Comprehensive Notes on Light: Reflection and Refraction
Reflection and Refraction of Light
Introduction
- We see objects because they reflect light into our eyes.
- Transparent mediums allow light to pass through.
- Light seems to travel in straight lines.
- Diffraction is the bending of light around small objects; in this case, the straight-line treatment fails.
- Light has properties of both waves and particles, according to the quantum theory of light.
- This chapter will cover reflection and refraction using the straight-line propagation of light.
Reflection of Light
- A polished surface, like a mirror, reflects most of the light that falls on it.
- 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 surfaces.
- Image formed by a plane mirror is always virtual and erect.
- The size of the image is equal to the size of the object.
- The image is as far behind the mirror as the object is in front of it.
- 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.
- Centre of Curvature (C): The center of the sphere of which the reflecting surface is a part; it is not part of the mirror.
- The center of curvature of a concave mirror lies in front of it.
- However, it lies behind the mirror in the case of a convex mirror.
- Radius of Curvature (R): The radius of the sphere of which the reflecting surface is a part; the distance PC is equal to R.
- Principal Axis: A straight line passing through the pole and the center of curvature; it is normal to the mirror at its pole.
- Principal Focus (F): The point on the principal axis where rays parallel to the principal axis converge after reflection from a concave mirror, or appear to diverge from after reflection from a convex mirror.
- Focal Length (f): The distance between the pole and the principal focus.
- Aperture: The diameter of the reflecting surface of the spherical mirror.
- Relationship between R and f: For spherical mirrors with small apertures, R = 2f.
- The nature, position, and size of the image formed by a concave mirror depend on the position of the object relative to points P, F, and C.
- Images can be real or virtual, magnified, diminished, or the same size.
Representation of Images Using Ray Diagrams
- Ray diagrams are used to study the formation of images by spherical mirrors.
- Two rays are sufficient to locate the image of an object.
- A ray parallel to the principal axis, after reflection, will pass through the principal focus (concave mirror) or appear to diverge from it (convex mirror).
- A ray passing through the principal focus (concave mirror) or directed towards it (convex mirror), after reflection, will emerge parallel to the principal axis.
- A ray passing through the center of curvature (concave mirror) or directed towards it (convex mirror), after reflection, is reflected back along the same path.
- A ray incident obliquely to the principal axis at the pole (P) is reflected obliquely, with the angle of incidence equal to the angle of reflection.
- At infinity: the image is at the focus F, highly diminished, point-sized, real, and inverted.
- Beyond C: the image is between F and C, diminished, real, and inverted.
- At C: the image is at C, same size, real, and inverted.
- Between C and F: the image is beyond C, enlarged, real, and inverted.
- At F: the image is at infinity.
- Between P and F: the image is behind the mirror, enlarged, virtual, and erect.
- At infinity: the image is at the focus F, behind the mirror, highly diminished, point-sized, virtual, and erect.
- Between infinity and the pole P: the image is between P and F, behind the mirror, diminished, virtual, and erect.
Uses of Concave Mirrors
- Torches, searchlights, and vehicle headlights.
- Shaving mirrors.
- Dentists' mirrors.
- Solar furnaces.
Uses of Convex Mirrors
- Rear-view mirrors in vehicles because they:
- Give an erect, diminished image.
- Have a wider field of view.
Sign Convention for Reflection by Spherical Mirrors
- New Cartesian Sign Convention:
- The 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 are positive, to the left are negative.
- Distances above the principal axis are positive, below are negative.
- Object distance (u): Distance of the object from the pole.
- Image distance (v): Distance of the image from the pole.
- Focal length (f): Distance of the principal focus from the pole.
- Mirror formula: \frac{1}{v} + \frac{1}{u} = \frac{1}{f}
- Magnification (m): Ratio of the height of the image to the height of the object.
- m = \frac{h'}{h}
- Also, m = - \frac{v}{u}
- A negative m indicates a real image, and a positive m indicates a virtual image.
Refraction of Light
- Refraction is the bending of light as it passes from one transparent medium to another.
- The extent of bending depends on the media involved.
Laws of Refraction
- 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.
- 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 is related to the speed of light in different media.
- Light travels fastest in a vacuum (3 \times 10^8 \text{ m/s}).
- Refractive index of medium 2 with respect to medium 1: n{21} = \frac{\text{Speed of light in medium 1}}{\text{Speed of light in medium 2}} = \frac{v1}{v_2}
- Refractive index of medium 1 with respect to medium 2: n{12} = \frac{\text{Speed of light in medium 2}}{\text{Speed of light in medium 1}} = \frac{v2}{v_1}
- Absolute refractive index of a medium: Refractive index with respect to vacuum.
- n_m = \frac{\text{Speed of light in air}}{\text{Speed of light in the medium}} = \frac{c}{v}
- Optical density: A medium's ability to refract light.
- A medium with a larger refractive index is optically denser.
- The speed of light is higher in a rarer medium than in a denser medium.
- Light traveling from a rarer to a denser medium slows down and bends towards the normal.
- Light traveling from a denser to a rarer medium speeds up and bends away from the normal.
Refraction by Spherical Lenses
- A lens is a transparent material bound by two surfaces, one or both of which are spherical.
- Convex lens: Thicker at the middle; converges light rays and is called a converging lens.
- Concave lens: Thicker at the edges; diverges light rays and is called a diverging lens.
- Each lens has two centers of curvature (C1 and C2).
- Principal axis: An imaginary straight line passing through the two centers of curvature.
- Optical center (O): The central point of a lens; a ray of light passing through it goes without deviation.
- Aperture: The effective diameter of the circular outline of a spherical lens.
- Principal focus (F): The point on the principal axis where rays parallel to the principal axis converge (convex lens) or appear to diverge from (concave lens).
- Each lens has two principal foci, F1 and F2.
- Focal length (f): The distance of the principal focus from the optical center.
- The nature, position, and size of the image formed by a convex lens depend on the position of the object.
- A concave lens always gives a virtual, erect, and diminished image.
- At infinity: image is at focus F2, highly diminished, point-sized, real, and inverted.
- Beyond 2F1: image is between F2 and 2F2, diminished, real, and inverted.
- At 2F1: image is at 2F2, same size, real, and inverted.
- Between F1 and 2F1: image is beyond 2F2, enlarged, real, and inverted.
- At focus F1: image is at infinity.
- Between focus F1 and optical center O: image is on the same side of the lens as the object, enlarged, virtual and erect.
- At infinity: image is at focus F1, highly diminished, point-sized, virtual, and erect.
- Between infinity and optical center O: image is between focus F1 and optical center O of the lens, diminished, virtual and erect.
- (i) 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. In the case of a concave lens, the ray appears to diverge from the principal focus located on the same side of the lens.
- (ii) A ray of light passing through a principal focus, after refraction from a convex lens, will emerge parallel to the principal axis. A ray of light appearing to meet at the principal focus of a concave lens, after refraction, will emerge parallel to the principal axis.
- (iii) A ray of light passing through the optical centre of a lens will emerge without any deviation.
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, and that of a concave lens is negative.
- Lens formula: \frac{1}{v} - \frac{1}{u} = \frac{1}{f}
- Magnification: m = \frac{h'}{h} = \frac{v}{u}
Power of a Lens
- The power of a lens is the reciprocal of its focal length.
- P = \frac{1}{f}
- The SI unit of power is dioptre (D).
- 1 D = 1 m^{-1}
- The power of a convex lens is positive, and that of a concave lens is negative.
- The net power of lenses placed in contact is the algebraic sum of the individual powers: P = P1 + P2 + P_3 + \dots