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


Introduction

Maxwell's equations are a set of four equations that form the foundation of classical electrodynamics, classical optics, and electrical circuits. These equations describe how electric and magnetic fields interact and how they are produced by charges and currents. James Clerk Maxwell first presented these equations in their complete form in the 1860s. Despite being more than a century old, they remain central to our understanding of electromagnetic phenomena.

Overview of Maxwell's equations

Maxwell's four equations are:

  1. Gauss's law for electricity
  2. Gauss's law for magnetism
  3. Faraday's law of induction
  4. Ampere's law with Maxwell's addition

Gauss's law for electricity

Gauss's law for electricity states that the electric flux through a closed surface is proportional to the charge enclosed by that surface. Mathematically, this can be expressed as:

∮ E · dA = Q_enclosed / ε₀

Here, E is the electric field, dA is the differential field across the enclosed surface, Q_is the total charge within Q_enclosed surface, and ε₀ is the permittivity of free space.

This law is a reflection of the idea that electric charge creates an electric field that radiates outward in all directions. It is especially useful in dealing with symmetrical objects such as spheres and cylinders.

Why

In the above visual example, a spherical surface encloses a charge Q The lines represent electric flux lines emerging out of the spherical surface proportional to the enclosing charge.

Gauss's law for magnetism

Gauss's law for magnetism states that the net magnetic flux through a closed surface is zero. In other words, magnetic monopoles do not exist. It can be expressed as:

∮ B · dA = 0

Where B is the magnetic field. This law emphasizes that magnetic field lines are closed loops that have no beginning or end.

The above visualization shows that the magnetic field lines (blue curve) form a closed loop through a closed surface, with no net magnetic flux passing through the surface.

Faraday's law of induction

Faraday's law of induction describes how a time-varying magnetic field induces an electromotive force (EMF) and thus an electric current in a closed loop of wire. It can be formulated as follows:

∮ E · dl = -dΦ_B/dt

Here, dl is the differential element of the loop, Φ_B is the magnetic flux through the loop, and dΦ_B/dt is the rate of change of magnetic flux.

Faraday's law is the principle behind electric generators and transformers. When the magnetic field changes through a loop, it produces electricity.

The rectangle represents a coil of wire. The curved lines represent a changing magnetic field inducing a current in the coil, which is represented by the red line with the arrow.

Ampere's law with Maxwell's addition

Ampere's law, with Maxwell's corrections, relates magnetic fields to electric currents and the rates of change of the electric fields that produce them. It is given as:

∮ B · dl = μ₀ (I + ε₀ dΦ_E/dt)

Here, B is the magnetic field, dl is the differential path element, I is the current passing through the loop, ε₀ is the permittivity of free space, dΦ_E/dt is the rate of change of electric flux, and μ₀ is the permeability of free space.

In this diagram, a continuous red arrow represents electric current flowing through a wire, and the coiled blue path represents the magnetic field loop formed around it, implementing Ampère's law.

Applications and significance

Maxwell's equations are quintessential in understanding how electromagnetic waves propagate. They not only show the relationship between electricity and magnetism but also explain that light itself is an electromagnetic wave that travels through space. Without Maxwell's equations, modern technological advancements, including radio broadcasting, radar, and power generation, would not be possible.

Historically, Maxwell's unification of electric and magnetic fields laid the foundation for Einstein's theory of relativity, which describes how these fields interact with moving objects.

Conclusion

Each of Maxwell's equations has its own significance and together they provide a complete theoretical description of classical electromagnetism. Accurately understanding and applying these equations provides a clear understanding of the nature of electric and magnetic fields and their interdependence. Their applications span many technologies, from microwaves to communication devices and reflect their vital relevance beyond the realm of visible light.


PHD → 2.1


U
username
0%
completed in PHD

← Prev (2)
Electrodynamics
Next (2.1.1) →
Gauss's law for electricity

Comments 



Maxwell's equations

https://www.physicsflow.com/phd/2.1?pfal=en