Light: Reflection and Refraction - Comprehensive Notes

Light: Reflection and Refraction

Introduction

We see objects due to light reflecting off them and entering our eyes.
Transparent mediums allow light to pass through.
Light appears to travel in straight lines, as evidenced by sharp shadows.
Diffraction: Light bends around small opaque objects, deviating from straight lines.
Wave theory of light explains diffraction but is inadequate for light-matter interactions.
Quantum theory reconciles wave and particle properties of light.
This chapter focuses on reflection and refraction using the straight-line propagation of light.

Reflection of Light

A highly polished surface, like a mirror, reflects most of the light.
Laws of Reflection:
- The angle of incidence equals the angle of reflection.
- The incident ray, the normal to the mirror at the point of incidence, and the reflected ray all lie in the same plane.
These laws apply to all reflecting surfaces, including spherical ones.
Image formed by a plane mirror is always virtual and erect.
The image size is equal to the object size.
The image is as far behind the mirror as the object is in front.
The image is laterally inverted.

Spherical Mirrors

Curved mirrors can be concave or convex.
Concave mirror: Reflecting surface is curved inwards.
Convex mirror: Reflecting surface is curved outwards.
Pole (P): The center of the reflecting surface of a spherical mirror.
Center of Curvature (C): The center of the sphere of which the mirror is a part; not part of the mirror.
- Concave mirror: C lies in front of the mirror.
- Convex mirror: C lies behind the mirror.
Radius of Curvature (R): Radius of the sphere of which the reflecting surface is a part; distance PC.
Principal Axis: Straight line through the pole and center of curvature; normal to the mirror at the pole.
Principal Focus (F):
- Concave mirror: Point where rays parallel to the principal axis converge after reflection.
- Convex mirror: Point from which rays parallel to the principal axis appear to diverge after reflection.
Focal Length (f): Distance between the pole and the principal focus.
Aperture: Diameter of the reflecting surface of the spherical mirror (MN in Fig.9.2).
Relationship between R and f for spherical mirrors with small apertures: R = 2f

Image Formation by Spherical Mirrors

The nature, position, and size of the image formed by a concave mirror depend on the object's position relative to P, F, and C.

Representation of Images Formed by Spherical Mirrors Using Ray Diagrams

Ray diagrams help visualize image formation.
Consider two rays for clarity:
- Ray parallel to the principal axis:
  - Concave mirror: Passes through the principal focus after reflection.
  - Convex mirror: Appears to diverge from the principal focus after reflection.
- Ray passing through the principal focus:
  - Concave mirror: Emerges parallel to the principal axis after reflection.
  - Convex mirror: Directed towards the principal focus, emerges parallel to the principal axis after reflection.
- Ray passing through the center of curvature:
  - Concave mirror: Reflected back along the same path.
  - Convex mirror: Directed towards the center of curvature, reflected back along the same path.
- Ray incident obliquely to the principal axis at the pole (P):
  - Concave/Convex mirror: Reflected obliquely, following laws of reflection (angle of incidence equals angle of reflection).

Image formation by Concave Mirror

Concave Mirrors are commonly used in torches, search-lights, vehicle headlights, shaving mirrors, and by dentists.
Large concave mirrors concentrate sunlight in solar furnaces.

Image formation by Convex Mirror

Convex Mirrors always give an erect, though diminished, image.
They have a wider field of view.
Used as rear-view mirrors in vehicles.

Sign Convention for Reflection by Spherical Mirrors (New Cartesian Sign Convention)

Pole (P) is the origin.
The principal axis is the x-axis.
Object is always placed to the left of the mirror.
Distances to the right of the origin (+$x$-axis) are positive.
Distances to the left of the origin (-$x$-axis) are negative.
Distances perpendicular to and above the principal axis (+$y$-axis) are positive.
Distances perpendicular to and below the principal axis (-$y$-axis) are negative.

Mirror Formula and Magnification

Object distance: u
Image distance: v
Focal length: f
Mirror formula: \frac{1}{v} + \frac{1}{u} = \frac{1}{f}
Magnification (m): m = \frac{h'}{h}, where h' is the height of the image and h is the height of the object.
Also, m = -\frac{v}{u}
Negative m: Real image.
Positive m: Virtual image.
Example 9.1: Convex mirror, R = +3.00 m, u = –5.00 m. Find v and h'.
Solution: f = R/2 = +1.50 m. Using the mirror formula, v = +1.15 m. m = +0.23 (virtual, erect, smaller).
Example 9.2: Concave mirror, h = +4.0 cm, u = –25.0 cm, f = –15.0 cm. Find v and h'.
Solution: Using the mirror formula, v = –37.5 cm. h' = –6.0 cm (real, inverted, enlarged).

Refraction of light

Refraction: The phenomenon of change in the direction of light propagation when it travels obliquely from one medium to another.
The bottom of a tank or pond containing water appears to be raised due to refraction.

Laws of Refraction

(i) The incident ray, the refracted ray, and the normal to the interface of two transparent media at the point of incidence, all lie in the same plane.
(ii) Snell's Law: The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant for a given color of light and pair of media.
\frac{\sin i}{\sin r} = \text{constant}
This constant is the refractive index of the second medium with respect to the first.

The Refractive Index

The refractive index expresses the extent of change in direction in a given pair of media.
It is related to the speed of light in different media.
Light travels fastest in vacuum (approximately 3 \times 10^8 m/s).
n{21} = \frac{\text{Speed of light in medium 1}}{\text{Speed of light in medium 2}} = \frac{v1}{v_2}
Absolute refractive index: Refractive index with respect to vacuum or air.
n_m = \frac{\text{Speed of light in air}}{\text{Speed of light in the medium}} = \frac{c}{v}
Optical density: The ability of a medium to refract light.
A medium with a larger refractive index is optically denser.
Light slows down and bends towards the normal when traveling from a rarer to a denser medium and vice versa.
The refractive index of water is nw = 1.33, and of crown glass, ng = 1.52.

Refraction by Spherical Lenses

Lens: A transparent material bound by two surfaces, one or both of which are spherical.
Convex lens: Thicker at the middle; converges light rays.
Concave lens: Thicker at the edges; diverges light rays.
Center of curvature (C): The center of the sphere of which the lens surface forms a part. A lens has two centers of curvature, C1 and C2.
Principal axis: Imaginary line passing through the two centers of curvature.
Optical center (O): Central point of the lens; a ray of light through O passes without deviation.
Aperture: Effective diameter of the circular outline of a spherical lens.
Focal length (f): Distance of the principal focus from the optical center.

Image Formation by Lenses

For drawing ray diagrams in lenses, alike of spherical mirrors, we consider any two of the following rays –
- A ray of light from the object, parallel to the principal axis, after refraction from a convex lens, passes through the principal focus on the other side of the lens, as shown in Fig. 9.13 (a). In case of a concave lens, the ray appears to diverge from the principal focus located on the same side of the lens, as shown in Fig. 9.13 (b).
- A ray of light passing through a principal focus, after refraction from a convex lens, will emerge parallel to the principal axis. This is shown in Fig. 9.14 (a). A ray of light appearing to meet at the principal focus of a concave lens, after refraction, will emerge parallel to the principal axis. This is shown in Fig.9.14 (b).
- A ray of light passing through the optical centre of a lens will emerge without any deviation. This is illustrated in Fig.9.15(a) and Fig.9.15 (b).

Sign Convention for Spherical Lenses

Similar to spherical mirrors, but all measurements are taken from the optical center of the lens.
Focal length of a convex lens is positive; concave lens is negative.

Lens Formula and Magnification

\frac{1}{v} - \frac{1}{u} = \frac{1}{f}
Magnification: m = \frac{h'}{h} = \frac{v}{u}

Power of a Lens

The degree of convergence or divergence of light rays achieved by a lens.
Power (P): Reciprocal of focal length: P = \frac{1}{f}
SI unit: Dioptre (D); 1 D = 1 m⁻¹.
Convex lens: Positive power.
Concave lens: Negative power.
Net power of lenses in contact: P = P1 + P2 + P_3 + …