The equivalence between energy and mass, encapsulated in Einstein's iconic equation $E = mc^2$, revolutionized our understanding of physics. This principle is fundamental to the study of nuclear physics, particularly in the context of mass defect and nuclear binding energy. For students of AS & A Level Physics (9702), grasping this concept is essential for comprehending the intricate balance within atomic nuclei and the vast energy transformations involved.

Key Concepts

Theoretical Foundations of $E = mc^2$

The equation $E = mc^2$ establishes a direct relationship between mass ($m$) and energy ($E$), with $c$ representing the speed of light in a vacuum ($3 \times 10^8 \, \text{m/s}$). This formula implies that mass can be converted into energy and vice versa, highlighting the interchangeable nature of these two fundamental entities in the universe. Einstein derived this equation as part of his Special Theory of Relativity, which fundamentally altered classical mechanics by introducing the concepts of spacetime and the constancy of the speed of light. The derivation begins with the principle that the laws of physics are the same in all inertial frames and that the speed of light remains constant regardless of the observer's state of motion.

Mass Defect and Nuclear Binding Energy

In atomic nuclei, protons and neutrons (nucleons) are held together by the strong nuclear force. However, the mass of a nucleus is often found to be less than the sum of its individual nucleons' masses. This difference is known as the **mass defect** ($\Delta m$). The mass defect arises because energy is released when nucleons bind together to form a nucleus, resulting in a lower total mass compared to the separate nucleons. The **nuclear binding energy** ($E_b$) quantifies the energy required to disassemble a nucleus into its constituent protons and neutrons. According to Einstein's equation, this energy can be calculated using the mass defect: $$ E_b = \Delta m \cdot c^2 $$ For example, consider the helium-4 nucleus ($^4\text{He}$). The mass of two protons and two neutrons separately is greater than the mass of the helium nucleus. The difference in mass, when multiplied by $c^2$, gives the binding energy that holds the nucleus together.

Applications of Mass-Energy Equivalence

The principle of mass-energy equivalence has profound implications in various fields:

Nuclear Power: Harnessing the energy released from nuclear fission or fusion processes relies on converting mass into energy.
Astronomy: Stellar phenomena, such as the energy production in stars, are explained through mass-energy transformations.
Particle Physics: Colliders convert kinetic energy into mass, creating new particles from energy.

Mathematical Derivation of $E = mc^2$

Deriving $E = mc^2$ involves understanding the relationship between energy, momentum, and mass in the framework of Special Relativity. Starting with the relativistic energy-momentum relation: $$ E^2 = (pc)^2 + (m_0 c^2)^2 $$ For a particle at rest ($p = 0$), this simplifies to: $$ E = m_0 c^2 $$ Here, $m_0$ is the rest mass of the particle, and $E$ represents its rest energy. This equation signifies that the energy inherent in an object's mass is a product of its mass and the square of the speed of light, a staggering number that underscores the vast energy potential contained within even small masses.

Conservation of Energy and Mass in Nuclear Reactions

In nuclear reactions, the conservation laws of physics dictate that both energy and mass are conserved. However, due to the mass-energy equivalence, it's more accurate to consider the **conservation of mass-energy**. During processes like nuclear fission and fusion, a small amount of mass is converted into a significant amount of energy, as dictated by $E = mc^2$. For example, in the fusion of hydrogen nuclei to form helium, the mass of the resultant helium nucleus is slightly less than the total mass of the original hydrogen nuclei. This mass difference is released as energy, which powers stars and hydrogen bombs alike.

Real-World Examples and Calculations

To illustrate the practical application of $E = mc^2$, let's consider the mass defect in the fusion of deuterium and tritium to form helium-4:

Mass of Deuterium (²H): 2.014102 u
Mass of Tritium (³H): 3.016049 u
Total Mass of Reactants: 5.030151 u
Mass of Helium-4 (⁴He): 4.002603 u
Mass Defect ($\Delta m$): 1.027548 u

Converting atomic mass units to kilograms ($1 \, \text{u} = 1.66053906660 \times 10^{-27} \, \text{kg}$): $$ \Delta m = 1.027548 \, \text{u} \times 1.66053906660 \times 10^{-27} \, \text{kg/u} = 1.706 \times 10^{-27} \, \text{kg} $$ Calculating the binding energy: $$ E_b = \Delta m \cdot c^2 = 1.706 \times 10^{-27} \, \text{kg} \times (3 \times 10^8 \, \text{m/s})^2 = 1.535 \times 10^{-10} \, \text{J} $$ This energy release is a testament to the immense power unlocked through nuclear reactions, pivotal in both energy generation and weaponry.

Limitations of the Mass-Energy Equivalence

While $E = mc^2$ is a cornerstone of modern physics, it operates within specific boundaries:

Non-inclusion of Binding Energy: The equation doesn't account for the energy required to assemble the mass from fundamental particles.
Neglect of Relativistic Mass: It considers rest mass, not accounting for mass changes at relativistic speeds.
Quantum Considerations: At quantum scales, mass-energy interactions become more complex, necessitating quantum field theories.

Experimental Evidence Supporting $E = mc^2$

Numerous experiments have validated the mass-energy equivalence:

Particle Accelerators: High-energy collisions convert kinetic energy into mass, creating new particles.
Nuclear Reactions: Observations of energy release in fission and fusion processes align with predictions from $E = mc^2$.
Annihilation Processes: Matter-antimatter annihilation results in energy release consistent with the total mass converted.

Impact on Modern Technology

The principles derived from $E = mc^2$ have led to technological advancements:

Nuclear Power Plants: Utilize controlled fission reactions to generate electricity.
Medical Imaging: Technologies like PET scans rely on annihilation energy from positron emission.
Space Exploration: Understanding energy-mass dynamics is crucial for propulsion systems.

Philosophical Implications

The realization that mass and energy are interchangeable reshapes our philosophical understanding of the universe. It underscores the dynamic nature of matter and energy, challenging classical notions of immutable mass and introducing a more fluid conception of physical reality.

Advanced Concepts

Relativistic Mass and Its Implications

Building upon $E = mc^2$, the concept of **relativistic mass** emerges when considering objects moving at velocities approaching the speed of light. Relativistic mass ($m_r$) increases with velocity ($v$) and is given by: $$ m_r = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}} $$ As $v$ approaches $c$, $m_r$ approaches infinity, illustrating why no object with mass can attain the speed of light. This relationship has profound implications for particle physics and cosmology, where particles often move at relativistic speeds.

Energy-Momentum Tensor in General Relativity

In General Relativity, the energy-momentum tensor ($T^{\mu\nu}$) extends the concept of mass-energy equivalence to include momentum and stress. It serves as the source of spacetime curvature, dictating how mass-energy influences the geometry of the universe. The tensor encompasses energy density, momentum density, and stress, providing a comprehensive description of the distribution and flow of energy and momentum in spacetime.

Nuclear Reactions and Binding Energy Calculations

Advanced nuclear physics involves calculating the binding energy of complex nuclei. This requires accounting for various interactions:

Strong Nuclear Force: The primary force binding nucleons, overcoming electromagnetic repulsion between protons.
Pairing Energy: Energy related to the pairing of nucleons, affecting the stability of nuclei.
Semi-Empirical Mass Formula: A tool for estimating binding energies, incorporating volume, surface, Coulomb, asymmetry, and pairing terms.

The semi-empirical mass formula (Weizsäcker formula) estimates the binding energy per nucleon ($B/A$): $$ B/A = a_v - a_s A^{-1/3} - a_c \frac{Z^2}{A^{4/3}} - a_a \frac{(A - 2Z)^2}{A^2} \pm a_p A^{-3/4} $$ where $a_v$, $a_s$, $a_c$, $a_a$, and $a_p$ are empirical coefficients, $A$ is the mass number, and $Z$ is the atomic number.

Quantum Field Theory and Mass Generation

In Quantum Field Theory (QFT), the origin of mass is explained through mechanisms like the Higgs mechanism. Particles acquire mass by interacting with the Higgs field, an omnipresent field permeating the universe. The Higgs boson, discovered in 2012, is a manifestation of this field. This interaction provides a fundamental understanding of why particles have the mass they do, complementing the mass-energy equivalence principle.

Energy-Mass Equivalence in Cosmology

In cosmology, mass-energy equivalence plays a critical role in understanding the evolution of the universe. Concepts like dark energy and dark matter involve forms of energy that contribute to the universe's expansion and structure without being directly observable. The interplay between mass and energy influences models of cosmic inflation, the big bang, and the ultimate fate of the cosmos.

High-Energy Astrophysical Processes

Astrophysical phenomena such as supernovae, neutron stars, and black holes involve extreme conditions where mass-energy equivalence is paramount. In supernova explosions, collapsing stellar cores convert mass into energy, ejecting material into space and dispersing heavy elements. Neutron stars, with densities exceeding that of atomic nuclei, showcase the limits of mass-energy interactions, while black holes represent regions where mass-energy curves spacetime to such an extent that not even light can escape.

Mass-Energy Equivalence and Particle Physics

In particle physics, collisions at high energies can create new particles, illustrating the conversion of kinetic energy into mass. Particle accelerators, like the Large Hadron Collider, exploit this principle to investigate fundamental particles and forces. The creation and annihilation of particles provide empirical support for $E = mc^2$, demonstrating the delicate balance between mass and energy in the subatomic realm.

Mass-Energy Equivalence in Modern Technology

Beyond nuclear energy, mass-energy equivalence influences modern technologies:

GPS Systems: Require relativistic corrections to account for time dilation effects, ensuring precise positioning.
Electric Power Generation: Involves mass-energy transformations in both conventional and renewable energy sources.
Medical Technologies: Techniques like ionizing radiation therapy utilize energy-mass principles to target cancer cells.

Philosophical and Ethical Considerations

The profound implications of mass-energy equivalence extend into philosophical and ethical domains. The ability to harness vast energy from minimal mass raises questions about responsible usage, environmental impact, and the moral responsibilities associated with powerful technologies like nuclear energy and weapons. Societal discourse must balance scientific advancement with ethical considerations to navigate the challenges posed by these capabilities.

Mathematical Challenges and Problem-Solving

Advanced studies involve complex problem-solving related to mass-energy equivalence:

Calculating Binding Energy: Given nuclear masses, determine the binding energy using $E = \Delta m c^2$.
Energy Conversion in Particle Collisions: Analyze collision outcomes to predict particle creation based on energy availability.
Relativistic Kinematics: Solve problems involving particles moving at speeds close to $c$, requiring relativistic energy and momentum calculations.

These challenges enhance analytical skills and deepen understanding of the interplay between mass and energy in various physical contexts.

Comparison Table

Aspect	Mass-Energy Equivalence ($E = mc^2$)	Classical Mechanics
Relationship	Direct equivalence between mass and energy.	Energy and mass are separate and conserved independently.
Speed of Light ($c$)	A fundamental constant linking mass and energy.	Not directly involved in energy-mass relationships.
Applicability	High-energy and relativistic contexts.	Low-energy, non-relativistic scenarios.
Equation Form	$E = mc^2$	Kinetic Energy: $KE = \frac{1}{2}mv^2$
Implications	Mass can be converted to energy and vice versa.	Mass and energy are conserved separately.

Summary and Key Takeaways

Einstein's $E = mc^2$ demonstrates the interchangeable nature of mass and energy.
Mass defect and nuclear binding energy are central to understanding nuclear stability.
Advanced concepts extend the principle to relativistic speeds, cosmology, and particle physics.
Practical applications range from nuclear power to medical technologies.
Understanding mass-energy equivalence is foundational for AS & A Level Physics studies.

Examiner Tip

Tips

1. Remember the Equation: Always include the square of the speed of light ($c^2$) when using $E = mc^2$.

2. Use Mnemonics: "Energy Mass Converts" can help recall that mass converts into energy via $E = mc^2$.

3. Practice Problems: Regularly solve binding energy and mass defect problems to strengthen your understanding and prepare for exams.

Did You Know

1. The total mass of the sun is converted into energy each second, producing the sunlight that sustains life on Earth.

2. In nuclear reactors, even a tiny amount of mass loss results in massive energy release, illustrating the power of $E = mc^2$.

3. The concept of mass-energy equivalence is not only theoretical; it's the foundation behind technologies like PET scans in medical diagnostics.

Common Mistakes

Mistake 1: Ignoring the speed of light squared in calculations.
Incorrect: $E = m \cdot c$
Correct: $E = mc^2$

Mistake 2: Confusing mass defect with binding energy.
Incorrect: Assuming mass defect is energy released.
Correct: Mass defect multiplied by $c^2$ gives the binding energy.

Mistake 3: Overlooking relativistic effects at high velocities.
Incorrect: Using classical kinetic energy formulas for particles moving near $c$.
Correct: Applying relativistic energy and momentum equations.

FAQ

What does each symbol in $E = mc^2$ represent?

In the equation, $E$ stands for energy, $m$ represents mass, and $c$ is the speed of light in a vacuum.

Why is $c^2$ used in the equation?

The speed of light squared ($c^2$) acts as the conversion factor, showing how a small amount of mass can be transformed into a large amount of energy.

Can mass be created from energy?

Yes, according to $E = mc^2$, energy can be converted into mass under the right conditions, such as in particle accelerators.

What is mass defect?

Mass defect is the difference between the mass of a nucleus and the sum of its individual protons and neutrons, representing the binding energy.

How is $E = mc^2$ applied in nuclear power?

In nuclear power, mass is converted into energy through fission or fusion reactions, releasing vast amounts of energy as predicted by $E = mc^2$.

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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Understand the Equivalence Between Energy and Mass (E = mc²)

Introduction