Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Electromagnetism ↓


Maxwell's equations


Maxwell's equations are a set of four fundamental equations that lay the foundation for classical electromagnetism, classical optics, and electrical circuits. These equations describe how electric and magnetic fields interact and propagate. Developed by James Clerk Maxwell in the mid-19th century, they have since become a cornerstone of physics, helping to explain a wide range of electromagnetic phenomena.

The four Maxwell equations

Maxwell's four equations can be grouped into two categories: two that describe how electric charges and currents give rise to electric and magnetic fields, and two that describe how electric and magnetic fields interact. Let us discuss each of them in detail.

Gauss's law

Gauss's law relates the electric field E to the distribution of electric charge. It states that the total electric flux flowing out of a closed surface is equal to the enclosed charge divided by the electric potential.

∇ · E = ρ/ε₀

Here, ρ represents the electric charge density, and ε₀ is the vacuum permittivity. This law implies that electric charges are the source and sink of the electric field.

Why Here, the charge Q is enclosed within the surface, giving rise to electric field lines, shown by arrows.

Gauss's law for magnetism

Gauss's law for magnetism states that the net magnetic flux through a closed surface is zero. This means that magnetic monopoles do not exist; magnetic field lines are continuous loops with no beginning or end.

∇ · B = 0

In this equation, B is the magnetic field. Since no magnetic monopoles are known, the magnetic field lines form closed loops.

This makes it clear that the magnetic field lines are closed loops, like the continuous circular path shown.

Faraday's law of induction

Faraday's law of induction describes how a changing magnetic field induces an electromotive force (EMF) or voltage in a closed loop. It is the principle behind transformers, electric generators, and inductors.

∇ × E = -∂B/∂t

Here, ∂B/∂t is the time rate of change of the magnetic field, and it induces an electric field E.

B Electric current flows through the loop due to the changing magnetic field B.

Ampere-Maxwell law

The Ampere–Maxwell law relates magnetic fields to electric currents and electric fields. It is a generalized form of Ampere's circuital law to include time-varying electric fields.

∇ × B = μ₀(J + ε₀∂E/∂t)

In this equation, J is the current density, μ₀ is the permeability of free space, and ∂E/∂t is the time rate of change of the electric field. This law shows that a changing electric field produces a magnetic field.

I The changing electric field, shown by the dashed lines, affects the magnetic field around it.

Applications of Maxwell's equations

Maxwell's equations have transformed our understanding of physics and have very wide applications in a variety of fields. These include technologies such as radio, microwaves, antennas, radar systems, wireless communications, and even quantum mechanics.

Electric circuits

Understanding how electric currents flow in circuits is a direct application of Maxwell's equations. For example, electromagnetic waves travel through wires and circuit components, and affect various electronic devices.

Consider a simple DC circuit in which a battery, a resistor, and wires form a loop. The battery provides a voltage that causes current to flow through the loop. Kirchhoff's circuit rules, derived from Maxwell's equations, can explain the behavior of such a circuit.

Electromagnetic waves

Maxwell's equations predict the existence of electromagnetic waves, describing how light and other forms of electromagnetic radiation propagate through space. These waves consist of an electric field and a magnetic field that oscillate perpendicular to each other and the direction of wave propagation.

IB The red wave represents the electric field E, and the blue wave represents the magnetic field B.

Optics and lighting

Maxwell's equations firmly established that light is an electromagnetic wave. This understanding of light has enabled the development of optics, leading to technologies such as lenses, cameras, telescopes, and more complex systems.

Due to the wave nature of light, phenomena such as reflection, refraction, interference, and diffraction can be explained by these fundamental equations.

Understanding magnetic materials

The concepts of magnetic fields and their interactions with matter fall under the purview of Maxwell's equations. These interactions lead to the development of motors, transformers, inductors, and many other important electronic components.

Wireless transmission

Wireless communication relies on transmitting electromagnetic waves through the air. The design of antennas and the efficiency of wireless communication are based on principles laid down by Maxwell's equations.

These equations help determine the optimal frequency and bandwidth for signals to avoid interference and maximize the quality of transmitted signals.

Closing thoughts on Maxwell's equations

Maxwell's equations are more than theoretical constructs; they are important laws of physics that have practical implications and applications in many modern technologies. Their impact ranges from the very small scale, such as quantum mechanics, to the very large scale, such as our understanding of the universe.

Their elegant and compact form highlights the interdependence of electric and magnetic fields and provides a unified framework that ensures the consistency of the laws of physics in different contexts.


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Maxwell's equations

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