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 ↓


Gauss's law for magnetism


Gauss's law for magnetism is one of the four cornerstone equations in Maxwell's equations, which form the foundation of classical electrodynamics, optics, and electrical circuits. These equations describe how electric and magnetic fields are generated and changed by each other, as well as by charges and currents. In particular, Gauss's law for magnetism deals with the nature of magnetic fields.

Introduction to magnetism and magnetic fields

Magnetism is a force that acts at a distance due to a magnetic field. It is produced by moving electric charges. We can find magnetic fields in everyday life, such as in compasses, magnets, and the Earth's own magnetic field.

The magnetic field is represented by the vector B. It is a vector field, meaning that it has both magnitude and direction at every point in space. Where the magnetic field lines are close together, the field is said to be strong; where they are far apart, it is weak.

Understanding Gauss's law for magnetism

Gauss's law for magnetism states that the net magnetic flux passing through any closed surface is zero. In mathematical form, it is expressed as:

∮ B • dA = 0

Here, B is the magnetic field, and dA is a vector representing an infinitesimal field over a closed surface, with a direction normal to the surface. The integration symbol ∮ represents the surface integral over a closed surface.

Illustration of Gauss's law for magnetism

Consider a spherical surface in a uniform magnetic field. Magnetic field lines pass through the surface, entering from one side and exiting from the opposite side. According to Gauss's law of magnetism, the total field lines entering and exiting the surface neutrally balance each other, giving a net flux of zero.

In the visual example above, the magnetic field lines enter and exit the spherical surface uniformly, which demonstrates the principle that magnetic monopoles do not exist.

Implications of Gauss's law for magnetism

The absence of magnetic monopoles is an important implication of Gauss's law for magnetism. In electrostatics, we have electric charges, which can exist independently as positive or negative charges. However, no isolated north or south magnetic poles have been observed, only dipoles. A consequence of the zero net flux is that every magnetic field line that enters a surface must also exit it, which reinforces the dipole nature of magnetism.

Applications and examples

A practical example of Gauss's law for magnetism is its application in magnetic circuit theory. When designing transformers and similar devices, this law ensures that magnetic flux conservation is considered. There are no "leakage" monopoles in the material, and designers use this principle to guarantee efficient operation through core design optimization.

Another example is the Earth's magnetic field, which behaves like a dipole. Imagine a simple closed surface around the Earth; according to Gauss's law of magnetism, magnetic field lines going into the closed surface at one pole must exit at the other pole, thus maintaining the balance of the law.

Conclusion

Gauss's law for magnetism is fundamental in understanding the symmetry and conservation properties of magnetic fields. Its expression in Maxwell's equations underlies the intrinsic dipole nature of magnets and helps in the practical design and analysis of systems using magnetic fields. Even in advanced studies, where quantum mechanics and relativity are considered, the principles contained in Gauss's law for magnetism provide a consistent and reliable framework for understanding magnetic phenomena.


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Gauss's law for magnetism

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