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 ↓


Ampere's Law


Ampere's law is one of the important cornerstones of electromagnetism, which is essential to the framework known as Maxwell's equations. Maxwell's equations describe the fundamentals of electricity and magnetism, and are thus crucial in understanding various physical phenomena. To gain a comprehensive understanding of Ampere's law, let's explore its definition, implications, and various examples illustrating its principles.

1. Introduction to Ampere's Law

Ampere's law is a mathematical relationship that relates the integrated magnetic field around a closed loop to the electric current passing through the loop. It forms the basis for understanding the behavior of magnetic fields in relation to electric currents. The law, in its integral form, is expressed as follows:

∮ B · dl = μ₀ (I + ε₀ dΦ E /dt)

In this equation:

  • ∮ denotes the closed line integral around a loop.
  • B is the magnetic field vector.
  • dl is an infinitesimal vector element of the loop.
  • μ₀ is the permittivity of free space.
  • I is the electric current passing through the loop.
  • ε₀ is the permittivity of free space.
  • dΦ E /dt is the rate of change of electric flux through the loop.

The term ε₀ dΦ E /dt is often referred to as displacement current. This term was introduced by James Clerk Maxwell to make the original Ampere's law consistent with conservation of charge.

I B

The above figure shows a current I flowing upward through a wire, and according to Ampere's law, a circular magnetic field B is surrounded around the wire.

2. Historical context and significance

Ampère's law was named after André-Marie Ampère, a French physicist and mathematician who played a key role in the discovery of electromagnetism. Ampère's contributions laid the groundwork for understanding the interaction between electric currents and magnetic fields. Although the original Ampère's law did not include the term displacement current, Maxwell's modification was necessary to accurately describe situations involving time-varying electric fields.

The inclusion of the displacement current term resolved inconsistencies about the continuity equation of charge in scenarios where the electric field changes with time. It was this insight that eventually integrated Ampere's law into the full set of Maxwell's equations and helped predict electromagnetic waves - a groundbreaking discovery that transformed technology through the development of radio, television, and many other applications.

3. Practical examples and applications

Let's look at some practical examples to see Ampere's law in action:

3.1 Example: Solenoid Magnetic Field

A solenoid is a coil of wire that produces a magnetic field when an electric current flows through it. Consider a long solenoid having n turns per unit length, carrying current I

By applying Ampere's law to the loop inside the solenoid:

∮ B · dl = B * L = μ₀ * n * I * L

Solving for B, we get:

B = μ₀ * n * I
I B B B B

Illustration of a solenoid with internal magnetic field lines. The magnetic field B inside the solenoid is uniform and parallel to its axis.

3.2 Example: Toroidal Magnetic Field

A toroid is a donut-shaped coil of wire. Let's analyze the magnetic field inside and outside the toroid.

For a toroid with N total turns and carrying a current I, the magnetic field inside the toroid can be found using Ampère's law:

∮ B · dl = B * 2πr = μ₀ * N * I

Solving for B, we get:

B = (μ₀ * N * I) / (2πr)

Here, r is the distance from the center of the toroid to the point where the magnetic field is being calculated.

B I

View of a toroid with magnetic field lines. The magnetic field B is concentrated inside the coil and diminishes outside the toroid.

4. Visualization of displacement current

To understand the concept of displacement current, consider a parallel plate capacitor. When an alternating current (AC) passes through the circuit, the capacitor charges and discharges, causing the electric field between the plates to change with time.

I I I I

The above diagram shows a parallel plate capacitor with charging current I and electric field between the plates E The changing electric field contributes to the displacement current ε₀ dΦ E /dt in Ampere’s law.

5. Implications in modern physics

Ampere's law as part of Maxwell's equations plays a key role not only in classical electromagnetism, but also in modern physics. It paves the way for understanding electromagnetic radiation, wave propagation, and even the fundamental interactions that govern our universe.

For example, the realization that changes in electric fields can generate magnetic fields (and vice versa) led to the notion of electromagnetic waves. Light itself is an electromagnetic wave, consisting of oscillating electric and magnetic fields propagating through space. This revelation was crucial to the development of quantum electrodynamics, one of the cornerstones of quantum field theory.

In addition, the broad scope of Ampere's law and Maxwell's equations have been instrumental in developing technologies such as antennas, transformers, and various wireless communication systems. They guide engineers and physicists in designing systems ranging from household electronic devices to sophisticated space communication interfaces.

6. Conclusion

Ampere's law, with its introduction of displacement current, is not merely a reflection of historical breakthroughs, but a dynamic tool that continues to influence physics and engineering. Its integral form allows for the analysis of complex magnetic and electric interactions across a spectrum of scenarios, ensuring its continuing relevance in many areas of scientific inquiry and technological innovation.

Understanding Ampere's law enriches our understanding of electromagnetic phenomena and increases our ability to harness these forces, thereby facilitating progress in both theoretical and applied science.


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Ampere's Law

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