Understanding power in electrical circuits is fundamental in physics, particularly when analyzing alternating currents (AC). This concept, specifically that the mean power in a resistive load is half the maximum power for sinusoidal current, is pivotal for students studying the AS & A Level Physics curriculum (9702). It bridges the gap between theoretical principles and real-world electrical applications, providing essential insights into energy distribution and efficiency in AC systems.

Key Concepts

Understanding Sinusoidal Current

Sinusoidal current is a type of alternating current where the flow of electric charge varies sinusoidally with time. Mathematically, it can be represented as:

$$ i(t) = I_{max} \sin(\omega t + \phi) $$

Where:

$i(t)$ is the instantaneous current at time $t$.
$I_{max}$ is the maximum current or amplitude.
$\omega$ is the angular frequency.
$\phi$ is the phase angle.

Power in Electrical Circuits

Power ($P$) in an electrical circuit is the rate at which energy is consumed or converted by the circuit elements. For a resistive load, power can be calculated using:

$$ P(t) = V(t) \cdot I(t) $$

Since $V(t) = I(t) \cdot R$, where $R$ is the resistance, substituting gives:

$$ P(t) = I(t)^2 \cdot R $$

For sinusoidal currents, this becomes:

$$ P(t) = (I_{max} \sin(\omega t))^2 \cdot R = I_{max}^2 R \sin^2(\omega t) $$

Mean (Average) Power

The mean power ($P_{mean}$) over a complete cycle is the average value of the instantaneous power. It is calculated using the integral of $P(t)$ over one period ($T$):

$$ P_{mean} = \frac{1}{T} \int_0^T I_{max}^2 R \sin^2(\omega t) dt $$

Using the identity $\sin^2(x) = \frac{1 - \cos(2x)}{2}$, the integral simplifies to:

$$ P_{mean} = \frac{I_{max}^2 R}{2} $$

Maximum Power

The maximum power ($P_{max}$) occurs when the sinusoidal current reaches its peak value. It is given by:

$$ P_{max} = I_{max}^2 R $$

Relationship Between Mean and Maximum Power

Comparing the expressions for mean and maximum power:

$$ P_{mean} = \frac{P_{max}}{2} $$

This indicates that the mean power delivered to a resistive load by a sinusoidal current is half of its maximum power. This relationship is crucial for designing and analyzing AC circuits, ensuring components are rated appropriately for their power handling capabilities.

Root Mean Square (RMS) Values

The RMS value of a sinusoidal current is a practical measure used to express the equivalent DC value that would produce the same power dissipation in a resistor. It is defined as:

$$ I_{rms} = \frac{I_{max}}{\sqrt{2}} $$

Using RMS values, the mean power can be expressed as:

$$ P_{mean} = I_{rms}^2 R $$>

This formulation simplifies power calculations in AC circuits, making it easier to compare with DC systems.

Implications in Electrical Engineering

The principle that mean power is half the maximum power has significant implications in electrical engineering, particularly in the sizing of components like resistors, capacitors, and inductors. Engineers must account for the RMS values to ensure components can handle the expected power without overheating or failing.

Examples and Applications

Consider an AC circuit with a resistive load where the maximum current is 10 A and the resistance is 5 Ω. The mean power can be calculated as:

$$ P_{mean} = \frac{I_{max}^2 R}{2} = \frac{10^2 \times 5}{2} = \frac{500}{2} = 250 \, \text{W} $$

Hence, the mean power delivered is 250 W, while the maximum power is:

$$ P_{max} = I_{max}^2 R = 10^2 \times 5 = 500 \, \text{W} $$

This example illustrates the practical application of the theoretical relationship between mean and maximum power in real-world scenarios.

Energy Efficiency and Power Management

Understanding this power relationship assists in energy efficiency and power management. It allows for accurate calculations of energy consumption over time, aiding in the design of energy-efficient systems and the implementation of power-saving measures.

Summary of Key Concepts

Sinusoidal currents vary sinusoidally with time, characterized by amplitude, frequency, and phase.
Instantaneous power in a resistive load is the product of instantaneous voltage and current.
Mean power is half the maximum power for sinusoidal currents in resistive loads.
RMS values provide a practical measure for comparing AC and DC power equivalents.
The relationship between mean and maximum power is essential for electrical component sizing and energy management.

Advanced Concepts

Mathematical Derivation of Mean Power

To derive the mean power for a sinusoidal current mathematically, consider the instantaneous power:

$$ P(t) = I^2(t) R = I_{max}^2 \sin^2(\omega t) R $$>

Using the trigonometric identity:

$$ \sin^2(\omega t) = \frac{1 - \cos(2\omega t)}{2} $$>

Substituting this into the power equation:

$$ P(t) = \frac{I_{max}^2 R}{2} (1 - \cos(2\omega t)) $$>

The mean power over one period ($T$) is:

$$ P_{mean} = \frac{1}{T} \int_0^T \frac{I_{max}^2 R}{2} (1 - \cos(2\omega t)) dt $$>

Integrating term by term:

$$ P_{mean} = \frac{I_{max}^2 R}{2} \left[ \frac{1}{T} \int_0^T 1 \, dt - \frac{1}{T} \int_0^T \cos(2\omega t) \, dt \right] $$>

The first integral is:

$$ \frac{1}{T} \int_0^T 1 \, dt = 1 $$>

The second integral evaluates to zero over one complete cycle:

$$ \frac{1}{T} \int_0^T \cos(2\omega t) \, dt = 0 $$>

Thus, the mean power simplifies to:

$$ P_{mean} = \frac{I_{max}^2 R}{2} $$>

This derivation confirms that the mean power is half the maximum power in a resistive load with sinusoidal current.

Complex Power and Phasor Representation

In AC circuit analysis, especially with inductive and capacitive components, the concept of complex power becomes significant. Complex power ($S$) combines both real power ($P$) and reactive power ($Q$), represented as:

$$ S = P + jQ $$>

Where $j$ is the imaginary unit. For purely resistive loads, reactive power $Q$ is zero, and thus:

$$ S = P $$>

However, in circuits with reactance, $Q$ is non-zero, necessitating the use of phasor representations to manage phase differences between voltage and current. This extends the basic concept of mean and maximum power to more complex scenarios involving power factor calculations and power triangle analyses.

Power Factor and Its Implications

Power factor is a measure of how effectively electrical power is being converted into useful work output. It is defined as the cosine of the phase angle ($\phi$) between the voltage and current:

$$ \text{Power Factor} = \cos(\phi) $$>

For purely resistive loads, $\phi = 0^\circ$, making the power factor unity (1). In inductive or capacitive loads, the power factor decreases, indicating a phase difference that results in inefficiencies. Understanding the mean power in relation to the power factor is crucial for optimizing electrical systems and minimizing energy loss.

Energy Storage in Reactive Components

In circuits with inductors and capacitors, energy is alternately stored and released, contributing to reactive power. This oscillation affects the mean power calculation as it introduces a component of power that does not contribute to net energy transfer over time. Analyzing mean power in such systems requires considering both real and reactive power to accurately assess overall power consumption and efficiency.

Thermal Effects and Resistive Heating

The mean power in resistive loads directly relates to thermal effects due to Joule heating. The heat generated ($Q$) in a resistor can be calculated using:

$$ Q = P_{mean} \cdot t = \frac{I_{max}^2 R}{2} \cdot t $$>

Where $t$ is the time duration. This relationship is fundamental in designing electrical systems to prevent overheating and ensure safe operation of components.

Applications in Power Transmission

In power transmission, managing mean power is essential for efficient energy distribution. High-voltage transmission minimizes current for a given power level, reducing resistive losses ($P = I^2 R$). Understanding the relationship between mean and maximum power aids in optimizing transmission parameters to enhance system reliability and reduce energy waste.

Advanced Problem-Solving Techniques

Complex problems involving mean and maximum power often require multi-step reasoning and the integration of multiple physics concepts. For instance, calculating power in circuits with both resistive and reactive components involves determining the impedance, calculating RMS values, and applying power factor corrections. Mastery of these techniques is crucial for tackling higher-level physics problems and real-world engineering challenges.

Interdisciplinary Connections

The principles governing mean and maximum power in resistive loads extend beyond physics into engineering disciplines like electrical engineering, where they inform the design of circuits, power systems, and electronic devices. Additionally, concepts like power efficiency and energy management are relevant to environmental science and sustainability efforts, highlighting the broad applicability of these fundamental physics principles.

Real-World Case Study: Residential Electrical Systems

In residential electrical systems, understanding mean power is vital for designing safe and efficient home wiring. Circuit breakers and fuses are rated based on the maximum expected current, ensuring they can handle transient spikes without tripping unnecessarily. Moreover, the mean power considerations guide the selection of appliances and their power ratings to prevent overloading circuits and maintain energy efficiency.

Future Trends and Innovations

Advancements in power electronics and smart grid technologies rely on precise power calculations, including mean and maximum power analyses. Innovations such as renewable energy integration, energy storage systems, and power optimization algorithms depend on accurate models of power behavior in various circuit configurations. Understanding the foundational concepts of mean versus maximum power paves the way for these technological developments.

Summary of Advanced Concepts

Mathematical derivation confirms that mean power is half the maximum power for sinusoidal currents.
Complex power and phasor representation extend power analysis to circuits with reactive components.
Power factor influences the efficiency and effectiveness of power delivery in AC systems.
Thermal effects from mean power are critical in preventing overheating in resistive loads.
Interdisciplinary applications highlight the broad relevance of mean and maximum power concepts.

Comparison Table

Aspect	Mean Power ($P_{mean}$)	Maximum Power ($P_{max}$)
Definition	Average power over one cycle of sinusoidal current.	Highest instantaneous power reached during the cycle.
Formula	$P_{mean} = \frac{I_{max}^2 R}{2}$	$P_{max} = I_{max}^2 R$
Significance	Represents the effective power delivered to the load.	Indicates the peak power handling capability of components.
Relation	Half of the maximum power.	Twice the mean power.
Application	Calculating energy consumption and thermal effects.	Designing components to withstand power spikes.

Summary and Key Takeaways

The mean power in a resistive load with sinusoidal current is half of its maximum power.
Mathematical derivations and RMS values simplify power calculations in AC circuits.
Understanding power relationships is crucial for designing efficient and safe electrical systems.
Advanced concepts like power factor and complex power extend these principles to more complex circuits.
Interdisciplinary applications highlight the broad relevance of power analysis in various fields.

Examiner Tip

Tips

- **Mnemonic for Mean Power**: Remember "Half Max Power" to recall that mean power is half of maximum power.
- **Use RMS Values**: Always convert to RMS values when analyzing AC circuits to simplify calculations.
- **Practice with Real-World Examples**: Apply concepts to everyday electrical devices to enhance understanding and retention.
- **Double-Check Factor of 1/2**: When calculating mean power, ensure the factor of one-half is included to avoid common mistakes.
- **Understand Phase Relations**: For AC circuits with reactance, always consider the phase angle to accurately determine power factor.

Did You Know

1. The concept of mean power being half of the maximum power isn't just theoretical—it plays a critical role in the design of household appliances, ensuring they operate safely without overheating.
2. This principle is fundamental in music production, where understanding power helps in designing speakers that handle varying current levels without distortion.
3. Renewable energy systems, such as solar panels and wind turbines, rely on accurate power calculations to optimize energy storage and distribution.

Common Mistakes

1. **Incorrect Calculation of Mean Power**: Students often forget to apply the factor of 1/2 when calculating mean power from maximum power.
**Incorrect**: $P_{mean} = I_{max}^2 R$
**Correct**: $P_{mean} = \frac{I_{max}^2 R}{2}$

2. **Confusing RMS and Mean Power**: Another frequent error is mistaking RMS values for mean power, leading to inaccurate power assessments.
3. **Neglecting Phase Angle in Non-Resistive Loads**: Students sometimes ignore the phase angle when dealing with inductive or capacitive loads, resulting in incorrect power factor calculations.

FAQ

What is mean power in a resistive load?

Mean power is the average power delivered to a resistive load over one complete cycle of a sinusoidal current. It is calculated as half of the maximum power.

How is maximum power calculated?

Maximum power is calculated using the formula $P_{max} = I_{max}^2 R$, where $I_{max}$ is the maximum current and $R$ is the resistance.

Why is mean power half of maximum power?

This relationship arises from the mathematical integration of the squared sinusoidal current over one period, resulting in the mean value being half of the peak value.

What is the RMS value of a sinusoidal current?

The RMS (Root Mean Square) value of a sinusoidal current is $I_{rms} = \frac{I_{max}}{\sqrt{2}}$, which represents the equivalent DC value delivering the same power to a resistor.

How does power factor affect mean power?

The power factor, which is the cosine of the phase angle between voltage and current, affects mean power by accounting for the real power delivered in the presence of reactive components. A lower power factor indicates more reactive power, reducing the effective mean power.

Can the mean power exceed maximum power?

No, the mean power is always half of the maximum power for sinusoidal currents in resistive loads. It cannot exceed the maximum power.

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

Get PDF

PDF

Explore

Your Flashcards are Ready!

Recall that the Mean Power in a Resistive Load is Half the Maximum Power for Sinusoidal Current

Introduction