Magnetic fields are fundamental concepts in physics, playing a crucial role in understanding electromagnetic phenomena. For students in the AS & A Level Physics course (9702), comprehending that a magnetic field is a field of force produced by moving charges or permanent magnets is essential. This knowledge not only underpins various technological applications but also lays the groundwork for advanced studies in electromagnetism and related disciplines.

Key Concepts

Definition of a Magnetic Field

A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. It is produced by moving charges (electric currents) or by the intrinsic magnetic moments of elementary particles, such as electrons. The magnetic field ($ \mathbf{B} $) at any point in space exerts a force on other moving charges and magnetic dipoles in the vicinity.

Sources of Magnetic Fields

Magnetic fields originate from two primary sources:

Moving Charges: Electric currents flowing through conductors generate magnetic fields. This relationship is quantitatively described by Ampère's Law.
Permanent Magnets: Objects like bar magnets have inherent magnetic domains aligned in a specific direction, creating a persistent magnetic field without the need for an external power source.

Magnetic Field Due to a Straight Current-Carrying Conductor

When an electric current ($ I $) flows through a straight conductor, it creates a magnetic field surrounding the conductor. The direction of the magnetic field can be determined using the right-hand rule: if the thumb points in the direction of the current, the fingers curl in the direction of the magnetic field lines.

The magnitude of the magnetic field at a distance ($ r $) from a long, straight conductor is given by:

$$ B = \frac{\mu_0 I}{2\pi r} $$

where $ \mu_0 = 4\pi \times 10^{-7} \, \text{T}\cdot\text{m}/\text{A} $ is the permeability of free space.

Magnetic Field of a Circular Current Loop

A circular loop of wire carrying a current generates a magnetic field similar to that of a bar magnet. The magnetic field at the center of the loop is given by:

$$ B = \frac{\mu_0 I}{2R} $$

where $ R $ is the radius of the loop. The field lines form concentric circles around the wire and pass through the center of the loop, creating a dipole pattern.

Magnetic Fields of Permanent Magnets

Permanent magnets produce magnetic fields due to the alignment of magnetic domains within the material. Each domain acts like a tiny current loop, and their collective alignment results in a macroscopic magnetic field. The field lines emerge from the north pole and enter the south pole of the magnet, forming closed loops.

Magnetic Flux and Gauss's Law for Magnetism

Magnetic flux ($ \Phi_B $) is a measure of the quantity of magnetism, considering the strength and the extent of a magnetic field. It is defined as the integral of the magnetic field over a given area ($ A $):

$$ \Phi_B = \int \mathbf{B} \cdot d\mathbf{A} $$

Gauss's Law for magnetism states that the net magnetic flux through any closed surface is zero:

$$ \oint \mathbf{B} \cdot d\mathbf{A} = 0 $$

This implies that there are no magnetic monopoles; magnetic field lines are continuous loops without a beginning or end.

Magnetic Force on Moving Charges

A moving charge in a magnetic field experiences a force known as the Lorentz force. The magnitude and direction of this force are given by:

$$ \mathbf{F} = q(\mathbf{v} \times \mathbf{B}) $$

where:

$ \mathbf{F} $ is the force vector
$ q $ is the electric charge
$ \mathbf{v} $ is the velocity of the charge
$ \mathbf{B} $ is the magnetic field

The force is perpendicular to both the velocity of the charge and the magnetic field, causing the charge to move in a circular or helical path around the field lines.

Biot-Savart Law

The Biot-Savart Law provides a method to calculate the magnetic field generated by a current-carrying conductor. For a small segment of the conductor ($ d\mathbf{l} $) carrying current ($ I $), the differential magnetic field ($ d\mathbf{B} $) at a point in space is:

$$ d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I \, d\mathbf{l} \times \mathbf{\hat{r}}}{r^2} $$

where:

$ \mathbf{\hat{r}} $ is the unit vector from the current element to the point of interest
$ r $ is the distance between them

By integrating this expression over the entire current distribution, the total magnetic field can be determined.

Magnetic Dipole Moment

The magnetic dipole moment ($ \mathbf{m} $) is a vector quantity that represents the strength and orientation of a magnet's or current loop's magnetic field. For a current loop, it is defined as:

$$ \mathbf{m} = I \mathbf{A} $$

where:

$ I $ is the current
$ \mathbf{A} $ is the area vector of the loop

The dipole moment plays a crucial role in determining the torque experienced by a magnetic dipole in an external magnetic field:

$$ \mathbf{\tau} = \mathbf{m} \times \mathbf{B} $$

Magnetic Field Lines

Magnetic field lines are a visual tool to represent the direction and strength of a magnetic field. Key properties include:

They exit from the north pole and enter the south pole of a magnet.
The density of the lines indicates the strength of the magnetic field; closer lines represent stronger fields.
They never intersect, as this would imply two directions for the magnetic field at a single point.
They form closed loops, aligning with Gauss's Law for magnetism.

Earth's Magnetic Field

The Earth itself acts as a giant magnet, generating a magnetic field that extends into space. This geomagnetic field is responsible for phenomena such as compass needle alignment and the auroras. The Earth's magnetic field is believed to result from the motion of molten iron alloys in its outer core, a process described by the dynamo theory.

Electromagnetic Induction

Electromagnetic induction refers to the generation of an electric current in a conductor due to a changing magnetic field. Faraday's Law quantifies this phenomenon:

$$ \mathcal{E} = -\frac{d\Phi_B}{dt} $$

where $ \mathcal{E} $ is the induced electromotive force (emf) and $ \Phi_B $ is the magnetic flux. This principle is the working mechanism behind transformers, electric generators, and inductors.

Applications of Magnetic Fields

Understanding magnetic fields is pivotal for various applications, including:

Electric Motors: Convert electrical energy into mechanical motion using magnetic forces.
Generators: Produce electrical energy from mechanical motion through electromagnetic induction.
Magnetic Storage: Use magnetic states to store data in devices like hard drives.
Magnetic Resonance Imaging (MRI): Employ strong magnetic fields for medical imaging.

Advanced Concepts

Maxwell's Equations and Magnetic Fields

Maxwell's Equations form the foundation of classical electromagnetism, unifying electricity and magnetism into a single framework. The relevant equations for magnetic fields include:

Gauss's Law for Magnetism:
$$\oint \mathbf{B} \cdot d\mathbf{A} = 0$$
This states that there are no magnetic monopoles and that magnetic field lines are continuous loops.
Faraday's Law of Induction:
$$\oint \mathbf{E} \cdot d\mathbf{l} = -\frac{d\Phi_B}{dt}$$
This describes how a time-varying magnetic field induces an electric field.
Biot-Savart Law (Integral Form):
$$\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int \frac{I \, d\mathbf{l}' \times (\mathbf{r} - \mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|^3}$$
This integral form provides the magnetic field generated by a steady current distribution.
Amperian Law (with Maxwell's Correction):
$$\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$$
This relates the magnetic field around a closed loop to the electric current and the rate of change of electric flux passing through the loop.

These equations are essential for advanced studies, enabling the derivation of wave equations for electromagnetic waves and exploring the interplay between electric and magnetic fields.

Electromagnetic Waves

Maxwell's Equations predict the existence of electromagnetic waves—oscillating electric and magnetic fields that propagate through space. These waves travel at the speed of light ($ c $) and include radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays.

The wave equation derived from Maxwell's Equations in free space is:

$$ \nabla^2 \mathbf{E} - \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0 $$ $$ \nabla^2 \mathbf{B} - \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{B}}{\partial t^2} = 0 $$

These equations demonstrate that changes in electric fields generate magnetic fields and vice versa, allowing the self-sustaining propagation of electromagnetic waves.

Magnetic Hysteresis

Magnetic hysteresis refers to the lag between changes in magnetization and the applied magnetic field in ferromagnetic materials. When such a material is exposed to a varying external magnetic field, its internal domains reorient, causing the magnetization to follow a path that depends on its history. This phenomenon is critical in applications like transformers and magnetic storage, where energy loss due to hysteresis must be minimized.

Magnetic Susceptibility and Permeability

Magnetic susceptibility ($ \chi_m $) measures how much a material will become magnetized in an applied magnetic field. It is defined as:

$$ \mathbf{M} = \chi_m \mathbf{H} $$

where $ \mathbf{M} $ is the magnetization and $ \mathbf{H} $ is the applied magnetic field strength.

Magnetic permeability ($ \mu $) quantifies a material's ability to support the formation of a magnetic field within itself. It is related to susceptibility by:

$$ \mu = \mu_0 (1 + \chi_m) $$

Magnetic Domains and Domain Theory

Magnetic domains are regions within a ferromagnetic material where the magnetic moments of atoms are aligned uniformly. Domain theory explains how materials become magnetized through the realignment of these domains. Understanding domain behavior is crucial for manipulating magnetic properties in materials science and engineering applications.

Advanced Problem-Solving: Calculating the Magnetic Field of a Solenoid

Consider a solenoid with $ n $ turns per unit length and carrying a current $ I $. The magnetic field inside the solenoid can be derived using the Biot-Savart Law or Ampère's Law.

Using Ampère's Law, the integral form for a solenoid is:

$$ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}} $$>

Choosing an Amperian loop that runs inside and outside the solenoid parallel to its axis, and assuming a long solenoid where edge effects are negligible, the magnetic field inside is uniform and parallel to the axis, while it is negligible outside.

The enclosed current $ I_{\text{enc}} = nL I $, where $ L $ is the length of the solenoid. Therefore:

$$ B \cdot L = \mu_0 n L I $$>

Solving for $ B $, we get:

$$ B = \mu_0 n I $$>

Interdisciplinary Connections: Electromechanical Systems

The principles of magnetic fields extend beyond pure physics into engineering and technology. For instance, in electromechanical systems like electric motors and generators, magnetic fields interact with electric currents to produce mechanical motion or electrical energy. Understanding the behavior of magnetic fields is essential for designing efficient and effective devices in various industries, including automotive, aerospace, and consumer electronics.

Magnetic Field Mapping and Visualization Techniques

Advanced studies involve mapping and visualizing magnetic fields using various techniques:

Magnetic Resonance Imaging (MRI): Utilizes strong magnetic fields and radio waves to create detailed images of the body's internal structures.
Magnetoencephalography (MEG): Measures the magnetic fields produced by neuronal activity in the brain, aiding in neurological research.
Finite Element Analysis (FEA): Computational methods used to model and simulate complex magnetic field distributions in engineering applications.

Relativistic Considerations of Magnetic Fields

From the perspective of special relativity, electric and magnetic fields are interrelated manifestations of the same electromagnetic force. A purely electric field in one inertial frame can appear as a combination of electric and magnetic fields in another frame moving relative to the first. This unification demonstrates the necessity of considering both fields together, especially at high velocities close to the speed of light.

The transformation of electric and magnetic fields between inertial frames is governed by the Lorentz transformations:

$$ B'_\parallel = B_\parallel $$ $$ B'_\perp = \gamma (B_\perp - \frac{v}{c^2} E_\perp) $$ $$ E'_\parallel = E_\parallel $$ $$ E'_\perp = \gamma (E_\perp + v \times B_\perp) $$

where:

$ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} $ is the Lorentz factor
$ v $ is the relative velocity between frames

Quantum Mechanical Perspectives on Magnetic Fields

At the quantum level, magnetic fields interact with particles through phenomena such as the Zeeman effect, where energy levels of atoms split in the presence of a magnetic field. Additionally, the concept of spin, an intrinsic form of angular momentum, gives rise to magnetic moments in elementary particles. Quantum Electrodynamics (QED) further explores these interactions, providing a detailed understanding of how magnetic fields influence particle behavior.

Magnetic Field Energy and Potential

The energy stored in a magnetic field is an important concept in both physics and engineering. The energy density ($ u $) of a magnetic field is given by:

$$ u = \frac{B^2}{2\mu_0} $$>

The total energy ($ U $) stored in a magnetic field within a volume ($ V $) is:

$$ U = \int_V \frac{B^2}{2\mu_0} \, dV $$>

Understanding magnetic energy is critical for designing inductive components, magnetic storage devices, and energy-efficient magnetic systems.

Magnetic Monopoles: Theoretical Considerations

While Gauss's Law for magnetism asserts the non-existence of magnetic monopoles, theoretical physics explores the possibility of their existence. Magnetic monopoles, if discovered, would have profound implications for field theory and symmetry principles. Various grand unified theories and extensions of the Standard Model predict the existence of monopoles, although none have been experimentally observed to date.

Research into magnetic monopoles continues, with ongoing experiments attempting to detect these elusive particles, potentially revolutionizing our understanding of magnetic phenomena.

Comparison Table

Aspect	Moving Charges	Permanent Magnets
Source of Magnetic Field	Electric currents (moving charges)	Aligned magnetic domains in materials
Field Persistence	Requires continuous current to maintain the field	Field remains without external power
Control	Easily adjustable by changing current	Fixed by material properties and magnetization
Applications	Electromagnets, electric motors, generators	Permanent magnets in compasses, MRIs, data storage
Energy Consumption	Consumes energy as long as current flows	No energy required to maintain field

Summary and Key Takeaways

A magnetic field is produced by moving charges or permanent magnets, influencing other charges and dipoles.
Key equations include Ampère's Law, Biot-Savart Law, and the Lorentz force equation.
Advanced concepts cover Maxwell's Equations, electromagnetic waves, and quantum perspectives.
Magnetic fields have diverse applications in technology and interdisciplinary fields.
Understanding magnetic fields is essential for both theoretical studies and practical applications in physics.

Examiner Tip

Tips

1. **Use Mnemonics for Right-Hand Rule:** Remember "Thumb for Current" and "Fingers for Field" to consistently determine the direction of magnetic fields around conductors.

2. **Practice Vector Cross Products:** Regularly solve problems involving the Lorentz force and Biot-Savart Law to become comfortable with vector calculations.

3. **Visualize Field Lines:** Drawing magnetic field lines can help in understanding complex field interactions and enhance retention for exams.

Did You Know

1. The Earth's magnetic field is not perfectly aligned with its rotational axis, causing the magnetic north pole to wander over time.

2. Magnetic fields can be used to levitate trains, as seen in maglev technology, which reduces friction and allows for high-speed travel.

3. Some animals, like migratory birds and sea turtles, use the Earth's magnetic field for navigation during their long journeys.

Common Mistakes

1. **Misapplying the Right-Hand Rule:** Students often confuse the orientation of their fingers and thumb, leading to incorrect magnetic field directions. *Incorrect:* Pointing the thumb in the field direction and curling fingers as current. *Correct:* Thumb in the direction of current and fingers curling around the conductor.

2. **Ignoring the Right Angle in Lorentz Force:** Forgetting that the force is perpendicular to both velocity and magnetic field vectors can result in wrong force calculations. Always ensure the angle between vectors is considered.

3. **Assuming Magnetic Monopoles Exist:** While exploring theoretical concepts, students might incorrectly treat monopoles as existent entities, leading to errors in applying Gauss's Law for magnetism.

FAQ

What is the primary difference between the magnetic fields produced by moving charges and permanent magnets?

Moving charges generate magnetic fields through electric currents, which require continuous energy input, whereas permanent magnets produce static magnetic fields due to the alignment of internal magnetic domains without the need for external power.

How does the right-hand rule help determine the direction of a magnetic field?

The right-hand rule allows you to align your thumb with the direction of current flow in a conductor and curl your fingers to indicate the direction of the resulting magnetic field around the conductor.

Why are there no magnetic monopoles according to Gauss's Law for magnetism?

Gauss's Law for magnetism states that the net magnetic flux through any closed surface is zero, implying that magnetic field lines are continuous loops without a starting or ending point, which means magnetic monopoles do not exist.

What role does the magnetic dipole moment play in magnetic interactions?

The magnetic dipole moment determines the torque a magnet experiences in an external magnetic field and is fundamental in understanding how magnets align and interact with each other.

How is electromagnetic induction utilized in everyday devices?

Electromagnetic induction is the principle behind electric generators, which convert mechanical energy into electrical energy, and transformers, which adjust voltage levels in power transmission.

Can magnetic fields exist in a vacuum?

Yes, magnetic fields can propagate through a vacuum as part of electromagnetic waves, such as light, which consist of oscillating electric and magnetic fields traveling through space.

1. Capacitance

1.1 Energy Stored in a Capacitor

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

1.1.2 Recall and use W = ½QV = ½CV²

1.2 Discharging a Capacitor

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

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

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

1.3 Capacitors and Capacitance

1.3.1 Define capacitance for isolated spherical conductors and parallel plate capacitors

1.3.2 Recall and use C = Q / V

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

1.3.4 Use capacitance formulae for capacitors in series and parallel

2. Nuclear Physics

2.1 Mass Defect and Nuclear Binding Energy

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

2.1.2 Explain nuclear fusion and nuclear fission

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

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

2.1.5 Define and use the terms mass defect and binding energy

2.2 Radioactive Decay

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

2.2.2 Understand that radioactive decay is spontaneous and random

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

2.2.4 Define half-life

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

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

3. Kinematics

3.1 Equations of Motion

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

3.1.2 Use graphical methods to represent motion

3.1.3 Determine displacement from velocity–time graph

3.1.4 Determine velocity using gradient of displacement–time graph

3.1.5 Determine acceleration using gradient of velocity–time graph

3.1.6 Derive equations for uniformly accelerated motion

3.1.7 Solve problems of uniformly accelerated motion

3.1.8 Describe experiment to determine acceleration due to gravity

3.1.9 Explain motion with uniform velocity and perpendicular acceleration

4. Deformation of Solids

4.1 Stress and Strain

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

4.1.2 Recall and use Hooke’s law

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

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

4.1.5 Describe an experiment to determine Young’s modulus

4.1.6 Understand deformation caused by tensile or compressive forces

4.2 Elastic and Plastic Behaviour

4.2.1 Understand elastic and plastic deformation and elastic limit

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

4.2.3 Calculate elastic potential energy using EP = ½kx²

5. Gravitational Fields

5.1 Gravitational Field of a Point Mass

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

5.1.2 Recall and use g = GM / r²

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

5.2 Gravitational Potential

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

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

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

5.3 Gravitational Field

5.3.1 Define gravitational field as force per unit mass

5.3.2 Represent a gravitational field using field lines

5.4 Gravitational Force Between Point Masses

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

5.4.2 Analyze circular orbits in gravitational fields

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

6. Electricity

6.1 Resistance and Resistivity

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

6.1.2 Define resistance and recall and use V = IR

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

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

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

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

6.2 Electric Current

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

6.2.2 Understand that the charge on charge carriers is quantised

6.2.3 Recall and use Q = It

6.2.4 Use I = Anvq for current-carrying conductors

6.3 Potential Difference and Power

6.3.1 Define potential difference as energy transferred per unit charge

6.3.2 Recall and use V = W / Q

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

7. D.C. Circuits

7.1 Practical Circuits

7.1.1 Recall and use the circuit symbols in the syllabus

7.1.2 Draw and interpret circuit diagrams using circuit symbols

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

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

7.1.5 Understand the effects of internal resistance on terminal potential difference

7.2 Kirchhoff’s Laws

7.2.1 Recall Kirchhoff’s first law: conservation of charge

7.2.2 Recall Kirchhoff’s second law: conservation of energy

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

7.2.4 Use the formula for combined resistance of resistors in series

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

7.2.6 Use the formula for combined resistance of resistors in parallel

7.2.7 Use Kirchhoff’s laws to solve simple circuit problems

7.3 Potential Dividers

7.3.1 Understand the principle of a potential divider circuit

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

7.3.3 Understand the use of a galvanometer in null methods

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

8. Thermal Equilibrium

8.1 Thermal Energy Transfer

8.1.1 Understand that thermal energy transfers from higher to lower temperature

8.1.2 Understand that regions of equal temperature are in thermal equilibrium

9. Temperature Scales

9.1 Temperature Measurement

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

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

10. Magnetic Fields

10.1 Concept of a Magnetic Field

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

10.1.2 Represent a magnetic field using field lines

10.2 Force on a Current-Carrying Conductor

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

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

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

10.3 Force on a Moving Charge

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

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

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

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

10.4 Magnetic Fields Due to Currents

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

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

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

10.5 Electromagnetic Induction

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

10.5.2 Recall and use Φ = BA for magnetic flux

10.5.3 Understand and use the concept of magnetic flux linkage

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

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

11. Specific Heat Capacity and Latent Heat

11.1 Specific Heat Capacity

11.1.1 Define and use specific heat capacity

11.2 Latent Heat

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

12. Dynamics

12.1 Momentum and Newton’s Laws of Motion

12.1.1 Mass resists change in motion

12.1.2 Recall F = ma and solve related problems

12.1.3 Define and use linear momentum

12.1.4 Force as rate of change of momentum

12.1.5 State and apply Newton’s laws

12.1.6 Describe weight as effect of gravitational field

12.1.7 Weight of an object is mass × gravitational acceleration

12.2 Non-uniform Motion

12.2.1 Understand frictional and drag forces

12.2.2 Describe motion in uniform gravitational field with air resistance

12.2.3 Objects moving against resistive force may reach terminal velocity

12.3 Linear Momentum and its Conservation

12.3.1 State conservation of momentum

12.3.2 Apply conservation of momentum to solve problems

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

12.3.4 Momentum is conserved, but some kinetic energy may change

13. Medical Physics

13.1 Production and Use of Ultrasound

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

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

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

13.1.4 Define specific acoustic impedance as Z = ρc

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

13.2 Production and Use of X-rays

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

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

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

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

13.3 PET Scanning

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

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

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

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

14. Waves

14.1 Progressive Waves

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

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

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

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

14.1.5 Recall and use v = f λ

14.1.6 Understand energy transfer by a progressive wave

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

14.2 Transverse and Longitudinal Waves

14.2.1 Compare transverse and longitudinal waves

14.2.2 Analyze and interpret graphical representations of transverse and longitudinal waves

14.3 Doppler Effect for Sound Waves

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

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

14.4 Electromagnetic Spectrum

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

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

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

14.5 Polarisation

14.5.1 Understand that polarisation is a phenomenon associated with transverse waves

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

15. The Mole

15.1 The Mole

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

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

16. Equation of State

16.1 Ideal Gas Law

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

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

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

16.1.4 Use the relationship k = R / NA

17. Kinetic Theory of Gases

17.1 Kinetic Theory

17.1.1 State the basic assumptions of the kinetic theory of gases

17.1.2 Explain pressure exerted by a gas due to molecular movement

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

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

18. Particle Physics

18.1 Atoms, Nuclei and Radiation

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

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

18.1.3 Distinguish between nucleon number and proton number

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

18.1.5 Use the notation AZX for nuclides representation

18.1.6 Understand conservation of nucleon number and charge in nuclear processes

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

18.1.8 Understand antiparticles and positrons as antiparticles of electrons

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

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

18.1.11 Use unified atomic mass unit (u) for mass

18.2 Fundamental Particles

18.2.1 Understand quarks as fundamental particles with six flavours

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

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

18.2.4 Describe protons and neutrons as composed of quarks

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

18.2.6 Recall electrons and neutrinos as fundamental particles (leptons)

19. Internal Energy

19.1 Internal Energy

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

19.1.2 Relate rise in temperature to an increase in internal energy

20. Forces, Density and Pressure

20.1 Turning Effects of Forces

20.1.1 Weight acts at a single point: centre of gravity

20.1.2 Define and apply the moment of a force

20.1.3 Understand the concept of a couple

20.1.4 Define and apply torque of a couple

20.2 Equilibrium of Forces

20.2.1 State and apply the principle of moments

20.2.2 System in equilibrium has no resultant force or torque

20.2.3 Use a vector triangle to represent coplanar forces in equilibrium

20.3 Density and Pressure

20.3.1 Define and use density

20.3.2 Define and use pressure

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

20.3.4 Use the equation ∆p = ρg∆h

20.3.5 Understand upthrust as the effect of hydrostatic pressure

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

21. Astronomy and Cosmology

21.1 Standard Candles

21.1.1 Understand luminosity as the total power radiated by a star

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

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

21.1.4 Use standard candles to determine distances to galaxies

21.2 Stellar Radii

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

21.2.2 Use Stefan–Boltzmann law L = 4πσr²T⁴

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

21.3 Hubble’s Law and Big Bang Theory

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

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

21.3.3 Explain why redshift suggests the Universe is expanding

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

22. First Law of Thermodynamics

22.1 First Law

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

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

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

23. Alternating Currents

23.1 Characteristics of Alternating Currents

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

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

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

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

23.2 Rectification and Smoothing

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

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

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

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

24. Superposition

24.1 Stationary Waves

24.1.1 Explain and use the principle of superposition

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

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

24.1.4 Determine wavelength from the positions of nodes or antinodes

24.2 Diffraction

24.2.1 Explain diffraction and show understanding of related experiments

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

24.3 Interference

24.3.1 Understand the terms interference and coherence

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

24.3.3 Understand the conditions required for two-source interference fringes

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

24.4 Diffraction Grating

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

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

25. Simple Harmonic Oscillations

25.1 Simple Harmonic Motion

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

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

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

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

26. Energy in Simple Harmonic Motion

26.1 Energy in SHM

26.1.1 Describe the interchange between kinetic and potential energy during SHM

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

27. Quantum Physics

27.1 Energy and Momentum of a Photon

27.1.1 Understand that electromagnetic radiation has a particulate nature

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

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

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

27.2 Photoelectric Effect

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

27.2.2 Understand and use the terms threshold frequency and threshold wavelength

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

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

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

27.3 Wave-Particle Duality

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

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

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

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

27.4 Energy Levels in Atoms and Line Spectra

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

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

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

28. Damped and Forced Oscillations, Resonance

28.1 Damped Oscillations

28.1.1 Understand that damping occurs due to resistive forces

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

28.2 Resonance

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

29. Work, Energy and Power

29.1 Energy Conservation

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

29.1.2 Apply the principle of conservation of energy

29.1.3 Understand efficiency as useful energy output / total energy input

29.1.4 Use efficiency concept to solve problems

29.1.5 Define power as work done per unit time

29.1.6 Solve problems using P = W / t

29.1.7 Derive P = Fv and solve problems

29.2 Gravitational Potential Energy and Kinetic Energy

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

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

29.2.3 Derive EK = ½mv² for kinetic energy

29.2.4 Recall and use EK = ½mv² for kinetic energy

30. Electric Fields

30.1 Electric Fields and Field Lines

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

30.1.2 Represent an electric field using field lines

30.2 Uniform Electric Fields

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

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

30.3 Electric Force Between Point Charges

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

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

30.4 Electric Field of a Point Charge

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

30.5 Electric Potential

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

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

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

31. Physical Quantities and Units

31.1 Physical Quantities

31.1.1 Physical quantities consist of magnitude and unit

31.1.2 Estimate physical quantities from the syllabus

31.2 SI Units

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

31.2.2 Express derived units from SI base units

31.2.3 Check homogeneity of physical equations

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

31.3 Errors and Uncertainties

31.3.1 Understand systematic and random errors

31.3.2 Assess uncertainty in derived quantities

31.4 uniform Electric Fields

31.4.1 Distinguish between precision and accuracy

31.5 Scalars and Vectors

31.5.1 Difference between scalar and vector quantities

31.5.2 Add and subtract coplanar vectors

31.5.3 Represent a vector as components

32. Motion in a Circle

32.1 Centripetal Acceleration

32.1.1 Understand centripetal acceleration and its role in circular motion

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

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

32.1.4 Understand force causing centripetal acceleration is always perpendicular to motion

32.2 Kinematics of Uniform Circular Motion

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

32.2.2 Understand and use angular speed

32.2.3 Define and express angular displacement in radians

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Understanding Magnetic Fields: Fields of Force Produced by Moving Charges and Permanent Magnets

Introduction

Key Concepts

Definition of a Magnetic Field

Sources of Magnetic Fields

Magnetic Field Due to a Straight Current-Carrying Conductor

Magnetic Field of a Circular Current Loop

Magnetic Fields of Permanent Magnets

Magnetic Flux and Gauss's Law for Magnetism

Magnetic Force on Moving Charges

Biot-Savart Law

Magnetic Dipole Moment

Magnetic Field Lines

Earth's Magnetic Field

Electromagnetic Induction

Applications of Magnetic Fields

Advanced Concepts

Maxwell's Equations and Magnetic Fields

Electromagnetic Waves

Magnetic Hysteresis

Magnetic Susceptibility and Permeability

Magnetic Domains and Domain Theory

Advanced Problem-Solving: Calculating the Magnetic Field of a Solenoid

Interdisciplinary Connections: Electromechanical Systems

Magnetic Field Mapping and Visualization Techniques

Relativistic Considerations of Magnetic Fields

Quantum Mechanical Perspectives on Magnetic Fields

Magnetic Field Energy and Potential

Magnetic Monopoles: Theoretical Considerations

Comparison Table

Summary and Key Takeaways

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