All Topics
physics-9702 | as-a-level
Your Flashcards are Ready!
15 Flashcards in this deck.
1.
Capacitance
2.
Nuclear Physics
3.
Kinematics
4.
Deformation of Solids
5.
Gravitational Fields
6.
Electricity
7.
D.C. Circuits
8.
Thermal Equilibrium
9.
Temperature Scales
10.
Magnetic Fields
11.
Specific Heat Capacity and Latent Heat
12.
Dynamics
13.
Medical Physics
14.
Waves
15.
The Mole
16.
Equation of State
17.
Kinetic Theory of Gases
18.
Particle Physics
19.
Internal Energy
20.
Forces, Density and Pressure
21.
Astronomy and Cosmology
22.
First Law of Thermodynamics
23.
Alternating Currents
24.
Superposition
25.
Simple Harmonic Oscillations
26.
Energy in Simple Harmonic Motion
27.
Quantum Physics
28.
Damped and Forced Oscillations, Resonance
29.
Work, Energy and Power
30.
Electric Fields
31.
Physical Quantities and Units
32.
Motion in a Circle
Define magnetic flux as the product of magnetic flux density and area perpendicular to flux directio
Topic 2/3
or
3
Still Learning
I know
12
Previous
Next
Magnetic Flux: The Product of Magnetic Flux Density and Area Perpendicular to Flux Direction
Introduction
Magnetic flux is a fundamental concept in the study of electromagnetic induction, pivotal to understanding how magnetic fields interact with materials. For students enrolled in the AS & A Level Physics course (9702), grasping the definition and implications of magnetic flux as the product of magnetic flux density and the area perpendicular to the flux direction is essential. This concept not only forms the basis for numerous applications in physics and engineering but also underpins key phenomena such as Faraday's Law of Induction and the operation of electric generators.
Key Concepts
Definition of Magnetic Flux
Magnetic flux ($\Phi$) quantifies the total magnetic field passing through a given area. It is defined as the product of the magnetic flux density ($B$) and the area ($A$) perpendicular to the direction of the magnetic field.
Mathematically, magnetic flux is expressed as: $$ \Phi = B \cdot A \cdot \cos(\theta) $$ where:
- $\Phi$ = Magnetic flux (Weber, Wb)
- $B$ = Magnetic flux density (Tesla, T)
- $A$ = Area perpendicular to the magnetic field (square meters, m²)
- $\theta$ = Angle between the magnetic field direction and the normal to the surface
In cases where the area is perpendicular to the magnetic field, $\theta = 0^\circ$ and $\cos(0^\circ) = 1$, simplifying the equation to:
$$ \Phi = B \cdot A $$Units and Dimensions
The unit of magnetic flux is the Weber (Wb), where:
$$ 1 \text{ Wb} = 1 \text{ Tesla} \cdot 1 \text{ m}^2 $$Breaking it down into fundamental SI units:
$$ 1 \text{ Wb} = 1 \frac{\text{kg} \cdot \text{m}}{\text{s}^2 \cdot \text{A}} $$Understanding Magnetic Flux Density
Magnetic flux density ($B$), also known simply as the magnetic field, represents the strength and direction of the magnetic field at a particular point. It is a vector quantity, having both magnitude and direction. The area ($A$) through which the magnetic field lines pass plays a crucial role in determining the magnetic flux.
Calculating Magnetic Flux
To calculate magnetic flux, one must consider both the magnitude of the magnetic field and the area through which it passes. The orientation of the area relative to the magnetic field direction significantly affects the flux.
For an area positioned at an angle $\theta$ to the magnetic field, the flux is given by:
$$ \Phi = B \cdot A \cdot \cos(\theta) $$**Example 1:** If a magnetic field of $2 \, \text{T}$ passes perpendicularly through an area of $3 \, \text{m}^2$, the magnetic flux is:
$$ \Phi = 2 \, \text{T} \times 3 \, \text{m}^2 = 6 \, \text{Wb} $$>**Example 2:** If the same magnetic field passes through the same area at an angle of $60^\circ$, the magnetic flux becomes:
$$ \Phi = 2 \, \text{T} \times 3 \, \text{m}^2 \times \cos(60^\circ) = 2 \times 3 \times 0.5 = 3 \, \text{Wb} $$>Magnetic Flux and Magnetic Field Lines
Magnetic flux is intrinsically linked to the concept of magnetic field lines, which visually represent the direction and strength of a magnetic field. The density of these lines indicates the magnetic flux density; areas with closely spaced lines exhibit higher flux density, leading to greater magnetic flux through a given area.
Magnetic Flux in Different Geometries
The calculation of magnetic flux varies with the geometry of the area through which the magnetic field passes.
- Flat Surface: For a flat surface entirely perpendicular to the magnetic field, the flux is maximum and given by $\Phi = B \cdot A$.
- Inclined Surface: For a surface inclined at an angle $\theta$ to the magnetic field, the flux is reduced by the factor $\cos(\theta)$, i.e., $\Phi = B \cdot A \cdot \cos(\theta)$.
- Curved Surface: For a curved surface, the flux calculation involves integrating the magnetic flux density over the entire surface area.
Faraday’s Law of Electromagnetic Induction
Understanding magnetic flux is essential for comprehending Faraday's Law, which states that a change in magnetic flux through a circuit induces an electromotive force (EMF) in the circuit. Mathematically, it's given by:
$$ \mathcal{E} = -\frac{d\Phi}{dt} $$>Where $\mathcal{E}$ is the induced EMF and $\frac{d\Phi}{dt}$ is the rate of change of magnetic flux. This principle is fundamental in the operation of transformers, electric generators, and inductors.
Gauss’s Law for Magnetism
Gauss’s Law for Magnetism states that the net magnetic flux through any closed surface is zero, implying that there are no magnetic monopoles. This is expressed as:
$$ \oint_{S} \mathbf{B} \cdot d\mathbf{A} = 0 $$>This means that magnetic field lines neither start nor end but form continuous loops.
Applications of Magnetic Flux
Magnetic flux has diverse applications across various fields:
- Electric Generators: Convert mechanical energy into electrical energy by rotating coils within a magnetic field, thereby changing the magnetic flux.
- Transformers: Transfer electrical energy between circuits through varying magnetic flux in the transformer core.
- Magnetic Storage: Store data in devices like hard drives using variations in magnetic flux.
- Inductive Charging: Utilizes changing magnetic flux to transfer energy wirelessly to devices.
Magnetic Flux in Electromagnetic Waves
In electromagnetic waves, changing magnetic flux is fundamental to the propagation of the wave. A time-varying magnetic field induces an electric field, and vice versa, allowing the wave to sustain itself as it travels through space.
Magnetic Flux and Inductance
Inductance is a property of an electrical conductor by which a change in current induces an EMF, quantified by the change in magnetic flux. The relationship is given by:
$$ \mathcal{E} = -L \frac{dI}{dt} $$>Where $L$ is the inductance and $\frac{dI}{dt}$ is the rate of change of current. High inductance means that a significant change in magnetic flux occurs for a given change in current, which is crucial for applications like inductors and transformers.
Magnetic Flux Quantization
In superconductors, magnetic flux becomes quantized, meaning it can only take on discrete values. This phenomenon is critical in the operation of devices like SQUIDs (Superconducting Quantum Interference Devices), which are used for sensitive magnetic field measurements.
Magnetic Flux and Lorentz Force
The interaction between magnetic flux and moving charges leads to the Lorentz force, which is the force exerted on a charged particle moving through a magnetic field. This force is pivotal in the operation of devices such as cyclotrons and mass spectrometers.
Magnetic Flux Density vs. Magnetic Field Strength
While often used interchangeably, magnetic flux density ($B$) and magnetic field strength ($H$) are related but distinct quantities. They are connected through the equation:
$$ B = \mu H $$>Where $\mu$ is the permeability of the medium, indicating how easily a material can support the formation of a magnetic field within itself.
Magnetic Flux in Electromagnetic Induction
Magnetic flux plays a central role in electromagnetic induction, where a change in flux induces an electromotive force (EMF) in a conductor. This principle is harnessed in the creation of electric currents in generators and the operation of transformers.
Magnetic Flux and Energy Storage
Magnetic flux is instrumental in energy storage systems such as inductive coils and transformers. The energy stored in a magnetic field is proportional to the square of the magnetic flux density, given by:
$$ E = \frac{1}{2} L I^2 = \frac{1}{2} \frac{\Phi^2}{L} $$>Where $E$ is the energy, $L$ is inductance, and $I$ is current.
Advanced Concepts
Mathematical Derivation of Magnetic Flux
To derive the expression for magnetic flux, consider a uniform magnetic field passing through a flat surface. Let the angle between the magnetic field and the perpendicular (normal) to the surface be $\theta$. The component of the magnetic field perpendicular to the surface is $B \cdot \cos(\theta)$. Therefore, the magnetic flux is:
$$ \Phi = B_\perp \cdot A = B \cdot A \cdot \cos(\theta) $$>For a non-uniform magnetic field, the total magnetic flux is obtained by integrating the magnetic flux density over the entire area:
$$ \Phi = \int_{A} \mathbf{B} \cdot d\mathbf{A} $$>This integral accounts for variations in both the magnitude and direction of the magnetic field across the surface.
Flux Through a Loop: Integral Formulation
Consider a loop of wire placed in a magnetic field. The magnetic flux through the loop is the integral of the magnetic flux density over the area enclosed by the loop. If the loop is in three-dimensional space with varying orientation, the flux calculation must consider the orientation of each infinitesimal area element.
$$ \Phi = \int_{S} \mathbf{B} \cdot d\mathbf{A} = \int_{S} B \cdot \cos(\theta) \cdot dA $$>Where:
- $\mathbf{B}$ = Magnetic flux density vector
- $d\mathbf{A}$ = Infinitesimal area vector
- $\theta$ = Angle between $\mathbf{B}$ and $d\mathbf{A}$
Magnetic Flux in Solenoids
A solenoid generates a uniform magnetic field within its core when an electric current passes through it. The magnetic flux through a solenoid can be calculated using the formula:
$$ \Phi = B \cdot A = \mu_0 n I \cdot \pi r^2 $$>Where:
- $\mu_0$ = Permeability of free space ($4\pi \times 10^{-7} \, \text{T}\cdot\text{m/A}$)
- $n$ = Number of turns per unit length
- $r$ = Radius of the solenoid
Variable Magnetic Fields and Time Dependence
When the magnetic flux through a surface changes with time, it induces an electromotive force (EMF) according to Faraday's Law. This time dependence is crucial for alternating current (AC) systems and electromagnetic wave propagation.
$$ \mathcal{E} = -\frac{d\Phi}{dt} $$>The negative sign indicates Lenz's Law, which states that the induced EMF and the resulting current will oppose the change in flux that produced them.
Electromagnetic Induction in Transformers
Transformers operate based on the principle of electromagnetic induction, utilizing two coils: the primary and the secondary. A varying current in the primary coil creates a changing magnetic flux, which induces an EMF in the secondary coil. The relationship between the number of turns in the coils and the induced voltages is given by:
$$ \frac{V_s}{V_p} = \frac{N_s}{N_p} $$>Where:
- $V_s$ = Secondary voltage
- $V_p$ = Primary voltage
- $N_s$ = Number of turns in the secondary coil
- $N_p$ = Number of turns in the primary coil
Maxwell’s Equations and Magnetic Flux
Magnetic flux is integral to Maxwell’s equations, which are the foundation of classical electromagnetism. Specifically, Gauss’s Law for Magnetism and Faraday’s Law involve magnetic flux in integral forms:
- Gauss’s Law for Magnetism: $$ \oint_{S} \mathbf{B} \cdot d\mathbf{A} = 0 $$
- Faraday’s Law of Induction: $$ \oint_{C} \mathbf{E} \cdot d\mathbf{l} = -\frac{d}{dt} \int_{S} \mathbf{B} \cdot d\mathbf{A} $$
These equations describe how electric and magnetic fields are interrelated and how they generate each other under changing conditions.
Magnetic Flux Density in Different Materials
The magnetic flux density varies depending on the medium through which the magnetic field passes. Materials are characterized by their permeability ($\mu$), which affects the relationship between $B$ and $H$ fields. High permeability materials, such as ferromagnets, significantly increase the magnetic flux density within them.
Flux Linkage in Multiple Loops
In coils with multiple turns, the total magnetic flux, or flux linkage ($\Lambda$), is the product of the magnetic flux through a single loop and the number of turns ($N$). It is given by:
$$ \Lambda = N \cdot \Phi = N \cdot B \cdot A $$>This concept is essential in applications like transformers and inductors, where multiple turns amplify the induced EMF.
Importance of Magnetic Flux in Magnetic Circuits
Magnetic flux is a key parameter in designing magnetic circuits, analogous to electric circuits. It helps in analyzing and optimizing components like cores, air gaps, and inductors to achieve desired magnetic properties and performance.
Magnetic Flux and Relativity
In the context of relativity, magnetic flux is part of the electromagnetic tensor, unifying electric and magnetic fields into a single framework. This unification is crucial for understanding electromagnetic phenomena at high velocities and in different inertial frames.
Magnetic Flux Quantization in Superconductors
In superconductors, magnetic flux can only take on discrete values, known as flux quanta. This quantization is described by:
$$ \Phi = n \cdot \Phi_0 $$>Where:
- $n$ = Integer
- $\Phi_0$ = Magnetic flux quantum ($2.067 \times 10^{-15} \, \text{Wb}$)
This phenomenon is the basis for the operation of superconducting quantum interference devices (SQUIDs), used for extremely sensitive magnetic field measurements.
Magnetic Flux and Energy Conservation
Changes in magnetic flux are governed by the principle of energy conservation. The induced EMF resulting from a changing flux does work on charges, converting magnetic energy into electrical energy or vice versa.
Non-uniform Magnetic Fields and Flux Calculations
In cases where the magnetic field is non-uniform across the area, flux calculations involve integrating the product of the magnetic flux density and the differential area element:
$$ \Phi = \int_{A} B(x, y, z) \cdot dA $$>This requires knowledge of the spatial variation of the magnetic field, often necessitating the use of calculus for precise determination.
Magnetic Flux and Induced Currents in Conductors
When a conductor moves through a magnetic field or experiences a change in magnetic flux, an induced current is generated. This principle is exploited in electric generators, where mechanical motion induces electrical energy.
Faraday’s Law in Moving Systems
In systems where the conducting loop is in motion, the rate of change of magnetic flux includes contributions from both the change in the magnetic field and the movement of the loop through the field. The induced EMF is given by:
$$ \mathcal{E} = - \frac{d\Phi}{dt} = - \left( \frac{\partial \Phi}{\partial t} + \mathbf{v} \times \mathbf{B} \cdot \mathbf{l} \right) $$>Where $\mathbf{v}$ is the velocity of the loop and $\mathbf{l}$ is the length of the conductor.
Magnetic Flux in Alternating Current (AC) Systems
In AC systems, the magnetic flux varies sinusoidally with time, leading to alternating EMFs and currents. The relationship between flux and voltage in such systems is essential for understanding power generation and distribution.
$$ \Phi(t) = \Phi_0 \sin(\omega t) $$>Where:
- $\Phi_0$ = Maximum flux
- $\omega$ = Angular frequency ($2\pi f$)
- $f$ = Frequency of the AC supply
The induced EMF is then:
$$ \mathcal{E}(t) = -\frac{d\Phi}{dt} = -\Phi_0 \omega \cos(\omega t) $$>Magnetic Flux and Electromagnetic Waves
Electromagnetic waves consist of oscillating electric and magnetic fields, with the changing magnetic flux contributing to the propagation of the wave. Maxwell's equations describe this interplay, highlighting the inseparable nature of electric and magnetic fluxes in wave propagation.
Fluxtronics: Advanced Applications
Fluxtronics is an emerging field that leverages the quantization of magnetic flux in superconductors for applications in quantum computing and ultra-sensitive magnetic sensors. It relies on precise control and measurement of magnetic flux quanta.
Magnetic Flux and Magnetic Resonance Imaging (MRI)
In MRI technology, magnetic flux gradients are applied to spatially encode the positions of protons in the body, enabling detailed imaging of internal structures. The precise control of magnetic flux is critical for the resolution and accuracy of MRI scans.
Interdisciplinary Connections: Magnetic Flux in Engineering and Technology
Magnetic flux intersects with various engineering disciplines, particularly in electrical and mechanical engineering. Its principles are applied in designing electric motors, generators, transformers, and inductive heating systems, showcasing its broad relevance beyond theoretical physics.
Challenging Problem: Calculating Flux in a Non-Uniform Field
Problem: A circular loop of radius $r$ lies in the xy-plane. A magnetic field varying with height $z$ is given by $B(z) = B_0 e^{-z^2/a^2} \hat{k}$. Calculate the total magnetic flux through the loop.
Solution:
The loop lies in the xy-plane, so each infinitesimal area element $dA = r dr d\theta \hat{k}$. The flux is:
$$ \Phi = \int_{0}^{2\pi} \int_{0}^{r} B(z) dA = \int_{0}^{2\pi} \int_{0}^{r} B_0 e^{-z^2/a^2} r dr d\theta $$>Since the magnetic field varies with $z$ and the loop is in the xy-plane ($z=0$), the flux simplifies to:
$$ \Phi = \int_{0}^{2\pi} \int_{0}^{r} B_0 e^{0} r dr d\theta = B_0 \int_{0}^{2\pi} d\theta \int_{0}^{r} r dr $$> $$ \Phi = B_0 \cdot 2\pi \cdot \left[ \frac{r^2}{2} \right]_{0}^{r} = B_0 \cdot 2\pi \cdot \frac{r^2}{2} = \pi r^2 B_0 $$>Answer: The total magnetic flux through the loop is $\Phi = \pi r^2 B_0$.
Advanced Problem: Induced EMF in a Moving Loop
Problem: A rectangular loop of width $w$ and height $h$ is moving with a constant velocity $v$ perpendicular to a uniform magnetic field $B$. Calculate the induced EMF in the loop as a function of time.
Solution:
As the loop moves through the magnetic field, the area exposed to the field changes. The magnetic flux through the loop at time $t$ is:
$$ \Phi(t) = B \cdot A(t) = B \cdot (w \cdot (vt)) $$>Thus, the flux is:
$$ \Phi(t) = Bwtv $$>The induced EMF is the negative rate of change of flux:
$$ \mathcal{E} = -\frac{d\Phi}{dt} = -\frac{d}{dt}(Bwtv) = -Bwv $$>Answer: The induced EMF in the loop is constant and equal to $\mathcal{E} = -B w v$.
Interdisciplinary Connections: Magnetic Flux in Electrical Engineering
In electrical engineering, magnetic flux is crucial in the design and operation of inductors and transformers. The efficiency of power transfer in transformers depends on maximizing magnetic flux linkage between primary and secondary coils. Additionally, magnetic flux considerations are essential in the development of electric motors, where flux variations result in mechanical motion.
Energy Stored in Magnetic Fields
The energy ($E$) stored in a magnetic field is related to the magnetic flux density and the volume of the space through which the flux passes. For an inductor, the energy is given by:
$$ E = \frac{1}{2} L I^2 = \frac{1}{2} \frac{\Phi^2}{L} $$>Where $L$ is inductance and $I$ is current. This relationship highlights the role of magnetic flux in energy storage mechanisms within electrical circuits.
Maxwell’s Addition to Ampère’s Law
Maxwell introduced the concept of displacement current to Ampère’s Law, allowing for a complete description of electromagnetic waves. This addition considers the rate of change of electric flux, which is complementary to the behavior of magnetic flux in sustaining wave propagation.
$$ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 (I + \epsilon_0 \frac{d\Phi_E}{dt}) $$>This formulation ensures the continuity of electromagnetic waves, emphasizing the interplay between electric and magnetic fluxes.
Magnetic Flux and Quantum Mechanics
In quantum mechanics, magnetic flux plays a role in phenomena such as the Aharonov-Bohm effect, where charged particles are affected by the electromagnetic potential in regions where the magnetic field is zero. This effect underscores the fundamental significance of magnetic flux in the quantum realm.
Advanced Simulation of Magnetic Flux
Modern physics employs computational simulations to model complex magnetic flux scenarios, especially in materials with non-linear magnetic properties or intricate geometries. Finite Element Analysis (FEA) is commonly used to solve Maxwell’s equations numerically, providing detailed insights into magnetic flux distribution and behavior.
Magnetic Flux in Relativistic Electrodynamics
In relativistic contexts, magnetic flux transforms under Lorentz transformations. An observer moving relative to a magnetic field may perceive a different electric and magnetic field distribution, illustrating the relativistic interdependence of electric and magnetic fluxes.
Applications in Renewable Energy
Magnetic flux principles are integral to renewable energy technologies, such as wind turbines and hydroelectric generators. The conversion of mechanical energy to electrical energy relies on changing magnetic flux through rotating coils, demonstrating the practical utility of magnetic flux in sustainable energy solutions.
Comparison Table
| Aspect | Magnetic Flux Density ($B$) | Magnetic Flux ($\Phi$) |
|---|---|---|
| Definition | Measure of the strength and direction of the magnetic field at a point. | Total magnetic field passing through a given area. |
| Units | Tesla (T) | Weber (Wb) |
| Formula | $B = \frac{\Phi}{A}$ | $\Phi = B \cdot A \cdot \cos(\theta)$ |
| Vector Quantity | Yes | No |
| Dependent On | Position in the magnetic field | Magnetic field and area orientation |
| Applications | Design of magnetic materials, sensors | Electric generators, transformers, inductors |
Summary and Key Takeaways
- Magnetic flux ($\Phi$) is the product of magnetic flux density ($B$) and the area ($A$) perpendicular to the magnetic field direction.
- It is measured in Webers (Wb) and is integral to understanding electromagnetic induction and Faraday’s Law.
- Magnetic flux plays a critical role in the operation of electric generators, transformers, and various electromagnetic devices.
- Advanced applications include MRI technology, quantum computing, and renewable energy systems.
- Understanding the relationship between $B$, $A$, and $\theta$ is essential for calculating and manipulating magnetic flux in diverse scenarios.
Coming Soon!
Examiner Tip
Tips
Use the mnemonic "BAcosθ" to remember the formula for magnetic flux: $\Phi = B \cdot A \cdot \cos(\theta)$. Visualize magnetic field lines passing through an area to better grasp flux orientation and magnitude.
Did You Know
Did You Know
1. The concept of magnetic flux was pivotal in the development of the first electric generators during the Industrial Revolution, enabling the widespread use of electricity.
2. Magnetic flux quantization in superconductors allows for the creation of extremely sensitive magnetometers, such as SQUIDs, which can detect minute magnetic fields from the human brain.
3. Earth's magnetic flux plays a crucial role in protecting the planet from solar wind, maintaining the integrity of our atmosphere.
Common Mistakes
Common Mistakes
1. **Incorrect Angle Usage:** Students often forget to account for the angle ($\theta$) between the magnetic field and the area. Always use $\Phi = B \cdot A \cdot \cos(\theta)$.
2. **Confusing Units:** Mixing up units of $B$ (Tesla) and $\Phi$ (Weber) can lead to calculation errors. Ensure consistent unit usage.
3. **Ignoring Area Orientation:** Assuming the area is always perpendicular to the magnetic field can simplify problems incorrectly. Always assess the orientation.
FAQ
What is magnetic flux?
Magnetic flux ($\Phi$) is the measure of the total magnetic field passing through a given area, calculated as the product of magnetic flux density ($B$) and the area perpendicular to the field.
How is magnetic flux density different from magnetic flux?
Magnetic flux density ($B$) refers to the strength and direction of the magnetic field at a point, while magnetic flux ($\Phi$) quantifies the total magnetic field passing through an area.
Why is the cosine of the angle important in the magnetic flux formula?
The cosine of the angle ($\theta$) accounts for the orientation of the area relative to the magnetic field, ensuring that only the component of the field perpendicular to the area contributes to the flux.
What is the unit of magnetic flux?
The unit of magnetic flux is the Weber (Wb).
How does magnetic flux relate to Faraday’s Law of Induction?
Faraday’s Law states that a change in magnetic flux through a circuit induces an electromotive force (EMF) in the circuit, driving electric current.
1.
Capacitance
2.
Nuclear Physics
3.
Kinematics
4.
Deformation of Solids
5.
Gravitational Fields
6.
Electricity
7.
D.C. Circuits
8.
Thermal Equilibrium
9.
Temperature Scales
10.
Magnetic Fields
11.
Specific Heat Capacity and Latent Heat
12.
Dynamics
13.
Medical Physics
14.
Waves
15.
The Mole
16.
Equation of State
17.
Kinetic Theory of Gases
18.
Particle Physics
19.
Internal Energy
20.
Forces, Density and Pressure
21.
Astronomy and Cosmology
22.
First Law of Thermodynamics
23.
Alternating Currents
24.
Superposition
25.
Simple Harmonic Oscillations
26.
Energy in Simple Harmonic Motion
27.
Quantum Physics
28.
Damped and Forced Oscillations, Resonance
29.
Work, Energy and Power
30.
Electric Fields
31.
Physical Quantities and Units
32.
Motion in a Circle
Get PDF
How would you like to practise?
