Class 12 Physics · Chapter 9 NotesRay Optics and Optical Instruments

Light is the part of the electromagnetic spectrum with wavelengths roughly between 400 nm and 750 nm, and its speed in vacuum is about 3 × 10⁸ m s⁻¹. Because the wavelength of light is very small compared with everyday objects, light can be treated as travelling in straight lines called rays. When a ray strikes a reflecting surface, the angle of reflection equals the angle of incidence, and the incident ray, reflected ray and normal at the point of incidence all lie in the same plane. These laws hold at every point of a plane or curved surface. For a spherical mirror, the normal at the point of incidence is along the radius, that is, the line joining the centre of curvature to that point.

• Angle of incidence i = angle of reflection r′
• Incident ray, reflected ray and normal are coplanar
• For a spherical mirror the normal passes through the centre of curvature

The Cartesian sign convention measures all distances from the pole of the mirror or the optical centre of the lens. Distances measured in the direction of the incident light are positive, and those measured opposite to it are negative; heights above the principal axis are positive and those below are negative. For paraxial rays, a concave mirror brings a parallel beam to a real focus, while a convex mirror makes the reflected rays appear to diverge from a virtual focus. The focal length f is half the radius of curvature, f = R/2. The mirror equation relates object distance u, image distance v and focal length f, and the linear magnification is m = h′/h = –v/u.

• f = R/2
• Mirror equation1/v + 1/u = 1/f
• Magnificationm = h′/h = –v/u

When light travels obliquely from one transparent medium into another, its direction changes at the interface; this is refraction. Snell's law states that the incident ray, refracted ray and normal are coplanar, and that sin i / sin r = n₂₁, the refractive index of the second medium with respect to the first. If n₂₁ > 1 the refracted ray bends towards the normal and medium 2 is optically denser; if n₂₁ < 1 it bends away from the normal. Optical density is not the same as mass density. A ray passing through a rectangular glass slab emerges parallel to the incident ray but is laterally shifted, and a tank of water appears shallower than it really is.

• Snell's lawsin i / sin r = n₂₁
• n₁₂ = 1/n₂₁
• Apparent depth = real depth / refractive index (for near-normal viewing)

When light travels from an optically denser medium to a rarer medium, the refracted ray bends away from the normal. As the angle of incidence increases, the angle of refraction also increases until, at the critical angle i_c, the refracted ray grazes the interface with r = 90°. For angles of incidence greater than i_c, refraction is not possible and the light is totally reflected back into the denser medium; this is total internal reflection. The critical angle satisfies sin i_c = n₂₁, where n₂₁ is the refractive index of the denser medium with respect to the rarer medium. Totally reflecting prisms bend light by 90° or 180°, and optical fibres use repeated total internal reflections to carry light signals with very little loss.

• sin i_c = n₂₁
• Critical angle for water is about 48.75°, for diamond about 24.41°
• Optical fibrecore of higher refractive index surrounded by cladding of lower refractive index

Refraction at a single spherical surface of radius R separating media of refractive indices n₁ and n₂ obeys n₂/v – n₁/u = (n₂ – n₁)/R. Applying this result to the two surfaces of a thin lens gives the lens maker's formula 1/f = (n₂ – n₁)/n₁ × (1/R₁ – 1/R₂), and the thin lens formula 1/v – 1/u = 1/f. These relations hold for convex and concave lenses and for real and virtual images when the sign convention is used correctly. Magnification for a lens is m = h′/h = v/u. The power of a lens is P = 1/f, measured in dioptres (1 D = 1 m⁻¹), and is positive for a converging lens and negative for a diverging lens.

• Refraction at a spherical surfacen₂/v – n₁/u = (n₂ – n₁)/R
• Thin lens formula1/v – 1/u = 1/f
• PowerP = 1/f, unit dioptre (D)

When two or more thin lenses are placed in contact, the image formed by one lens acts as the object for the next. Adding the thin lens equations for the individual lenses shows that the reciprocal of the effective focal length of the combination equals the sum of the reciprocals of the individual focal lengths: 1/f = 1/f₁ + 1/f₂ + 1/f₃ + ... In terms of power, P = P₁ + P₂ + P₃ + ..., where the sum is algebraic, so converging and diverging lenses partly cancel each other. The total magnification of the combination is the product of the individual magnifications, m = m₁ m₂ m₃ ... Such combinations are used in cameras, microscopes and telescopes to obtain the desired focal length and sharper images.

• 1/f = 1/f₁ + 1/f₂ + 1/f₃ + ...
• P = P₁ + P₂ + P₃ + ...
• m = m₁ m₂ m₃ ...

A triangular glass prism has two refracting surfaces. If A is the angle of the prism, i the angle of incidence, e the angle of emergence and r₁, r₂ the angles of refraction inside the prism, then r₁ + r₂ = A and the angle of deviation is δ = i + e – A. The deviation varies with the angle of incidence and is minimum when the ray passes symmetrically, that is, when i = e and r₁ = r₂. At minimum deviation D_m, r = A/2 and i = (A + D_m)/2, so the refractive index of the prism material is n₂₁ = sin[(A + D_m)/2] / sin(A/2). For a thin prism, D_m = (n₂₁ – 1)A, so a thin prism deviates light only slightly.

• r₁ + r₂ = A
• δ = i + e – A
• n₂₁ = sin[(A + D_m)/2] / sin(A/2)

A simple microscope is a converging lens of short focal length held close to the object so that it forms an erect, magnified, virtual image. When the image is at the near point D = 25 cm, the magnification is m = 1 + D/f; when the image is at infinity, m = D/f. A compound microscope uses an objective of short focal length to form a real, inverted, magnified first image, which the eyepiece then magnifies. Its total magnification is m = (L/f_o)(D/f_e), where L is the tube length and f_o, f_e are the focal lengths of the objective and eyepiece. A telescope uses an objective of large focal length and large aperture with an eyepiece of small focal length; its magnifying power is m = f_o/f_e, and the tube length is f_o + f_e. Reflecting telescopes use a concave mirror objective to avoid chromatic aberration.

• Simple microscopem = 1 + D/f (image at near point), m = D/f (image at infinity)
• Compound microscopem = (L/f_o)(D/f_e)
• Telescopem = f_o/f_e, tube length = f_o + f_e