PHD
  1. Classical mechanics  1.1. Newtonian mechanics  1.1.1. Laws of motion in Newtonian mechanics  1.1.2. Dynamics of particles and systems  1.1.3. Conservation laws  1.1.4. Central force motion  1.2. Lagrangian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Constraints and normalized coordinates  1.2.4. Noether's theorem  1.3. Hamiltonian mechanics  1.3.1. Hamilton's equations  1.3.2. Canonical conversion  1.3.3. Poisson bracket  1.3.4. Action-angle variable  1.4. Rigid body mobility  1.4.1. Rotation of rigid bodies  1.4.2. Moment of inertia tensor  1.4.3. Euler's equations in rigid body dynamics  1.4.4. Gyroscopic motion  1.5. Chaos and nonlinear dynamics  1.5.1. Stage space and attractive  1.5.2. Bifurcation and chaos theory  1.5.3. Hamiltonian Chaos  1.5.4. Lyapunov exponent
  2. Electrodynamics  2.1. Maxwell's equations  2.1.1. Gauss's law for electricity  2.1.2. Gauss's law for magnetism  2.1.3. Faraday's Law  2.1.4. Ampere's Law  2.1.5. Boundary conditions in Maxwell's equations  2.2. Electromagnetic waves  2.2.1. Wave equation  2.2.2. Polarization  2.2.3. Reflection and Refraction  2.2.4. Waveguides and resonators  2.3. Special relativity  2.3.1. Lorentz transformations  2.3.2. Relativistic energy and momentum  2.3.3. Relativistic electrodynamics  2.3.4. Minkowski spacetime  2.4. Radiation and scattering  2.4.1. Dipole radiation  2.4.2. Multipole expansion  2.4.3. Compton scattering  2.4.4. Thomson and Rayleigh scattering  2.5. Plasma Physics  2.5.1. Debye Screening  2.5.2. Magnetohydrodynamics  2.5.3. Plasma instability  2.5.4. Fusion Plasma
  3. Quantum mechanics  3.1. Foundations of quantum mechanics  3.1.1. Principles of quantum mechanics  3.1.2. Wave function and probability interpretation  3.1.3. Uncertainty principle  3.1.4. Quantum Tunneling  3.2. Schrödinger Equation  3.2.1. Time-dependent Schrödinger equation  3.2.2. Time-independent Schrödinger equation  3.2.3. Eigenvalues and eigenfunctions in the Schrödinger equation  3.2.4. Path integrals in quantum mechanics  3.3. Quantum operators  3.3.1. Commutators and observables  3.3.2. Angular momentum operator  3.3.3. Ladder Operator  3.3.4. Pauli matrices  3.4. Quantum entanglement and measurement  3.4.1. Bell's theorem  3.4.2. Quantum Teleportation  3.4.3. Quantum entanglement and quantum decoherence in measurement  3.4.4. Quantum entanglement and measurement in quantum mechanics  3.5. Relativistic quantum mechanics  3.5.1. Klein–Gordon equation  3.5.2. Dirac equation  3.5.3. Quantum field theory  3.5.4. Feynman path integral
  4. Statistical mechanics and thermodynamics  4.1. Classical thermodynamics  4.1.1. Laws of Thermodynamics  4.1.2. Carnot cycle  4.1.3. Entropy and free energy  4.1.4. Thermodynamic efficiency  4.2. Kinetic theory of gases  4.2.1. Maxwell–Boltzmann distribution  4.2.2. Transport phenomenon  4.2.3. Mean free path  4.2.4. Non-equilibrium systems in the kinetic theory of gases  4.3. Statistical mechanics  4.3.1. Microstates and Macrostates  4.3.2. Partition function  4.3.3. Bose–Einstein and Fermi–Dirac statistics  4.3.4. Fluctuations and correlations  4.4. Phase transition  4.4.1. Important events  4.4.2. Landau theory  4.4.3. Ising model  4.4.4. Renormalization group theory
  5. Quantum field theory  5.1. Second Quantization  5.1.1. Quantum harmonic oscillator in second quantization  5.1.2. Creation and annihilation operators in second quantization  5.1.3. Path Integral in Second Quantization  5.1.4. Fock space  5.2. Quantum Electrodynamics  5.2.1. Feynman diagrams  5.2.2. Renormalization in quantum electrodynamics  5.2.3. Gauge invariance in quantum electrodynamics  5.2.4. Vacuum polarization  5.3. Quantum Chromodynamics  5.3.1. Quarks and gluons  5.3.2. Color range  5.3.3. Lattice QCD  5.3.4. Asymptotic freedom  5.4. Standard model of particle physics  5.4.1. Electroweak theory  5.4.2. Higgs mechanism  5.4.4. Grand Unified Theories
  6. General relativity and gravity  6.1. Einstein's field equations  6.1.1. Ricci tensor and scalar curvature  6.1.2. Schwarzschild solution  6.1.3. Kerr metric  6.1.4. Gravitational waves  6.2. Cosmology  6.2.1. Friedmann equation  6.2.2. Cosmic inflation  6.2.3. Dark matter and dark energy  6.2.4. Large scale structure  6.3. Black holes and wormholes  6.3.1. Event horizon in black holes and wormholes  6.3.2. Hawking radiation  6.3.3. Penrose process  6.3.4. Information paradox in black holes and wormholes
  7. Condensed matter physics  7.1. Crystal structure and lattice  7.1.1. Bravais lattices  7.1.2. Band theory in crystal structure and lattices  7.1.3. Phonons in crystal structure and lattice  7.1.4. Block's theorem  7.2. Superconductivity  7.2.1. BCS principle  7.2.2. Meissner effect  7.2.3. High Temperature Superconductor  7.2.4. Josephson effect

PHD → Electrodynamics → Maxwell's equations ↓


Faraday's Law


Faraday's law of electromagnetic induction is a fundamental principle in electromagnetism and is one of Maxwell's four equations that lay the foundation for classical electrodynamics, optics, and electrical circuits. In this detailed explanation, we'll dive deep into the details of Faraday's law, its mathematical formulation, practical implications, and how it fits into the larger framework of Maxwell's equations.

Basic concept

Faraday's law describes how a changing magnetic field can induce an electromotive force (EMF) or voltage in a closed circuit. This principle is the working basis behind many electrical devices and technologies, such as generators, transformers and inductors.

Mathematical representation

The mathematical expression of Faraday's law is:

    EMF = -dΦ/dt
    EMF = -dΦ/dt

Here, EMF is the electromotive force, and dΦ/dt is the rate of change of magnetic flux Φ through the circuit.

Visual example

Let's visualize Faraday's law using a simple loop of wire placed in a magnetic field. Consider the following example:

B Wire Loop

In this illustration, a magnetic field B is passing through a loop of wire. If the strength or direction of the magnetic field changes with time, it induces an EMF in the loop of wire according to Faraday's law.

Detailed description

Magnetic flux

Before we proceed further, let us understand the concept of magnetic flux. The magnetic flux Φ through a surface is defined as the product of the magnetic field B and the area A through which it passes, which is perpendicular to the field. Mathematically, it is expressed as:

    Φ = B · A = B * A * cos(θ)
    Φ = B · A = B * A * cos(θ)

Here, θ is the angle between the magnetic field and the normal (perpendicular) to the surface. In simple terms, magnetic flux measures how much magnetic field is passing through a given area.

Induced EMF and Lenz's law

According to Faraday's law, the induced EMF is proportional to the rate of change of magnetic flux with time. The negative sign in the formula EMF = -dΦ/dt is explained by Lenz's law. According to Lenz's law, the direction of the induced EMF is such that it opposes the change in magnetic flux that produces it. This means that the induced current will create a magnetic field opposing the initial change.

Practical example

Consider a simple generator, which is a rotating coil within a magnetic field. As the coil rotates, the angle θ between the magnetic field lines and the normal to the field of the coil changes, causing a change in the magnetic flux Φ through the coil:

    Φ = B * A * cos(ωt)
    Φ = B * A * cos(ωt)

where ω is the angular velocity of rotation. As a result, the induced EMF in the coil is:

    EMF = -d(B * A * cos(ωt))/dt = B * A * ω * sin(ωt)
    EMF = -d(B * A * cos(ωt))/dt = B * A * ω * sin(ωt)

This expression shows how the induced emf in a generator varies as sinusoidal functions of time, which is the basis of alternating current (AC) electricity.

Implication in electrical circuits

Faraday's law has a significant impact on the functioning of inductors and transformers. In an inductor, a change in the current flowing through the coil causes a change in the magnetic flux, which in turn induces an EMF that opposes the change in current (again according to Lenz's law). This property of inductors is used in various electrical and electronic applications.

SVG example with transformers

A transformer uses Faraday's law to convert voltage. Here's a simplified example:

primary Secondary main

In a transformer, when an alternating current passes through the primary coil, it produces a changing magnetic field. This changing magnetic field passes through the secondary coil and induces an EMF in it, which may vary in magnitude depending upon the ratio of the coils.

Structure of Maxwell's equations

Faraday's law is one of Maxwell's four equations. Here's a quick look at where it fits in the larger context.

    ∇×E = -∂B/∂t
    ∇×E = -∂B/∂t

Expressed in differential form this shows that the curl of the electric field E is equal to the negative rate of change of the magnetic field B with time. This relation is essential for linking the separate elements of electromagnetism into a unified theory.

Faraday's law describes how time-varying magnetic fields can generate electric fields. This interaction is part of the propagation of electromagnetic waves, a phenomenon that is the basis of wireless communication technologies and many other aspects of modern life.

Conclusion

Faraday's law is the basis of electromagnetism that provides a quantitative understanding of how magnetic fields can affect electrical circuits. It elegantly explains how electric power can be generated, manipulated, and transmitted across distances, making it important in the field of electrical engineering and beyond.

Its integration into Maxwell's equations represents a significant advance in physics, providing profound insight into the nature of electromagnetic interactions. By fostering a thorough understanding of Faraday's law, students and practitioners will appreciate the complex workings of the electromagnetic world around us.


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Faraday's Law

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