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 ↓


Electromagnetic waves


Electromagnetic waves are waves that are formed as a result of vibrations between an electric field and a magnetic field. In other words, electromagnetic waves are made up of oscillating electric and magnetic fields. The essential aspect of electromagnetic waves is that they do not require a medium to travel; they can travel even in the vacuum of space.

Introduction to Maxwell's equations

Maxwell's equations are a set of four fundamental equations that describe how electric fields and magnetic fields interact. They form the foundation of classical electrodynamics, optics, and electrical circuits. James Clerk Maxwell formulated these equations in the 19th century by combining previous work by Gauss, Faraday, and Ampere.

The four Maxwell equations

Maxwell's equations are as follows:

1. Gauss's Law for Electricity: ∇ • E = ρ/ε₀

This equation states that the electric flux through a closed surface is proportional to the charge contained in it.

2. Gauss's Law for Magnetism: ∇ • B = 0

This equation implies that there are no magnetic monopoles; the net magnetic flux through any closed surface is zero.

3. Faraday's Law of Induction: ∇ x E = -∂B/∂t

This law states that a changing magnetic field produces an electric field.

4. Ampere's Law (with Maxwell's addition): ∇ x B = μ₀(J + ε₀ ∂E/∂t)

This equation states that magnetic fields are produced by moving charges or currents and changing electric fields.

Generation of electromagnetic waves

Electromagnetic waves arise from solutions of Maxwell's equations. To understand electromagnetic waves, consider a charge that is accelerating. When a charge accelerates, it disturbs the electric and magnetic fields around it. According to Maxwell's equations, a changing electric field induces a magnetic field and vice versa.

Let's see how electromagnetic waves are formed:

  1. A vibrating or accelerating charge produces an oscillating electric field.
  2. According to Faraday's law of induction, this oscillating electric field produces an oscillating magnetic field.
  3. The oscillating magnetic field, in turn, generates a new electric field cycle, as described by a modification of Ampere's law.

The result is a wave propagating through space. These vibrations reinforce each other and travel outward, forming electromagnetic waves.

Characteristics of electromagnetic waves

Electromagnetic waves have several essential characteristics:

  • Transverse Waves: Oscillations of both electric and magnetic fields are perpendicular to the direction of wave propagation.
  • Speed of light: Electromagnetic waves in a vacuum travel at the speed of light, which is represented by the following equation:
    • c = 1 / √(μ₀ε₀)
  • where c is the speed of light, μ₀ is the permittivity of free space, and ε₀ is the permittivity of free space.
  • No medium required: Unlike sound waves, electromagnetic waves do not require a medium. They can travel in the vacuum of space.

Visualization of electromagnetic waves

E-Region B-region

In the above visualization, the red line represents the oscillating electric field (E-field), and the blue line represents the oscillating magnetic field (B-field). Both fields are perpendicular to each other and perpendicular to the direction of travel of the wave.

Frequency and wavelength

Like all waves, electromagnetic waves are characterized by their frequency and wavelength. The wavelength is the distance between successive crests (or troughs), and the frequency is the number of waves passing a point per second. The relationship between the speed ( c ), frequency ( f ), and wavelength ( λ ) of an electromagnetic wave is given by:

c = λf

Examples of electromagnetic wave frequencies include:

  • Radio waves: These have long wavelength and low frequency and are used in broadcasting and communication.
  • Microwaves: Used for heating food and in some communications technologies.
  • Infrared radiation: Sensed as heat and used in remote controls.
  • Visible light: The range of electromagnetic waves that can be seen by the human eye.
  • Ultraviolet radiation: This is high in energy and can be harmful, but it is also used to sterilize equipment.
  • X-rays: These are used in medical imaging because they can pass through soft tissues but are absorbed by denser materials such as bones.
  • Gamma rays: With the shortest wavelength and highest energy, these arise from nuclear reactions and radioactive decay.

Example of generating electromagnetic waves

Consider a simple oscillating circuit known as a dipole antenna. The dipole antenna consists of two metal rods with a small gap between them. When alternating current is applied, the electrons oscillate back and forth along the rods. This oscillation creates a changing electric field and a corresponding magnetic field. As a result, electromagnetic waves are emitted.

These waves can be used to transmit information, such as in radio and television broadcasting. The frequency of oscillation determines the frequency of the emitted electromagnetic wave.

Conclusion

Electromagnetic waves are a fundamental aspect of both nature and technology. From allowing us to see the world around us to enabling the transmission of information across the globe, understanding electromagnetic waves is essential. James Clerk Maxwell's contributions through his set of equations provided a path to understanding how electricity and magnetism are unified. Electromagnetic waves, revealing the interaction between electric and magnetic fields, travel through space at extraordinary speeds to bridge vast distances without the need for a medium. From the colours of the rainbow to X-rays revealing hidden structures within us, electromagnetic waves continue to inspire and revolutionise our understanding of the universe.


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Electromagnetic waves

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