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


Geometrical optics, also called ray optics, is a simplified model of optics that describes light propagation in terms of rays. The basic principles include the laws of reflection and refraction, which allow us to understand and predict how light interacts with surfaces, lenses, and mirrors. This field plays an essential role in understanding the basics of optical systems, where analysis of the interaction of light with objects of a scale larger than the wavelength of light is often sufficient.

Ray-like light

In geometrical optics, we assume that light travels in the form of rays. These rays can be thought of as narrow beams of light that travel in straight lines in a uniform medium. A great way to visualize light rays is to use arrows indicating the direction of light propagation.

The main assumptions underlying geometrical optics are as follows:

  • Light travels in straight lines in a homogeneous medium: This assumption implies that when light enters a medium with a uniform refractive index, it does not change direction.
  • Light can be modeled by rays: this assumption simplifies the analysis because we deal with straight lines rather than waves.
  • Wavelengths are negligible compared to the dimensions of the optical elements: this is why geometrical optics is accurate for lenses and mirrors much larger than the wavelength of light.

Let us begin by discussing the laws of reflection and refraction.

Laws of reflection

Reflection is the process in which light returns back when it hits a surface. The rules governing this phenomenon are simple, but powerful in predicting the path of light.

  • The angle of incidence is equal to the angle of reflection.
  • The incident ray, the reflected ray and the normal to the surface all lie in the same plane.

Given below is a diagram showing reflection of light:

angle of incidence angle of reflection General

In the above picture:

  • The incident ray is the light ray coming towards the reflecting surface.
  • The reflected ray is the light ray that goes back after hitting the surface.
  • Both the incident and reflected rays make equal angles with the line perpendicular to the surface at the point of incidence, i.e. the normal.

Laws of refraction

Refraction occurs when light passes from one medium to another, causing its direction to change due to a change in speed. The laws of refraction, or Snell's law, describe this bending of light.

  • The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant, which is the refractive index.
  • The incident ray, the refracted ray and the normal to the boundary of the medium all lie in the same plane.

Mathematically, Snell's law is expressed as:

n1 * sin(θ1) = n2 * sin(θ2)

Where:

  • n1 and n2 are the refractive indices of medium 1 and medium 2 respectively.
  • θ1 and θ2 are the angles of incidence and refraction, respectively.

Given below is a diagram showing refraction of light:

angle of incidence angle of refraction General

In this diagram:

  • The incident ray enters the boundary between the two media, which causes it to bend.
  • The refracted ray is the light ray in the second medium.
  • The angle of refraction may be smaller or larger than the angle of incidence, depending upon the refractive index and the nature of the medium involved.

Applications of geometrical optics

Geometrical optics is used to design various types of optical instruments. Let us study some specific examples to understand its application:

Mirror

Mirrors reflect light, and form images using the laws of reflection. Common mirror types include:

  • Plane mirrors: These plane mirrors form virtual images which are erect and of the same size as the object.
  • Concave mirrors: These inwardly curved mirrors can form a real, inverted image if the object is outside the focal point, and a virtual, erect image if the object is inside the focal point.
  • Convex mirrors: These outwardly curved mirrors always form virtual, smaller and erect images.

Lens

Lenses refract light and are usually classified as convex or concave.

  • Convex lenses: These lenses, thick at the centre, converge light rays, forming real, inverted images or virtual, upright images, depending on the distance of the object.
  • Concave lens: These lenses which are thin at the centre diverge the light rays and form mainly virtual, erect and diminished images.

Lens formula

The lens formula, useful in lens-based calculations, is represented as:

1/f = 1/v + 1/u

Where:

  • f is the focal length of the lens.
  • v is the image distance.
  • u is the distance of the object.

Practical example

To further understand the concept of geometrical optics, let us look at two real-life examples:

Example 1: Using a magnifying glass

A magnifying lens uses a convex lens to magnify objects placed within its focal length. This enables you to see details more clearly, because the lens forms a larger, virtual image.

Example 2: Car side mirror

Convex mirrors are used in car side mirrors to provide a wider field of view. Convex mirrors create virtual images that are smaller than they appear, giving drivers a view of more of the area behind them.

Conclusion

Geometrical optics provides a simplified but powerful model for understanding light propagation and interaction with objects. By understanding the laws of reflection and refraction, you can predict and use optical behavior in a variety of applications, such as lenses and mirrors. Although it makes assumptions about the nature of light, geometrical optics remains an invaluable tool for designing and analyzing everyday optical systems.


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Geometrical Optics

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