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 → Electromagnetic waves ↓


Reflection and Refraction


In the study of electrodynamics, it is important to understand how electromagnetic waves interact with different media. The two fundamental concepts that govern this interaction are reflection and refraction. Both phenomena are beautifully explained by the wave nature of electromagnetic radiation, such as light.

Electromagnetic waves

Before delving deeper into reflection and refraction, it is important to have a basic understanding of electromagnetic waves. They are waves of electric and magnetic fields that propagate through space. The speed of these waves in a vacuum is about (3 times 10^8) meters per second, commonly known as the speed of light, (c).

These waves are described by Maxwell's equations, which are central to classical electrodynamics. In general, an electromagnetic wave is characterized by its wavelength ((lambda)), frequency ((f)), and speed ((v)), which are related by the formula:

c = lambda cdot f

Reflection

Reflection occurs when electromagnetic waves hit the surface of a medium and return back to the original medium. The angle at which the waves hit the surface is called the angle of incidence ((theta_i)), and the angle at which they are reflected is called the angle of reflection ((theta_r)). According to the law of reflection, these angles are equal:

theta_i = theta_r

This fundamental principle can be understood as follows:

incident rayReflected ray θ i θR

In this diagram, the incoming wave, or incident ray, strikes a surface and reflects at an angle equal to its angle of incidence, demonstrating the principle of reflection.

Reflection from different surfaces

The nature of the surface greatly affects the reflection of electromagnetic waves:

  • Reflection: This type of reflection occurs on a smooth surface like a mirror, where the rays reflect in a highly organized manner, preserving the wavefront.
  • Diffuse reflection: On rough surfaces the rays are scattered in many directions, destroying the organised wavefronts, which can be seen in objects such as concrete walls.

Refraction

When electromagnetic waves travel from one medium to another, they bend, a process called refraction. The angle of incidence ((theta_i)) and the angle of refraction ((theta_t)) are related through Snell's law:

n_1 cdot sin(theta_i) = n_2 cdot sin(theta_t)

Where (n_1) and (n_2) are the refractive indices of the first and second medium respectively. The refractive index is a measure of how much the speed of light decreases inside a medium. The speed of light in a medium is given by:

v = frac{c}{n}

The bending of light can be represented as follows:

incident rayRefracted ray θ i θ t

In this scenario, a ray of light enters water from air. The speed decreases, causing the ray to bend toward the normal, demonstrating the principle of refraction.

Applications of reflection and refraction

The principles of reflection and refraction are applied in many optical devices and technologies:

  • Mirrors: use the law of reflection to direct light, allowing us to see ourselves and our surroundings.
  • Lenses: use refraction to focus or diverge light rays, making magnification possible in eyeglasses, cameras, and microscopes.
  • Optical fibers: use both reflection and refraction to transmit light over long distances with minimal loss.

Mathematical derivations

For a more in-depth understanding, let us derive Snell's law using the wave theory of light. Consider two media with refractive indices (n_1) and (n_2). When a wavefront hits the interface at an angle, it changes the speed and wavelength, but not the frequency. According to the wavefront theory of light, this can be formulated as:

n_1 cdot frac{lambda_1}{lambda_2} = frac{sin(theta_i)}{sin(theta_t)}

Since the speed of the wave is related to the refractive index as ( v = frac{c}{n} ), and the wavelength (lambda) in the medium is related to the speed and frequency as ( v = lambda cdot f ), the wavelengths (lambda_1) and (lambda_2) must obey:

frac{v_1}{v_2} = frac{n_2}{n_1}

Thus we recover the familiar form of Snell's law from wavefront analysis.

Conclusion

Reflection and refraction are fundamental aspects of how electromagnetic waves interact with mediums, governed by well-established laws of physics. These principles not only explain everyday optical phenomena, but also enable advanced technologies that enhance our understanding of the universe. By embracing both the theoretical basis and practical applications, we gain a much richer understanding of the role of the electromagnetic spectrum in our lives.


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Reflection and Refraction

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