Graduate
  1. Classical mechanics  1.1. Advanced Kinematics  1.1.1. Generalized coordinate systems  1.1.2. Parametric equations of motion  1.1.3. Non-inertial frames and fictitious forces  1.1.4. Covariant formulation of momentum  1.1.5. Action-angle variable  1.2. Lagrangian and Hamiltonian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Noether's theorem and conservation laws  1.2.4. Normalized Speed  1.2.5. Canonical conversion  1.2.6. Poisson bracket and Hamilton–Jacobi theory  1.2.7. Hamiltonian chaos and integrability  1.3. Rigid body mobility  1.3.1. Euler's equations of motion  1.3.2. Principal moments of inertia  1.3.3. Gyroscopic motion and precession  1.3.4. Inertia tensor  1.3.5. Stability of rotational speed  1.4. Nonlinear dynamics and chaos  1.4.1. Phase space and stability analysis  1.4.2. Poincaré sections and bifurcation theory  1.4.3. Strange Attractors and Fractals  1.4.4. Lyapunov exponent  1.4.5. KAM theorem and quasi-periodic motion
  2. Electromagnetism  2.1. Advanced Electrodynamics  2.1.1. Green's function and potential theory  2.1.2. Multipole expansion  2.1.3. Image charging method  2.1.4. Boundary conditions and uniqueness theorem  2.2. Electromagnetic wave propagation  2.2.1. Plane waves in dielectrics and conductors  2.2.2. Waveguides and cavity resonators  2.2.3. Radiation pressure and optical tweezers  2.2.4. Polarization and Jones calculus  2.3. Relativistic electrodynamics  2.3.1. Covariant formulation of electrodynamics  2.3.2. Lienard–Wiechert potentials  2.3.3. Synchrotron radiation and bremsstrahlung  2.3.4. Electromagnetic field tensor and Lorentz transformations  2.4. Plasma physics  2.4.1. Magnetohydrodynamics  2.4.2. Debye shielding and plasma oscillations  2.4.3. Alfvén waves and tokamak confinement  2.4.4. Plasma instabilities and turbulence
  3. Statistical mechanics and thermodynamics  3.1. Advanced Thermodynamics  3.1.1. Legendre transforms and thermodynamic potentials  3.1.2. Fluctuation–dissipation theorem  3.1.3. Onsager interpersonal relations  3.1.4. Critical events and phase transitions  3.2. Quantum statistical mechanics  3.2.1. Density Matrix and Ensemble Theory  3.2.2. Bose–Einstein condensates  3.2.3. Quantum phase transition  3.2.4. Fermi–Dirac and Bose–Einstein statistics  3.2.5. Partition functions and grand canonical ensembles
  4. Quantum mechanics  4.1. Advanced wave mechanics  4.1.1. WKB approximation  4.1.2. Path integral formulation  4.1.3. Quantum harmonic oscillator and coherent states  4.2. Angular momentum and spin  4.2.1. Spherical Harmonics  4.2.2. Clebsch–Gordan coefficient  4.2.3. Wigner–Eckart theorem  4.2.4. Spin–orbit coupling  4.3. Quantum scattering theory  4.3.1. Born approximation  4.3.2. Partial wave analysis  4.3.3. Optical theorem  4.3.4. S-matrix theory  4.4. Quantum field theory  4.4.1. Second Quantization  4.4.2. Feynman diagrams and propagators  4.4.3. Renormalization theory  4.4.4. Gauge theory and Yang–Mills fields  4.4.5. Spontaneous symmetry breaking and the Higgs mechanism
  5. General relativity and cosmology  5.1. Tensor calculus and differential geometry  5.1.1. Einstein field equations  5.1.2. Schwarzschild and Kerr metrics  5.1.3. Geodesics and Christoffel symbols  5.2. Cosmology and the Universe  5.2.1. The FLRW metric and cosmic inflation  5.2.2. Dark energy and structure formation  5.2.3. Cosmic microwave background radiation
  6. Condensed matter physics  6.1. Band structure and transport theory  6.1.1. Block theorem and the Kronig–Penney model  6.1.2. Fermi surface topology  6.1.3. Quantum Hall Effect  6.2. Superconductivity  6.2.1. BCS theory and Cooper pairs  6.2.2. Meissner effect and flux quantization  6.2.3. The Josephson Effect and SQUIDs  6.3. Topological phases of matter  6.3.1. Topological Insulators  6.3.2. Majorana fermions in topological phases of matter
  7. Nuclear and Particle Physics  7.1. Quantum Chromodynamics (QCD)  7.1.1. Quark–gluon plasma  7.1.2. Asymptotic freedom  7.1.3. Confinement and hadronization  7.2. Beyond the Standard Model  7.2.1. Grand Unified Theories  7.2.2. Supersymmetry and extra dimensions  7.2.3. Neutrino oscillations

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Electromagnetism


Electromagnetism is one of the fundamental forces of nature and is essential to understanding not only classical physics but also modern technology. It is the study of electric charges, magnetic fields, and the way they interact. Unlike other forces, the electromagnetic force manifests in many forms, including light, electricity, and magnetism. This vast subject affects many disciplines, from electronics to quantum mechanics and beyond. Let's take a deep dive into its key components and principles.

Electric charges

At the core of electromagnetism are electric charges. Electric charges can be of two types: positive and negative. These charges exert forces on each other. Like charges repel, and unlike charges attract. The unit of charge is the coulomb (C). The interaction of these charges is governed by Coulomb's law, which can be expressed as:

F = k * |q1 * q2| / r^2

Where:

  • F is the force between the charges.
  • q1 and q2 are the quantities of two charges.
  • r is the distance between the centres of the two charges.
  • k is the Coulomb constant, approximately 8.99 x 10^9 N m^2/C^2.

Electric field

Electric fields are vector fields around a charge that exert forces on other charges. The concept of electric field helps us understand how a charge affects another charge in its vicinity without physical contact. The electric field E at a point in space is defined as the force F experienced by the charge per unit charge:

E = F / q

This field has both magnitude and direction, pointing away from positive charges and toward negative charges. The electric field can be represented graphically by field lines, which indicate the direction in which a positive test charge would move. Let's visualize this:

Magnetic field

Like electric fields, magnetic fields represent the effect a magnetic object has on its environment. Magnets have north and south poles, where, like electric charges, opposite poles attract and like poles repel. Magnetic fields are manifested by moving charges or magnetic objects, represented by lines extending from the north to the south pole.

The magnetic field produced by a current-carrying wire can be represented by concentric circles around the wire. Using the right-hand rule, if the thumb of the right hand points in the direction of the conventional current, the fingers turn in the direction of the magnetic field lines.

Electromagnetic induction

Electromagnetic induction is the process by which a changing magnetic field within a closed loop induces an electric current. This fundamental principle is central to many technologies, from electric generators to transformers.

Faraday's law of induction describes this phenomenon quantitatively. The electromotive force (emf) induced in a loop is equal to the negative of the rate of change of the magnetic flux through the loop:

emf = -dΦ/dt

Where:

  • emf is the electric driving force.
  • Φ is the magnetic flux.

Maxwell's equations

Maxwell's equations are a set of four differential equations that form the foundation of classical electromagnetism. These equations describe how electric and magnetic fields arise from changes in charges, currents, and fields.

  1. Gauss's law for electricity: It relates the electric field to the charge distribution.
    2. ∇•E = ρ/ε₀
  2. Gauss's law for magnetism: It states that there are no magnetic monopoles.
    3. ∇•B = 0
  3. Faraday's law of induction: It states how a magnetic field varying with time induces an electric current.
    4. ∇×E = -∂B/∂t
  4. Ampere-Maxwell law: It modifies Ampere's law to take into account the change in electric field with time.
    5. ∇×B = μ₀J + μ₀ε₀∂E/∂t

Electromagnetic waves

Electromagnetic waves are waves of electric and magnetic fields that propagate through space. According to Maxwell's equations, a time-varying electric field produces a magnetic field, and a time-varying magnetic field produces an electric field. These maintain each other, resulting in electromagnetic waves propagating at the speed of light c, about 3 x 10^8 m/s.

Electromagnetic waves can be viewed as perpendicular oscillations of electric and magnetic fields. The wave travels in a direction perpendicular to both fields:

I B

Applications of electromagnetism

Electromagnetism is not just a theoretical construct; it has real-world applications everywhere around us. Some examples include:

  • Electric motors: They convert electrical energy into mechanical energy using electromagnetic interactions.
  • Transformers: Use electromagnetic induction to transfer electrical energy between circuits.
  • Radio and television: Use electromagnetic waves to transmit audio and video signals over long distances.
  • Medical imaging: Techniques such as MRI use magnetic fields for detailed imaging of the interior of the body.

In short, understanding electromagnetism is vital to the development of modern technologies and scientific progress. Its principles govern many aspects of the natural world and are integral to advances in physics and engineering.


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Electromagnetism

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