Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Electromagnetism → Electromagnetic induction ↓


Faraday's Law


Introduction

Faraday's law is a fundamental principle of electromagnetism that describes how electric currents are produced by changes in magnetic fields. This law is the cornerstone of electromagnetic theory and is important for understanding how various electrical devices work, such as transformers, electric generators, and inductors.

Basic Concept

According to Faraday's law, a change in the magnetic environment of a coil of wire will induce an electromotive force (EMF) in the coil. If the circuit is closed, the induced EMF drives the current, and this process is called electromagnetic induction.

Mathematically, Faraday's law is described by the following equation:

EMF = -dΦB/dt

Here, EMF is the electromotive force in volts, and ΦB is the magnetic flux in webers. The negative sign indicates the direction of the induced EMF and is based on Lenz's law, which states that the induced EMF will oppose the change that produced it.

Magnetic flux

Magnetic flux, denoted by ΦB, is a measure of the magnitude of the magnetic field passing through a given area. It is calculated as:

ΦB = B * A * cos(θ)

Where:

  • B is the magnetic field strength in Tesla.
  • A is the area in square metres through which the field lines pass.
  • θ is the angle between the field lines and the perpendicular to the surface.

The magnetic flux depends on the strength of the magnetic field, the size of the region, and the orientation angle of the region relative to the magnetic field.

Visualization of magnetic flux

B A θ

This diagram above shows the region of space representing the magnetic field vectors, with B being the magnetic field and A being the area the lines pass through. The angle θ is between the field and the perpendicular to the surface (A), which affects the magnetic flux.

Faraday's law and induced emf

Electromagnetic induction, as described by Faraday's law, is about how a change in magnetic flux leads to the generation of an electrical driving force. To see this, consider a simple coil moving through a magnetic field, or when the magnetic field around a stationary coil changes.

Example: a moving loop

Consider a simple circular loop that moves in and out of a magnetic field. As the volume of the loop within the field changes, the magnetic flux through the loop changes, inducing an EMF according to Faraday's law.

For example, if a loop passes through a space with a constant magnetic field:

EMF = -N * (dΦB/dt)

Where N is the number of turns in the coil. This shows that the induced EMF depends not only on the rate of change of magnetic flux (dΦB/dt) but also on the number of turns in the coil.

Moving coil visualization

B Coil

In this simplified diagram, the coil is shown as a rectangle with circular segments, and it moves in a uniform magnetic field (blue arrows). Changes in the field within the coil lead to changes in the magnetic flux through the coil, causing an EMF to be induced.

Example: rotating a coil

Another practical example is a coil rotating within a magnetic field, such as in an electric generator. The angle θ between the magnetic field and the surface of the coil changes with time, affecting the flux and inducing an EMF.

The EMF in a rotating coil can be described as follows:

EMF(t) = NBAω * sin(ωt)

Where:

  • N is the number of turns
  • B is the magnetic field strength
  • A is the area of the coil
  • ω is the angular frequency of rotation
  • t is the time

This expression shows that the magnitude of the induced e.m.f. depends on the frequency of rotation and varies sinusoidally with time.

Role of Lenz's law

Lenz's law is important in determining the direction of the induced current. It states that the direction of the induced current will be such that it will oppose the change in the magnetic flux that produces it. This is why the minus sign appears in Faraday's equation.

Consider a magnetic field whose strength increases. The induced current will generate its own magnetic field in the opposite direction to minimize the change.

Example: magnetic damping

As a second application of Lenz's law, consider a metal ring placed in a changing magnetic field. The induced current in the ring will oppose the change, and this opposition is used in an electromagnetic braking system called magnetic damping.

When a magnetic field passes through a conductor while relative motion is present, eddy currents are produced. These currents by their nature resist changes, thanks to Lenz's law.

Applications of Faraday's law

Faraday's law plays a vital role in the functioning of transformers, electric generators and many types of sensors. The principle of electromagnetic induction is used in these devices for various technological applications.

In transformers, the idea is to transfer electrical energy between two or more circuits through electromagnetic induction. Generators convert mechanical energy into electrical energy using the principles of changing magnetic fields.

Conclusion

Faraday's law of electromagnetic induction is fundamental to understanding many of the technologies around us. By discovering and applying this principle, innovations in power generation and electromagnetism have reshaped our world.

This theory not only forms the basis of a vast range of technologies, but also provides a rich understanding of the interaction between electricity and magnetism, giving us profound insights into physical phenomena.


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Faraday's Law

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