Understanding Reflection, Refraction & Snell's Law

• Ray optics concern reflection, refraction, and dispersion of light.
• Refraction is how light bends when moving from one medium from another

Light

• Light travels in a straight line and is called a ray of light.
• A group of light rays is called a beam of light.
• The speed of light in a vacuum is 3 x 10^8ms⁻¹, the fastest speed attainable in nature.

Laws of Reflection

• The angle of reflection is the angle between the reflected ray and the normal to the reflecting surface.
• The angle of incidence is the angle between the incident ray and the normal.
• The angle of reflection equals the angle of incidence, i.e., ∠i = ∠r.
• The incident ray, the normal to the mirror at the point of incidence, and the reflected ray all lie in the same plane.

Snell's Law

• Snell's law for refraction is given by sin i / sin r = n, where:
  • i is the angle of incidence
  • r is the angle of refraction
  • n is the refractive index of the medium

Critical Angle

• The critical angle is the angle of incidence at which a ray traveling from a denser to a rarer medium makes an angle of refraction of 90°, denoted by i_c.

Cartesian Sign Convention

• Positive signs are for distances measured in the same direction as the incident light.
• Negative signs are used for distances measured opposite to the incident light.
• All distances are measured from the pole of the mirror or the optical center of the lens.
• Heights measured upwards relative to the x-axis and normal to the principal axis of the mirror/lens are positive.
• Heights measured downwards relative to the x-axis are negative.

Mirror Formula

• If u is the object distance, v is the image distance, and f is the focal length of the mirror.
• The mirror formula is: 1/v + 1/u = 1/f

Focal Length

• Concave mirror focal length is negative.
• Convex mirror focal length is positive.

Magnification

• The magnification produced by a mirror is given by m = h'/h = -v/u, where:
  • h' is the height of the image
  • h is the height of the object

Total Internal Reflection

• When light travels from an optically denser medium to a rarer medium at the interface, it is partly reflected back into the same medium and partly refracted to the second medium.
• Internal reflection: When light is reflected back.
• Total internal reflection: When all light is reflected back.
• Applications of total internal reflection include mirage, diamond, prism, and optical fibers.

Refraction Through a Glass Slab

• The emergent ray through a glass slab is parallel to the incident ray but laterally displaced.
• The angle of incidence equals the angle of emergence.

Refraction at Spherical Surfaces

• For rays incident from a medium with refractive index n₁ to another with refractive index n₂: (n₂/v) - (n₁/u) = (n₂ - n₁)/R

Refraction Through a Prism

• For a prism with angle A and refractive index n₂, placed in a medium with refractive index n₁, and D being the angle of minimum deviation: n₂₁ = n₂ / n₁ = sin[(A + Dm) / 2] / sin(A / 2)

Lens Formula

• If u is the object distance, v is the image distance, and f is the focal length of the lens: 1/v - 1/u = 1/f

Focal Length of Lenses

• The focal length f is positive for a converging lens and negative for a diverging lens.

Lens Magnification

• The magnification produced by a lens is given by m = h'/h = v/u, where h' is the height of the image and h is the height of the object.

Power of a Lens

• The power (P) of a lens is given by P = 1/f, where f is the focal length of the lens.
• The SI unit of power is diopter (D); 1 D = 1 m⁻¹

Effective Focal Length

• The effective focal length for a combination of thin lenses with focal lengths f₁, f₂, f₃, ..., is given by: 1/f = 1/f₁ + 1/f₂ + 1/f₃ + ...
• The effective power of the combination is given by: P = P₁ + P₂ + P₃ + ...