StudySmarter - The all-in-one study app.

4.8 • +11k Ratings

More than 3 Million Downloads

Free

StudySmarter AI is coming soon!

- :00Days
- :00Hours
- :00Mins
- 00Seconds

A new era for learning is coming soonSign up for free

Suggested languages for you:

Americas

Europe

Fields exist everywhere in space, unlike the forces that we use in simple problems of physics where they act only on a certain body. There is useful information we can extract from considering them in extended regions. For both the electric and magnetic fields it is useful to consider flux, which is a measure of the amount of field that…

Content verified by subject matter experts

Free StudySmarter App with over 20 million students

Explore our app and discover over 50 million learning materials for free.

Magnetic Flux and Magnetic Flux Linkage

- Astrophysics
- Absolute Magnitude
- Astronomical Objects
- Astronomical Telescopes
- Black Body Radiation
- Classification by Luminosity
- Classification of Stars
- Cosmology
- Doppler Effect
- Exoplanet Detection
- Hertzsprung-Russell Diagrams
- Hubble's Law
- Large Diameter Telescopes
- Quasars
- Radio Telescopes
- Reflecting Telescopes
- Stellar Spectral Classes
- Telescopes
- Atoms and Radioactivity
- Fission and Fusion
- Medical Tracers
- Nuclear Reactors
- Radiotherapy
- Random Nature of Radioactive Decay
- Thickness Monitoring
- Circular Motion and Gravitation
- Applications of Circular Motion
- Centripetal and Centrifugal Force
- Circular Motion and Free-Body Diagrams
- Fundamental Forces
- Gravitational and Electric Forces
- Gravity on Different Planets
- Inertial and Gravitational Mass
- Vector Fields
- Conservation of Energy and Momentum
- Dynamics
- Application of Newton's Second Law
- Buoyancy
- Drag Force
- Dynamic Systems
- Free Body Diagrams
- Normal Force
- Springs Physics
- Superposition of Forces
- Tension
- Electric Charge Field and Potential
- Charge Distribution
- Charged Particle in Uniform Electric Field
- Conservation of Charge
- Electric Field Between Two Parallel Plates
- Electric Field Lines
- Electric Field of Multiple Point Charges
- Electric Force
- Electric Potential Due to Dipole
- Electric Potential due to a Point Charge
- Electrical Systems
- Equipotential Lines
- Electricity
- Ammeter
- Attraction and Repulsion
- Basics of Electricity
- Batteries
- Capacitors in Series and Parallel
- Circuit Schematic
- Circuit Symbols
- Circuits
- Current Density
- Current-Voltage Characteristics
- DC Circuit
- Electric Current
- Electric Generators
- Electric Motor
- Electrical Power
- Electricity Generation
- Emf and Internal Resistance
- Kirchhoff's Junction Rule
- Kirchhoff's Loop Rule
- National Grid Physics
- Ohm's Law
- Potential Difference
- Power Rating
- RC Circuit
- Resistance
- Resistance and Resistivity
- Resistivity
- Resistors in Series and Parallel
- Series and Parallel Circuits
- Simple Circuit
- Static Electricity
- Superconductivity
- Time Constant of RC Circuit
- Transformer
- Voltage Divider
- Voltmeter
- Electricity and Magnetism
- Benjamin Franklin's Kite Experiment
- Changing Magnetic Field
- Circuit Analysis
- Diamagnetic Levitation
- Electric Dipole
- Electric Field Energy
- Magnets
- Oersted's Experiment
- Voltage
- Electromagnetism
- Electrostatics
- Energy Physics
- Big Energy Issues
- Conservative and Non Conservative Forces
- Efficiency in Physics
- Elastic Potential Energy
- Electrical Energy
- Energy and the Environment
- Forms of Energy
- Geothermal Energy
- Gravitational Potential Energy
- Heat Engines
- Heat Transfer Efficiency
- Kinetic Energy
- Mechanical Power
- Potential Energy
- Potential Energy and Energy Conservation
- Pulling Force
- Renewable Energy Sources
- Wind Energy
- Work Energy Principle
- Engineering Physics
- Angular Momentum
- Angular Work and Power
- Engine Cycles
- First Law of Thermodynamics
- Moment of Inertia
- Non-Flow Processes
- PV Diagrams
- Reversed Heat Engines
- Rotational Kinetic Energy
- Second Law and Engines
- Thermodynamics and Engines
- Torque and Angular Acceleration
- Famous Physicists
- Fields in Physics
- Alternating Currents
- Capacitance
- Capacitor Charge
- Capacitor Discharge
- Coulomb's Law
- Electric Field Strength
- Electric Fields
- Electric Potential
- Electromagnetic Induction
- Energy Stored by a Capacitor
- Equipotential Surface
- Escape Velocity
- Gravitational Field Strength
- Gravitational Fields
- Gravitational Potential
- Magnetic Fields
- Magnetic Flux Density
- Magnetic Flux and Magnetic Flux Linkage
- Moving Charges in a Magnetic Field
- Newton’s Laws
- Operation of a Transformer
- Parallel Plate Capacitor
- Planetary Orbits
- Synchronous Orbits
- Fluids
- Absolute Pressure and Gauge Pressure
- Application of Bernoulli's Equation
- Archimedes' Principle
- Conservation of Energy in Fluids
- Fluid Flow
- Fluid Systems
- Force and Pressure
- Force
- Conservation of Momentum
- Contact Forces
- Elastic Forces
- Force and Motion
- Gravity
- Impact Forces
- Moment Physics
- Moments Levers and Gears
- Moments and Equilibrium
- Pressure
- Resultant Force
- Safety First
- Time Speed and Distance
- Velocity and Acceleration
- Work Done
- Fundamentals of Physics
- Further Mechanics and Thermal Physics
- Bottle Rocket
- Charles law
- Circular Motion
- Diesel Cycle
- Gas Laws
- Heat Transfer
- Heat Transfer Experiments
- Ideal Gas Model
- Ideal Gases
- Kinetic Theory of Gases
- Models of Gas Behaviour
- Newton's Law of Cooling
- Periodic Motion
- Rankine Cycle
- Resonance
- Simple Harmonic Motion
- Simple Harmonic Motion Energy
- Temperature
- Thermal Equilibrium
- Thermal Expansion
- Thermal Physics
- Volume
- Work in Thermodynamics
- Geometrical and Physical Optics
- Kinematics Physics
- Air Resistance
- Angular Kinematic Equations
- Average Velocity and Acceleration
- Displacement, Time and Average Velocity
- Frame of Reference
- Free Falling Object
- Kinematic Equations
- Motion in One Dimension
- Motion in Two Dimensions
- Rotational Motion
- Uniformly Accelerated Motion
- Linear Momentum
- Magnetism
- Ampere force
- Earth's Magnetic Field
- Fleming's Left Hand Rule
- Induced Potential
- Magnetic Forces and Fields
- Motor Effect
- Particles in Magnetic Fields
- Permanent and Induced Magnetism
- Magnetism and Electromagnetic Induction
- Eddy Current
- Faraday's Law
- Induced Currents
- Inductance
- LC Circuit
- Lenz's Law
- Magnetic Field of a Current-Carrying Wire
- Magnetic Flux
- Magnetic Materials
- Monopole vs Dipole
- RL Circuit
- Measurements
- Mechanics and Materials
- Acceleration Due to Gravity
- Bouncing Ball Example
- Bulk Properties of Solids
- Centre of Mass
- Collisions and Momentum Conservation
- Conservation of Energy
- Density
- Elastic Collisions
- Force Energy
- Friction
- Graphs of Motion
- Linear Motion
- Materials
- Materials Energy
- Moments
- Momentum
- Power and Efficiency
- Projectile Motion
- Scalar and Vector
- Terminal Velocity
- Vector Problems
- Work and Energy
- Young's Modulus
- Medical Physics
- Absorption of X-Rays
- CT Scanners
- Defects of Vision
- Defects of Vision and Their Correction
- Diagnostic X-Rays
- Effective Half Life
- Electrocardiography
- Fibre Optics and Endoscopy
- Gamma Camera
- Hearing Defects
- High Energy X-Rays
- Lenses
- Magnetic Resonance Imaging
- Noise Sensitivity
- Non Ionising Imaging
- Physics of Vision
- Physics of the Ear
- Physics of the Eye
- Radioactive Implants
- Radionuclide Imaging Techniques
- Radionuclide Imaging and Therapy
- Structure of the Ear
- Ultrasound Imaging
- X-Ray Image Processing
- X-Ray Imaging
- Modern Physics
- Bohr Model of the Atom
- Disintegration Energy
- Franck Hertz Experiment
- Mass Energy Equivalence
- Nuclear Reaction
- Nucleus Structure
- Quantization of Energy
- Spectral Lines
- The Discovery of the Atom
- Wave Function
- Nuclear Physics
- Alpha Beta and Gamma Radiation
- Binding Energy
- Half Life
- Induced Fission
- Mass and Energy
- Nuclear Instability
- Nuclear Radius
- Radioactive Decay
- Radioactivity
- Rutherford Scattering
- Safety of Nuclear Reactors
- Oscillations
- Energy Time Graph
- Energy in Simple Harmonic Motion
- Hooke's Law
- Kinetic Energy in Simple Harmonic Motion
- Mechanical Energy in Simple Harmonic Motion
- Pendulum
- Period of Pendulum
- Period, Frequency and Amplitude
- Phase Angle
- Physical Pendulum
- Restoring Force
- Simple Pendulum
- Spring-Block Oscillator
- Torsional Pendulum
- Velocity
- Particle Model of Matter
- Physical Quantities and Units
- Converting Units
- Physical Quantities
- SI Prefixes
- Standard Form Physics
- Units Physics
- Use of SI Units
- Physics of Motion
- Acceleration
- Angular Acceleration
- Angular Displacement
- Angular Velocity
- Centrifugal Force
- Centripetal Force
- Displacement
- Equilibrium
- Forces of Nature Physics
- Galileo's Leaning Tower of Pisa Experiment
- Inclined Plane
- Inertia
- Mass in Physics
- Speed Physics
- Static Equilibrium
- Radiation
- Antiparticles
- Antiquark
- Atomic Model
- Classification of Particles
- Collisions of Electrons with Atoms
- Conservation Laws
- Electromagnetic Radiation and Quantum Phenomena
- Isotopes
- Neutron Number
- Particles
- Photons
- Protons
- Quark Physics
- Specific Charge
- The Photoelectric Effect
- Wave-Particle Duality
- Rotational Dynamics
- Angular Impulse
- Angular Kinematics
- Angular Motion and Linear Motion
- Connecting Linear and Rotational Motion
- Orbital Trajectory
- Rotational Equilibrium
- Rotational Inertia
- Satellite Orbits
- Third Law of Kepler
- Scientific Method Physics
- Data Collection
- Data Representation
- Drawing Conclusions
- Equations in Physics
- Uncertainties and Evaluations
- Space Physics
- Thermodynamics
- Heat Radiation
- Thermal Conductivity
- Thermal Efficiency
- Thermodynamic Diagram
- Thermodynamic Force
- Thermodynamic and Kinetic Control
- Torque and Rotational Motion
- Centripetal Acceleration and Centripetal Force
- Conservation of Angular Momentum
- Force and Torque
- Muscle Torque
- Newton's Second Law in Angular Form
- Simple Machines
- Unbalanced Torque
- Translational Dynamics
- Centripetal Force and Velocity
- Critical Speed
- Free Fall and Terminal Velocity
- Gravitational Acceleration
- Kinetic Friction
- Object in Equilibrium
- Orbital Period
- Resistive Force
- Spring Force
- Static Friction
- Turning Points in Physics
- Cathode Rays
- Discovery of the Electron
- Einstein's Theory of Special Relativity
- Electromagnetic Waves
- Electron Microscopes
- Electron Specific Charge
- Length Contraction
- Michelson-Morley Experiment
- Millikan's Experiment
- Newton's and Huygens' Theories of Light
- Photoelectricity
- Relativistic Mass and Energy
- Special Relativity
- Thermionic Electron Emission
- Time Dilation
- Wave Particle Duality of Light
- Waves Physics
- Acoustics
- Applications of Ultrasound
- Applications of Waves
- Diffraction
- Diffraction Gratings
- Doppler Effect in Light
- Earthquake Shock Waves
- Echolocation
- Image Formation by Lenses
- Interference
- Light
- Longitudinal Wave
- Longitudinal and Transverse Waves
- Mirror
- Oscilloscope
- Phase Difference
- Polarisation
- Progressive Waves
- Properties of Waves
- Ray Diagrams
- Ray Tracing Mirrors
- Reflection
- Refraction
- Refraction at a Plane Surface
- Resonance in Sound Waves
- Seismic Waves
- Snell's law
- Spectral Colour
- Standing Waves
- Stationary Waves
- Total Internal Reflection in Optical Fibre
- Transverse Wave
- Ultrasound
- Wave Characteristics
- Wave Speed
- Waves in Communication
- X-rays
- Work Energy and Power
- Conservative Forces and Potential Energy
- Dissipative Force
- Energy Dissipation
- Energy in Pendulum
- Force and Potential Energy
- Force vs. Position Graph
- Orbiting Objects
- Potential Energy Graphs and Motion
- Spring Potential Energy
- Total Mechanical Energy
- Translational Kinetic Energy
- Work Energy Theorem
- Work and Kinetic Energy

Lerne mit deinen Freunden und bleibe auf dem richtigen Kurs mit deinen persönlichen Lernstatistiken

Jetzt kostenlos anmeldenNie wieder prokastinieren mit unseren Lernerinnerungen.

Jetzt kostenlos anmeldenFields exist everywhere in space, unlike the forces that we use in simple problems of physics where they act only on a certain body. There is useful information we can extract from considering them in extended regions. For both the electric and magnetic fields it is useful to consider flux, which is a measure of the amount of field that is crossing a certain surface.

As we already know, magnetic phenomena can be described using a time-dependent field extended in space. We will denote this field by the letter B.

Since the field is extended in space we can actually restrict ourselves to a certain surface and only consider the effect of the magnetic field. As we will see in the following section, Faraday's law concerns magnetic fluxes, so we now present its definition for the case of a uniform magnetic field.

The magnetic flux is the amount of magnetic field that crosses perpendicular to a certain surface.

The magnetic flux can be computed as follows:

\[\Phi = \vec{B} \cdot \vec{A} = |\vec{B}| \cdot |\vec{A}| \cdot \cos(\theta)\]

Here, the dot indicates a scalar product and the vector *A* carries the value of a certain area and is directed in the direction of the normal vector of the surface. The symbol | | indicates the modulus of the vector and *θ* represents the angle between the normal vector and the magnetic field vector. See the image below for clarification:

Orientation-dependent magnetic flux through a flat surface. www.physicsbootcamp.org

In complex settings, the magnetic field is not uniform and the surface is not flat (which leads to the use of integrals and characterisations that are out of the scope of this article). We will only consider flat surfaces and uniform magnetic fields. This will result in a dependence of the magnetic flux on the angle between the magnetic field and the surface.

Faraday's law is an experimental law that was later mathematically formalised and incorporated as part of what we now know as Maxwell's laws. It relates a concept from the electric field, the potential difference, with magnetic flux.

In particular, it relates the electromotive force (EMF) to the rate of change of magnetic flux. The electromotive force is the energy needed per unit of charge to establish a certain electric potential difference between two points and is usually denoted by the letter *ε*.

The mathematical description of Faraday's law is:

\[\varepsilon = - \frac{d \phi}{dt}\]

where there is a derivation with respect to the time of the flux. Although this description is very general, if we restrict ourselves to the aforementioned case of uniform magnetic field and a fixed area, we arrive, due to the expression of the scalar product, at the following equation:

\[\varepsilon = \omega \cdot |\vec{B}| \cdot |\vec{A}| \cdot \sin(\theta)\]

where *ω* is the angular velocity of the changing angle. The image below is an experimental setup for producing an electromotive force using a certain moving surface and a uniform magnetic field.

Experimental set-up for Faraday's law. openpress.usask.ca

The equations that govern the behaviour of the electromagnetic field (Maxwell's laws) are linear, which means that we can consider the superposition of different fields that fulfill the same equations. If we are considering an experimental setup that generates an electromotive force, a simple quantity can help in increasing the output of electromotive force; this is what we call linkage. The magnetic flux linkage is measured in units of Webers \(\mathrm{Wb}\) just like magnetic flux.

Imagine the setting we had before: a coil rotating in the presence of a magnetic field. The variation of magnetic flux induces an electromotive force. If we now take the same setting with *N* coils, we can create *N* different surfaces so the electromotive force is multiplied by a factor of *N*. This is what we call flux linkage.

The mathematical description of flux linkage is based on Faraday's law. Again, since we are considering simple settings, we'll restrict ourselves to the case where we have *N* identical coils and this number remains constant. Furthermore, they are all synchronised and have the same three-dimensional orientation. This leads to the following increase of the flux:

\[\phi_L = N \cdot \phi \Rightarrow \varepsilon_L = N \cdot \varepsilon\]

where ϕ_{L} is the total magnetic flux linkage resulting from *N* coils and ε_{L} is the total electromotive force associated. Combining this with Faraday's law gives us the equation for magnetic flux linkage\[\phi_L=N\cdot|B|\cdot|A|\cdot\sin\left(\theta\right)\]

By doing this, we can manage to increase the potential difference with a simple addition of similar coils we can connect to the same circuit setup.

We are now going to consider several examples of experimental setups. The magnetic field present has a value of 10 Teslas, while the area of the coils we are using is 1 m^{2}. We are rotating the coil with an angular velocity of 2 rad/s.

Imagine the magnetic field is directed in the x-axis, that is:

\(\vec{B} = (10,0,0)T\)

On the other hand, the normal vector evolves in the following way:

\(\vec{A} = (\cos(2 \cdot t), \sin(2 \cdot t), 0) m^2\)

where *t* is the time. This yields the following expression for the magnetic flux:

\(\phi = \vec{B} \cdot \vec{A} = 10 \cdot \cos(2 \cdot t) Wb\)

This allows us to easily compute:

\(\varepsilon = - \frac{d \phi}{dt} = - \frac{d}{dt} (10 \cdot \cos(2 \cdot t)) = 20 \cdot \sin(2 \cdot t) V\)

Below you will find a graph showing the time evolution of the magnetic flux and the generated electromotive force.

Temporal evolution of the magnetic flux (red) and the electromotive force (blue).

If we had managed to increase the magnetic field or make the surface of the coil bigger, we could also have generated an electromotive force, since we are varying the magnetic flux over time.

If we now consider 20 identical coils rotating synchronously, the graph of the time dependence for the magnetic flux density and the electromotive force would look like this:

Comparison between a set-up with 1 coil and with 20 coils. In the horizontal axis, time is represented and in the vertical axis, the electromotive force

We see here that the values of the total flux (and, then, of the EMF) have significantly grown by using only 19 extra coils.

We briefly turn to the case now for a static surface and a varying magnetic field. If now the field starts with an initial value of 0 Teslas, but continues to grow with time in the following manner:

\(\vec{B} = (10 \cdot t, 0,0) T\)

Consider a surface whose normal vector is:

\(\vec{A} = (1,0,0)m^2\)

We should arrive at the following expression for the magnetic flux:

\(\phi = \vec{B} \cdot \vec{A} = 10 \cdot t \space Wb\)

This expression's time derivative gives the expression for the electromotive force, that is:

\(\varepsilon = -\frac{d \phi}{dt} = - \frac{d}{dt}(10 \cdot t) = -10 \space V\)

It would generate a constant electromotive force between the extreme points of the coil. Of course, we could use several coils to build a magnetic flux linkage and increase the output.

In fact, when we use several coils, it is usual to vary the magnetic field and not the orientation to generate an electromotive force. This is the reason why we usually associate the concept of magnetic flux and Faraday's law with only one rotating coil, while the concept of flux linkage usually designates several static coils in the presence of a magnetic field.

- Magnetic flux is a quantity that measures the amount of magnetic field crossing perpendicular to a certain surface.
- Faraday's law establishes a relationship between a force creating a difference in electromagnetic potential and the variation of magnetic flux over time.
- Faraday's law applies whenever at least one of these three varies over time: the intensity of the magnetic field, the area it goes through, or the orientation of the surface with respect to the field.
- The situation where there are several coils through which a magnetic field goes is called flux linkage. The flux increases proportionally.

*B.* The magnetic flux linkage is the growth of magnetic flux by having different surfaces crossed by a magnetic field.

_{L}=N⋅Φ, where Φ is the magnetic flux and N is the number of coils.

Magnetic flux linkage is measured in Webers (Wb).

More about Magnetic Flux and Magnetic Flux Linkage

60%

of the users don't pass the Magnetic Flux and Magnetic Flux Linkage quiz! Will you pass the quiz?

Start QuizHow would you like to learn this content?

Creating flashcards

Studying with content from your peer

Taking a short quiz

94% of StudySmarter users achieve better grades.

Sign up for free!94% of StudySmarter users achieve better grades.

Sign up for free!How would you like to learn this content?

Creating flashcards

Studying with content from your peer

Taking a short quiz

Free physics cheat sheet!

Everything you need to know on . A perfect summary so you can easily remember everything.

Be perfectly prepared on time with an individual plan.

Test your knowledge with gamified quizzes.

Create and find flashcards in record time.

Create beautiful notes faster than ever before.

Have all your study materials in one place.

Upload unlimited documents and save them online.

Identify your study strength and weaknesses.

Set individual study goals and earn points reaching them.

Stop procrastinating with our study reminders.

Earn points, unlock badges and level up while studying.

Create flashcards in notes completely automatically.

Create the most beautiful study materials using our templates.

Sign up to highlight and take notes. It’s 100% free.