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 → Optics → Geometrical Optics ↓


Fermat's principle


Fermat's principle is a fundamental concept in the study of geometric optics, which is an essential part of undergraduate physics. This principle helps explain how light travels in different mediums and what paths it takes when going from one point to another. Fermat's principle can be used to derive the laws of reflection and refraction, which are important for understanding lenses, mirrors, and various optical devices. In this detailed explanation, we will explore the principle in detail, use many visual examples to aid our understanding and show how the principle applies to real-world situations.

Understanding Fermat's principle

Fermat's principle states that the path taken by light between two points is the path that can be traversed in the shortest time. This principle was first expressed by Pierre de Fermat in the 17th century and has become one of the central ideas in optics.

Fermat's principle can be mathematically formulated by considering that light travels faster in a less optically dense medium than in a denser medium. Therefore, the path that requires the least time will deviate through these mediums based on the respective speeds of light.

Time = Distance / Speed

Given two points A and B, the path of light is determined by the speed of light in each medium between A and B and the distance travelled by it in each medium.

Examples of Fermat's principle

First, let's test Fermat's principle in a simple, one-medium situation. Imagine light traveling in a straight line through a uniform medium such as air. According to Fermat's principle, light must take the shortest path or the path with the shortest time. In a uniform medium, the shortest path is simply a straight line.

Now suppose there is a block of glass between point A and point B.

Medium 1 (Air) -> Glass block -> Medium 1 (Air)

The speed of light will slow down when it enters the glass block because its optical density is higher. Accordingly, the light will bend at the boundary according to Fermat's principle, so it will still take a minimum amount of time to travel from A to B.

Explanation of the law of reflection by Fermat's principle

Let us apply this concept to a plane mirror to understand how Fermat's principle explains the law of reflection.

When a ray of light hits a mirror, it gets reflected. According to the law of reflection, the angle of incidence is equal to the angle of reflection. Why is that so?

Angle of Incidence (θi) = Angle of Reflection (θr)

This can be explained by Fermat's principle. The incident and reflected rays and the point of incidence are arranged such that the total distance travelled by the light, i.e. the transit time, is minimum. Equal angles ensure that both parts of the path (in air towards the mirror and from the mirror to the eye) are as short as possible given the constraint of the range where reflection takes place.

Explanation of the law of refraction by Fermat's principle

The law of refraction (Snell's law) can also be derived using Fermat's principle. It states that if a ray passes from one medium to another, it bends according to the ratio of their refractive indices.

n1 * sin(θ1) = n2 * sin(θ2)

Where:

  • n1 and n2 are the refractive indices of the initial and second medium.
  • θ1 is the angle of incidence.
  • θ2 is the angle of refraction.

To understand this using Fermat's principle, consider light traveling from air (medium 1) to water (medium 2). Since light travels faster in air than in water, it bends at the boundary so that the total time it takes to travel from its starting point in the air to its final point through the water is minimized. Snell's law mathematically describes the ideal angles according to the speed changes between the media.

Visual example of refraction

┌───────┐ medium 1 (air) │ │ │ A │ Ray bending here │ │ └───────┘ boundary │ │ │ B │ medium 2 (water) └───────┘

In the diagram above, light travels from point A (in air) to point B (in water). At the boundary, it bends according to Snell's law to reduce the travel time in the two media.

Applications of Fermat's principle

Understanding Fermat's principle has profound implications in the design and understanding of optical instruments such as lenses, telescopes, and microscopes. It helps us analyze how lenses focus light to form images. Let's dive into a practical example.

Lens

Lenses are used to converge or diverge light rays. Using Fermat's principle, we can understand why certain lens shapes bring light to a point focus.

Convex Lens: Converging Concave Lens: Diverging

In a convex lens the outer portion of the lens is thicker than the center. As the thickness of the lens increases it takes longer for the light to pass through the edges of the lens, causing the central light rays to meet at a common focal point, reducing the time taken by the thickness of the lens.

Visual example of lens focusing

Convex Lens . | . →─────|─────→ Focus point `. | .'  / ─────F─────

Concave lenses diverge light rays. When a distant object is viewed through a concave lens, Fermat's principle states that the central rays diverge because they are aligned to avoid the thick edge without unnecessary bias due to the shape of the lens.

Concept of complex systems

Fermat's principle helps us simulate complex optical systems by helping us predict how light will behave in different configurations, using known indices of refraction and speed in different mediums. This ability is crucial for designing technology such as fiber optics, where efficient, precise light transmission is required.

Fiber optics

Fiber optics work by moving light along a long, continuous path with minimal loss. The light is internally reflected in a fine alignment determined by the critical angle obtained through Fermat's principle.

Internal Reflection in Fiber Optics → █ █ → █ → █ →

The pathway for signals, like the network structure inside a fibre optic cable, must follow reflection rules, which are easily explained through Fermat’s time-driven path – ensuring that data is exchanged efficiently without unnecessary detours.

Conclusion

Fermat's principle is not just an abstract concept but a powerful tool for understanding the behaviour of light in a variety of situations. Whether it is analysing reflections in a simple mirror, designing a complex lens, or ensuring efficient signal transmission in fibre optics, Fermat's minimum time path for light provides a fundamental understanding of many optical phenomena.

By exploring this principle and its implications in a variety of applications, students can gain a deeper understanding of the interplay of physics, mathematics, and engineering that underlies the world of optics. Fermat's contributions and the further development of his principle continue to illuminate the way forward in understanding and shaping the world of light.


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Fermat's principle

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