

11.1 Specific Heat Capacity

11.1.1 Define and use specific heat capacity

11.2 Latent Heat

11.2.1 Define and use specific latent heat; distinguish between fusion and vaporisation

12. Dynamics

12.1 Momentum and Newton’s Laws of Motion

12.1.1 Mass resists change in motion

12.1.2 Recall F = ma and solve related problems

12.1.3 Define and use linear momentum

12.1.4 Force as rate of change of momentum

12.1.5 State and apply Newton’s laws

12.1.6 Describe weight as effect of gravitational field

12.1.7 Weight of an object is mass × gravitational acceleration

12.2 Non-uniform Motion

12.2.1 Understand frictional and drag forces

12.2.2 Describe motion in uniform gravitational field with air resistance

12.2.3 Objects moving against resistive force may reach terminal velocity

12.3 Linear Momentum and its Conservation

12.3.1 State conservation of momentum

12.3.2 Apply conservation of momentum to solve problems

12.3.3 For elastic collision, kinetic energy and relative speed are conserved

12.3.4 Momentum is conserved, but some kinetic energy may change

13. Medical Physics

13.1 Production and Use of Ultrasound

13.1.1 Understand that a piezo-electric crystal changes shape when a p.d. is applied and generates an e.m.f

13.1.2 Understand how ultrasound waves are generated and detected by a piezoelectric transducer

13.1.3 Understand how ultrasound reflection at boundaries provides diagnostic information about internal st

13.1.4 Define specific acoustic impedance as Z = ρc

13.1.5 Use the equation IR / I₀ = (Z₁ – Z₂)² / (Z₁ + Z₂)² for the intensity reflection coefficient

13.2 Production and Use of X-rays

13.2.1 Explain that X-rays are produced by electron bombardment of a metal target and calculate the minimum

13.2.2 Understand the use of X-rays in imaging internal body structures and the term 'contrast' in X-ray im

13.2.3 Recall and use I = I₀e^(-μx) for the attenuation of X-rays in matter

13.2.4 Understand how computed tomography (CT) produces 3D images by combining multiple 2D X-ray images

13.3 PET Scanning

13.3.1 Understand that a tracer contains radioactive nuclei and is introduced into the body for imaging

13.3.2 Understand that β+ decay is used in positron emission tomography (PET)

13.3.3 Understand that annihilation occurs when a particle interacts with its antiparticle, conserving mass

13.3.4 Explain how gamma-ray photons from annihilation are detected to create an image of tracer concentrat

14. Waves

14.1 Progressive Waves

14.1.1 Describe wave motion in ropes, springs, and ripple tanks

14.1.2 Understand and use the terms displacement, amplitude, phase difference, period, frequency, wavelengt

14.1.3 Use time-base and y-gain on a cathode-ray oscilloscope (CRO) to determine frequency and amplitude

14.1.4 Derive and use v = f λ for the wave equation

14.1.5 Recall and use v = f λ

14.1.6 Understand energy transfer by a progressive wave

14.1.7 Recall and use intensity = power / area, intensity ∝ (amplitude)²

14.2 Transverse and Longitudinal Waves

14.2.1 Compare transverse and longitudinal waves

14.2.2 Analyze and interpret graphical representations of transverse and longitudinal waves

14.3 Doppler Effect for Sound Waves

14.3.1 Understand the change in frequency when a sound source moves relative to a stationary observer

14.3.2 Use the expression f₀ = fₛ(v / (v ± vₛ)) for the observed frequency

14.4 Electromagnetic Spectrum

14.4.1 State all electromagnetic waves are transverse waves traveling at the same speed in free space

14.4.2 Recall the range of wavelengths from radio waves to γ-rays

14.4.3 Recall that wavelengths 400–700 nm are visible to the human eye

14.5 Polarisation

14.5.1 Understand that polarisation is a phenomenon associated with transverse waves

14.5.2 Recall and use Malus's law (I = I₀ cos²θ) to calculate intensity of a plane-polarised wave after pas

15. The Mole

15.1 The Mole

15.1.1 Understand that amount of substance is an SI base quantity with the unit mol

15.1.2 Use molar quantities where one mole contains Avogadro's constant of particles

16. Equation of State

16.1 Ideal Gas Law

16.1.1 Understand that a gas obeying pV ∝ T is an ideal gas

16.1.2 Recall and use the equation pV = nRT for an ideal gas

16.1.3 Recall pV = NkT, where N = number of molecules and k is Boltzmann constant

16.1.4 Use the relationship k = R / NA

17. Kinetic Theory of Gases

17.1 Kinetic Theory

17.1.1 State the basic assumptions of the kinetic theory of gases

17.1.2 Explain pressure exerted by a gas due to molecular movement

17.1.3 Derive and use pV = (3/2)NmkT for pressure of a gas

17.1.4 Compare pV = (3/2)NmkT with pV = NkT to deduce average kinetic energy of a molecule

18. Particle Physics

18.1 Atoms, Nuclei and Radiation

18.1.1 Infer the existence and small size of the nucleus from α-particle scattering

18.1.2 Describe the nuclear atom model: protons, neutrons, and electrons

18.1.3 Distinguish between nucleon number and proton number

18.1.4 Understand isotopes as forms of the same element with different neutrons

18.1.5 Use the notation AZX for nuclides representation

18.1.6 Understand conservation of nucleon number and charge in nuclear processes

18.1.7 Describe the composition, mass, and charge of α-, β-, and γ-radiations

18.1.8 Understand antiparticles and positrons as antiparticles of electrons

18.1.9 State that (electron) antineutrinos are produced during β- decay and neutrinos during β+ decay

18.1.10 Represent α- and β-decay using radioactive decay equations

18.1.11 Use unified atomic mass unit (u) for mass

18.2 Fundamental Particles

18.2.1 Understand quarks as fundamental particles with six flavours

18.2.2 Understand hadrons as baryons (three quarks) or mesons (one quark and one antiquark)

18.2.3 Describe changes in quark composition during β- and β+ decay

18.2.4 Describe protons and neutrons as composed of quarks

18.2.5 Recall and use the charge of each quark and its respective antiquark

18.2.6 Recall electrons and neutrinos as fundamental particles (leptons)

19. Internal Energy

19.1 Internal Energy

19.1.1 Understand that internal energy is the sum of kinetic and potential energies of molecules in a syste

19.1.2 Relate rise in temperature to an increase in internal energy

20. Forces, Density and Pressure

20.1 Turning Effects of Forces

20.1.1 Weight acts at a single point: centre of gravity

20.1.2 Define and apply the moment of a force

20.1.3 Understand the concept of a couple

20.1.4 Define and apply torque of a couple

20.2 Equilibrium of Forces

20.2.1 State and apply the principle of moments

20.2.2 System in equilibrium has no resultant force or torque

20.2.3 Use a vector triangle to represent coplanar forces in equilibrium

20.3 Density and Pressure

20.3.1 Define and use density

20.3.2 Define and use pressure

20.3.3 Derive equation for hydrostatic pressure ∆p = ρg∆h

20.3.4 Use the equation ∆p = ρg∆h

20.3.5 Understand upthrust as the effect of hydrostatic pressure

20.3.6 Calculate upthrust using F = ρgV (Archimedes' principle)

21. Astronomy and Cosmology

21.1 Standard Candles

21.1.1 Understand luminosity as the total power radiated by a star

21.1.2 Recall and use the inverse square law F = L / (4πd²) for radiant flux intensity

21.1.3 Understand that an object with known luminosity is a standard candle

21.1.4 Use standard candles to determine distances to galaxies

21.2 Stellar Radii

21.2.1 Recall and use Wien’s displacement law λmax ∝ 1 / T to estimate the surface temperature of a star

21.2.2 Use Stefan–Boltzmann law L = 4πσr²T⁴

21.2.3 Use Wien’s law and Stefan–Boltzmann law to estimate the radius of a star

21.3 Hubble’s Law and Big Bang Theory

21.3.1 Understand redshift and the increase in wavelength of light from distant objects

21.3.2 Use ∆λ / λ = ∆f / f = v / c for redshift

21.3.3 Explain why redshift suggests the Universe is expanding

21.3.4 Recall and use Hubble’s law v = H₀d to explain the Big Bang theory

22. First Law of Thermodynamics

22.1 First Law

22.1.1 Recall and use W = p∆V for work done when the volume of a gas changes at constant pressure

22.1.2 Understand the difference between work done by a gas and work done on a gas

22.1.3 Recall and use the first law of thermodynamics: ∆U = q + W

23. Alternating Currents

23.1 Characteristics of Alternating Currents

23.1.1 Understand and use terms period, frequency, and peak value for alternating current or voltage

23.1.2 Use x = x₀ sin(ωt) for sinusoidal alternating current or voltage

23.1.3 Recall that the mean power in a resistive load is half the maximum power for sinusoidal current

23.1.4 Distinguish between root-mean-square (r.m.s.) and peak values and recall I₀r.m.s = I₀ / √2 and V₀r.m

23.2 Rectification and Smoothing

23.2.1 Distinguish graphically between half-wave and full-wave rectification

23.2.2 Explain the use of a single diode for half-wave rectification

23.2.3 Explain the use of four diodes (bridge rectifier) for full-wave rectification

23.2.4 Analyze the effect of a capacitor in smoothing, including the effect of capacitance and load resista

24. Superposition

24.1 Stationary Waves

24.1.1 Explain and use the principle of superposition

24.1.2 Understand experiments demonstrating stationary waves using microwaves, stretched strings, and air c

24.1.3 Explain the formation of stationary waves using graphical methods and identify nodes and antinodes

24.1.4 Determine wavelength from the positions of nodes or antinodes

24.2 Diffraction

24.2.1 Explain diffraction and show understanding of related experiments

24.2.2 Understand the effect of gap width relative to wavelength in diffraction experiments

24.3 Interference

24.3.1 Understand the terms interference and coherence

24.3.2 Demonstrate two-source interference using water waves, sound, light, and microwaves

24.3.3 Understand the conditions required for two-source interference fringes

24.3.4 Recall and use λ = ax / D for double-slit interference with light

24.4 Diffraction Grating

24.4.1 Recall and use d sin θ = nλ for diffraction grating calculations

24.4.2 Describe the use of a diffraction grating to determine the wavelength of light

25. Simple Harmonic Oscillations

25.1 Simple Harmonic Motion

25.1.1 Understand and use terms displacement, amplitude, period, frequency, angular frequency, and phase di

25.1.2 Understand that simple harmonic motion occurs when acceleration is proportional to displacement from

25.1.3 Use a = -ω²x and recall x = x₀ sin(ωt) as the solution

25.1.4 Use v = v₀ cos(ωt) and v = ±ω√(x₀² - x²) for velocity

26. Energy in Simple Harmonic Motion

26.1 Energy in SHM

26.1.1 Describe the interchange between kinetic and potential energy during SHM

26.1.2 Recall and use E = ½mω²x₀² for the total energy of a system undergoing SHM

27. Quantum Physics

27.1 Energy and Momentum of a Photon

27.1.1 Understand that electromagnetic radiation has a particulate nature

27.1.2 Recall and use E = hf for the energy of a photon

27.1.3 Use the electronvolt (eV) as a unit of energy

27.1.4 Understand that a photon has momentum and use p = E / c for photon momentum

27.2 Photoelectric Effect

27.2.1 Understand that photoelectrons are emitted from a metal surface when illuminated by electromagnetic

27.2.2 Understand and use the terms threshold frequency and threshold wavelength

27.2.3 Explain photoelectric emission in terms of photon energy and work function

27.2.4 Recall and use hf = Φ + ½mv² for the maximum kinetic energy of photoelectrons

27.2.5 Explain why the maximum kinetic energy of photoelectrons is independent of intensity, while current

27.3 Wave-Particle Duality

27.3.1 Understand that the photoelectric effect shows the particulate nature of light, while diffraction sh

27.3.2 Describe and interpret electron diffraction as evidence for the wave nature of particles

27.3.3 Understand the de Broglie wavelength as the wavelength associated with a moving particle

27.3.4 Recall and use λ = h / p for de Broglie wavelength

27.4 Energy Levels in Atoms and Line Spectra

27.4.1 Understand discrete electron energy levels in atoms (e.g., atomic hydrogen)

27.4.2 Understand the appearance and formation of emission and absorption line spectra

27.4.3 Recall and use hf = E₁ – E₂ for the energy difference between electron energy levels

28. Damped and Forced Oscillations, Resonance

28.1 Damped Oscillations

28.1.1 Understand that damping occurs due to resistive forces

28.1.2 Understand the types of damping: light, critical, and heavy damping

28.2 Resonance

28.2.1 Understand that resonance occurs when an oscillating system is forced to oscillate at its natural fr

29. Work, Energy and Power

29.1 Energy Conservation

29.1.1 Understand the concept of work, and recall W = F × s

29.1.2 Apply the principle of conservation of energy

29.1.3 Understand efficiency as useful energy output / total energy input

29.1.4 Use efficiency concept to solve problems

29.1.5 Define power as work done per unit time

29.1.6 Solve problems using P = W / t

29.1.7 Derive P = Fv and solve problems

29.2 Gravitational Potential Energy and Kinetic Energy

29.2.1 Derive ∆EP = mg∆h for gravitational potential energy changes

29.2.2 Recall and use ∆EP = mg∆h for gravitational potential energy changes

29.2.3 Derive EK = ½mv² for kinetic energy

29.2.4 Recall and use EK = ½mv² for kinetic energy

30. Electric Fields

30.1 Electric Fields and Field Lines

30.1.1 Understand that an electric field is a field of force and define it as force per unit positive charg

30.1.2 Represent an electric field using field lines

30.2 Uniform Electric Fields

30.2.1 Recall and use E = ∆V / ∆d to calculate electric field strength between charged parallel plates

30.2.2 Describe the effect of a uniform electric field on the motion of charged particles

30.3 Electric Force Between Point Charges

30.3.1 Understand that, for a point outside a spherical conductor, the charge may be considered a point mas

30.3.2 Recall and use Coulomb’s law F = Q₁Q₂ / (4πε₀r²) for the force between two point charges

30.4 Electric Field of a Point Charge

30.4.1 Recall and use E = Q / (4πε₀r²) for the electric field strength due to a point charge

30.5 Electric Potential

30.5.1 Define electric potential as work done per unit positive charge in bringing a test charge from infin

30.5.2 Recall and use V = Q / (4πε₀r) for electric potential due to a point charge

30.5.3 Understand how electric potential leads to electric potential energy of two point charges and use EP

31. Physical Quantities and Units

31.1 Physical Quantities

31.1.1 Physical quantities consist of magnitude and unit

31.1.2 Estimate physical quantities from the syllabus

31.2 SI Units

31.2.1 SI base units: mass, length, time, current, temperature

31.2.2 Express derived units from SI base units

31.2.3 Check homogeneity of physical equations

31.2.4 Use prefixes (pico, nano, micro, kilo, etc.)

31.3 Errors and Uncertainties

31.3.1 Understand systematic and random errors

31.3.2 Assess uncertainty in derived quantities

31.4 uniform Electric Fields

31.4.1 Distinguish between precision and accuracy

31.5 Scalars and Vectors

31.5.1 Difference between scalar and vector quantities

31.5.2 Add and subtract coplanar vectors

31.5.3 Represent a vector as components

32. Motion in a Circle

32.1 Centripetal Acceleration

32.1.1 Understand centripetal acceleration and its role in circular motion

32.1.2 Recall and use a = rω² and a = v² / r

32.1.3 Recall and use F = mrω² and F = mv² / r

32.1.4 Understand force causing centripetal acceleration is always perpendicular to motion

32.2 Kinematics of Uniform Circular Motion

32.2.1 Recall and use ω = 2π / T and v = rω

32.2.2 Understand and use angular speed

32.2.3 Define and express angular displacement in radians

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Understanding Malus's Law and Its Application in Polarisation

Introduction