Understanding the electric potential energy stored in a capacitor is fundamental in the study of electrostatics and circuit theory within the AS & A Level Physics curriculum (9702). This concept not only elucidates how energy is stored and managed in electronic components but also forms the basis for various applications in modern technology. By analyzing the area under the potential–charge graph, students can gain a deeper insight into the relationship between charge, potential difference, and energy in capacitive systems.

Key Concepts

Fundamental Definitions

A **capacitor** is an electronic component that stores energy in the form of an electric field created between a pair of conductors separated by an insulator. The **capacitance (C)** of a capacitor is defined as the ratio of the magnitude of charge (Q) on each conductor to the potential difference (V) between them: $$C = \frac{Q}{V}$$ The unit of capacitance is the farad (F), where 1 farad equals 1 coulomb per volt. **Electric Potential Energy (U)** is the energy stored in the electric field of a capacitor and can be determined from the area under the potential–charge (V–Q) graph. This graphical representation illustrates how the potential difference varies with the charge stored on the capacitor.

The Potential–Charge Relationship

The potential–charge graph for a capacitor is a straight line passing through the origin, assuming a constant capacitance. The slope of this line is the inverse of the capacitance: $$V = \frac{Q}{C}$$ Graphically, $V$ is plotted on the y-axis, and $Q$ on the x-axis. The area under this linear graph between 0 and charge $Q$ represents the electric potential energy stored in the capacitor.

Calculating Electric Potential Energy

To determine the electric potential energy stored in a capacitor from the V–Q graph, integrate the potential with respect to charge: $$U = \int_{0}^{Q} V dQ$$ Substituting $V = \frac{Q}{C}$ into the equation: $$U = \int_{0}^{Q} \frac{Q}{C} dQ = \frac{1}{C} \int_{0}^{Q} Q dQ = \frac{1}{C} \left[\frac{Q^2}{2}\right]_0^Q = \frac{Q^2}{2C}$$ Alternatively, using the relationship $Q = CV$, the energy can also be expressed as: $$U = \frac{1}{2} CV^2$$ Both expressions are equivalent and commonly used depending on the known quantities in a given problem.

Energy Stored in Different Capacitor Configurations

When capacitors are arranged in series or parallel, the total capacitance changes accordingly, affecting the total stored energy. **Series Configuration:** For capacitors in series, the reciprocal of the total capacitance ($C_{total}$) is the sum of the reciprocals of the individual capacitances: $$\frac{1}{C_{total}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots + \frac{1}{C_n}$$ The charge stored in each capacitor remains the same, but the potential differences add up. **Parallel Configuration:** For capacitors in parallel, the total capacitance is the sum of the individual capacitances: $$C_{total} = C_1 + C_2 + \dots + C_n$$ The potential difference across each capacitor is identical, while the charges add up. Understanding these configurations is crucial for analyzing real-world circuits where multiple capacitors are used together.

Practical Examples

Consider a capacitor with a capacitance of 5 F charged to a potential difference of 10 V. The electric potential energy stored can be calculated as: $$U = \frac{1}{2} CV^2 = \frac{1}{2} \times 5 \, \text{F} \times (10 \, \text{V})^2 = 250 \, \text{J}$$ Graphically, this energy corresponds to the area under the linear V–Q graph from 0 to Q, where $Q = CV = 50 \, \text{C}$. Another example involves a series configuration of two capacitors, 3 F and 6 F. The total capacitance is: $$\frac{1}{C_{total}} = \frac{1}{3} + \frac{1}{6} = \frac{1}{2} \Rightarrow C_{total} = 2 \, \text{F}$$ If connected to a 12 V source, the total stored energy is: $$U = \frac{1}{2} \times 2 \, \text{F} \times (12 \, \text{V})^2 = 144 \, \text{J}$$ This energy is distributed across the capacitors based on their capacitances and the potential difference each experiences.

Graphical Interpretation

The V–Q graph is a powerful tool for visualizing the relationship between potential difference and charge in a capacitor. For an ideal capacitor, the graph is a straight line with slope $1/C$. The area under this curve up to a charge $Q$ represents the electric potential energy: $$U = \frac{1}{2} CV^2 = \text{Area under the V–Q curve}$$ This graphical method provides an intuitive understanding of energy storage and can be extended to analyze non-linear capacitors by evaluating the integral of $V$ with respect to $Q$.

Units and Dimensional Analysis

Ensuring dimensional consistency is vital when performing calculations related to electric potential energy. The units for each quantity are as follows:

Charge ($Q$): Coulombs (C)
Potential Difference ($V$): Volts (V)
Capacitance ($C$): Farads (F)
Electric Potential Energy ($U$): Joules (J)

Using dimensional analysis, the energy equation can be verified: $$U = \frac{Q^2}{2C} \Rightarrow \frac{\text{C}^2}{\text{F}} = \frac{\text{C}^2}{\text{C}/\text{V}} = \text{C} \times \text{V} = \text{J}$$ This confirms the correctness of the formula in terms of units.

Energy Density in Capacitors

The **energy density** (energy per unit volume) in a capacitor is an important consideration, especially in applications requiring miniaturization and high energy storage. For a parallel-plate capacitor, the energy density ($u$) can be expressed as: $$u = \frac{U}{V} = \frac{1}{2} \epsilon \frac{V^2}{d^2}$$ where $\epsilon$ is the permittivity of the dielectric material, and $d$ is the separation between the plates. Optimizing energy density involves selecting materials with high permittivity and minimizing plate separation without compromising the dielectric integrity.

Limitations and Real-World Considerations

While the theoretical models provide precise relationships, real-world capacitors exhibit non-ideal behaviors such as leakage current, finite breakdown voltage, and temperature dependence. These factors can affect the actual energy storage capacity and efficiency. Understanding these limitations is essential for designing reliable electronic systems and ensuring safety in practical applications.

Advanced Concepts

Mathematical Derivation of Energy Stored

To derive the expression for electric potential energy stored in a capacitor, consider the work done ($dW$) to move an infinitesimal charge ($dQ$) against the potential difference ($V$): $$dW = V dQ$$ Substituting $V = \frac{Q}{C}$: $$dW = \frac{Q}{C} dQ$$ Integrating from 0 to $Q$: $$U = \int_{0}^{Q} \frac{Q}{C} dQ = \frac{1}{C} \int_{0}^{Q} Q dQ = \frac{1}{C} \left[\frac{Q^2}{2}\right]_0^Q = \frac{Q^2}{2C}$$ This integral represents the total work done in charging the capacitor and thus the energy stored.

Energy in Dielectric Materials

Introducing a dielectric material between the plates of a capacitor affects its capacitance and energy storage. The capacitance with a dielectric ($C_d$) is: $$C_d = \kappa C$$ where $\kappa$ is the dielectric constant. The energy stored becomes: $$U_d = \frac{Q^2}{2C_d} = \frac{Q^2}{2\kappa C}$$ The presence of a dielectric reduces the electric field for a given charge, thereby allowing more charge to be stored for the same energy or the same charge to be stored with less energy.

Electric Field and Energy Density

The energy density ($u$) in the electric field between the capacitor plates is given by: $$u = \frac{1}{2} \epsilon E^2$$ where $E$ is the electric field strength and $\epsilon$ is the permittivity of the medium. This expression shows that the energy density increases with the square of the electric field, highlighting the importance of controlling field strength in high-energy applications.

Energy Storage Efficiency

The efficiency of energy storage in capacitors is influenced by factors such as dielectric losses, resistive heating, and leakage currents. The theoretical maximum energy storage is given by the ideal equations, but practical limitations reduce the efficiency. Minimizing these losses involves selecting appropriate materials and designing capacitors to withstand operating conditions without significant degradation.

Nonlinear Capacitors and Energy Storage

In certain materials, the relationship between $V$ and $Q$ is not linear, leading to nonlinear capacitors. In such cases, the potential–charge graph deviates from a straight line, and the energy stored must be calculated using the integral: $$U = \int_{0}^{Q} V(Q) dQ$$ Nonlinear capacitors are used in applications like varactors and tunable filters, where variable capacitance is required.

Interdisciplinary Connections

The principles of energy storage in capacitors extend beyond physics into various engineering disciplines:

Electrical Engineering: Capacitors are integral in power supply systems, signal processing, and energy storage solutions like supercapacitors.
Mechanical Engineering: Capacitive sensors utilize changes in capacitance to measure physical quantities such as displacement or force.
Materials Science: Developing advanced dielectric materials with higher permittivity enhances capacitor performance and energy density.

Understanding energy storage in capacitors thus provides a foundational knowledge applicable across multiple fields.

Complex Problem-Solving: Multi-Step Reasoning

**Problem 1:** A parallel-plate capacitor with plate area $A = 0.1 \, \text{m}^2$ and plate separation $d = 0.005 \, \text{m}$ is filled with a dielectric material of permittivity $\epsilon = 4\epsilon_0$. If the capacitor is charged to a potential difference of 12 V, calculate the electric potential energy stored. **Solution:** First, calculate the capacitance with the dielectric: $$C = \kappa \epsilon_0 \frac{A}{d} = 4 \times 8.854 \times 10^{-12} \, \text{F/m} \times \frac{0.1}{0.005} = 7.0832 \times 10^{-9} \, \text{F}$$ Next, calculate the energy stored: $$U = \frac{1}{2} CV^2 = \frac{1}{2} \times 7.0832 \times 10^{-9} \, \text{F} \times (12)^2 \, \text{V}^2 = 5.099 \times 10^{-6} \, \text{J}$$ **Problem 2:** Two capacitors, 5 F and 10 F, are connected in parallel and charged to a potential difference of 15 V. Determine the total electric potential energy stored in the system. **Solution:** Total capacitance in parallel: $$C_{total} = 5 + 10 = 15 \, \text{F}$$ Energy stored: $$U = \frac{1}{2} \times 15 \, \text{F} \times (15 \, \text{V})^2 = \frac{1}{2} \times 15 \times 225 = 1687.5 \, \text{J}$$

Experimental Techniques for Measuring Energy Stored

Several experimental methods are employed to determine the electric potential energy stored in capacitors:

Charge-Discharge Method: By measuring the time constant during charging and discharging through a known resistor, capacitance and energy can be inferred.
Potential–Charge Graph Plotting: Directly plotting $V$ against $Q$ and calculating the area under the curve to determine energy.
Oscilloscope Measurements: Using oscilloscopes to capture voltage and charge variations over time enables precise energy calculations.

Accurate measurements require careful control of environmental factors and calibration of equipment to minimize errors.

Comparison Table

Aspect	Ideal Capacitor	Real Capacitor
Energy Storage Formula	$U = \frac{1}{2} CV^2$	Similar, but includes loss factors
Leakage Current	None	Present, leading to energy loss
Dielectric Material	Perfect insulator	Practical materials with finite permittivity
Breakdown Voltage	Theoretical infinite	Finite, determined by material properties
Energy Density	Depends solely on capacitance and voltage	Influenced by physical size and material constraints

Summary and Key Takeaways

Electric potential energy in a capacitor is determined by the area under the V–Q graph.
Key formulas: $U = \frac{Q^2}{2C}$ and $U = \frac{1}{2} CV^2$.
Capacitor configurations (series and parallel) significantly affect total energy storage.
Advanced concepts include energy density, dielectric effects, and real-world limitations.
Interdisciplinary applications highlight the broad relevance of capacitive energy storage.

Examiner Tip

Tips

To remember the energy formula, think of the "half" in $U = \frac{1}{2} CV^2$ as the average value when charging a capacitor from 0 to V volts.

Use dimensional analysis to verify your equations. Ensuring that your final units are in joules (J) can help catch calculation errors.

When working with capacitor configurations, draw the circuit and label each capacitor’s voltage and charge to better visualize how they interact.

Did You Know

Capacitors are not just passive components in electronic circuits; they play a crucial role in modern energy storage solutions. For instance, supercapacitors can store up to 10,000 times more energy than traditional capacitors, making them essential in applications like regenerative braking in electric vehicles.

Another interesting fact is that the concept of capacitors dates back to the 18th century with the invention of the Leyden jar, one of the earliest forms of capacitors. This early discovery paved the way for the development of countless electronic devices we rely on today.

Common Mistakes

Ignoring Units: Students often forget to convert units properly. For example, mixing farads with microfarads without conversion can lead to incorrect energy calculations.

Misapplying Formulas: Another common error is using $U = CV^2$ instead of the correct $U = \frac{1}{2} CV^2$. This mistake results in calculating twice the actual energy stored.

Overlooking Capacitor Configurations: When dealing with capacitors in series or parallel, failing to adjust the total capacitance accordingly can lead to incorrect energy storage values.

FAQ

What is the significance of the area under the V–Q graph?

The area under the potential–charge (V–Q) graph represents the electric potential energy stored in the capacitor. It provides a visual and mathematical way to calculate energy based on charge and voltage.

How does adding a dielectric affect the energy stored in a capacitor?

Adding a dielectric increases the capacitance, allowing more charge to be stored at the same voltage, which can either increase the energy stored or reduce the required voltage for a given energy level.

Can the electric potential energy in a capacitor be negative?

No, electric potential energy in a capacitor is always positive because it depends on the square of the charge or voltage, both of which are positive quantities.

What happens to the energy stored when capacitors are connected in series?

When capacitors are connected in series, the total capacitance decreases, which affects the total energy stored. The energy depends on the individual capacitances and the applied voltage across the series combination.

Is the energy stored in a real capacitor always equal to the theoretical calculation?

No, real capacitors have imperfections like leakage currents and dielectric losses, which cause the actual energy stored to be less than the theoretical value calculated using ideal formulas.

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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Determine Electric Potential Energy Stored in a Capacitor from the Area Under the Potential–Charge Graph

Introduction