CBSE Class 12 Physics Ray Optics and Optical Instruments Notes Set 04

1. Reflection of Light by Spherical Mirrors

A spherical mirror is a part of a hollow sphere whose one surface is polished. If the reflecting surface curves inward it is a concave mirror; if it bulges outward it is a convex mirror.
(i) Laws of reflection: The angle of reflection equals the angle of incidence, and the incident ray, the reflected ray and the normal at the point of incidence lie in one plane. For a spherical mirror the normal at any point is along the radius through that point.
(ii) Terms: The pole P is the centre of the mirror surface; the centre of curvature C is the centre of the sphere; the radius of curvature R = PC; the principal axis is the line through P and C.
(iii) Focus: Rays parallel to the principal axis (paraxial rays) meet, or appear to meet, at the principal focus F after reflection. The focal length f = PF.
f = \( \frac{R}{2} \)

Cartesian sign convention:
1. All distances are measured from the pole (or the optical centre of a lens).
2. Distances in the direction of the incident light are positive; distances opposite to it are negative.
3. Heights above the principal axis are positive; heights below it are negative.
With light travelling from left to right, a real object always has u negative. A concave mirror has f negative and a convex mirror has f positive.

2. Mirror Equation and Magnification

For an object at distance u and its image at distance v from the pole,
\( \frac{1}{v} + \frac{1}{u} = \frac{1}{f} \)
Linear magnification: m = \( \frac{h'}{h} = -\frac{v}{u} \)
1. m negative: the image is real and inverted. m positive: the image is virtual and erect.
2. |m| > 1: the image is magnified; |m| < 1: the image is diminished.
3. A concave mirror gives a real, inverted image for an object beyond F, and a virtual, erect, magnified image for an object between P and F (used as a shaving mirror and by dentists).
4. A convex mirror always gives a virtual, erect and diminished image, and gives a wide field of view (used as a rear-view mirror in vehicles).
5. The focal length of a mirror depends only on its curvature, so it does not change when the mirror is placed in water.

3. Refraction of Light

When light passes obliquely from one medium to another, its direction changes at the boundary. This is refraction.
Snell's law: \( \frac{\sin i}{\sin r} = n_{21} = \frac{n_2}{n_1} \), where n₂₁ is the refractive index of medium 2 with respect to medium 1. Equivalently, n₁ sin i = n₂ sin r.
Absolute refractive index: n = \( \frac{c}{v} \), where c = 3 × 10⁸ m s⁻¹ is the speed of light in vacuum and v is its speed in the medium. Since v < c in every material medium, n > 1. Refractive index has no unit.
1. A ray going from a rarer to a denser medium bends towards the normal; from denser to rarer it bends away from the normal.
2. The frequency of light does not change on refraction; its speed and wavelength change: λ_medium = \( \frac{\lambda_{\text{vacuum}}}{n} \).
3. n₁₂ = \( \frac{1}{n_{21}} \), and n₃₂ = n₃₁ × n₁₂.
4. A glass slab with parallel faces does not deviate a ray; it only shifts it sideways (lateral shift).

Real and apparent depth: An object at the bottom of water appears raised when seen from above, almost along the normal.
Apparent depth = \( \frac{\text{Real depth}}{n} \), and the object appears raised by d\( \left(1 - \frac{1}{n}\right) \), where d is the real depth.

4. Total Internal Reflection

When light travels from a denser medium to a rarer medium, the refracted ray bends away from the normal. As the angle of incidence increases, the angle of refraction reaches 90°. The angle of incidence in the denser medium for which the angle of refraction in the rarer medium is 90° is called the critical angle C.
For a denser medium of refractive index n with respect to air (or vacuum): sin C = \( \frac{1}{n} \)
In general, for light going from medium 1 (denser) to medium 2 (rarer): sin C = \( \frac{n_2}{n_1} \)

If the angle of incidence is greater than C, no refraction takes place and all the light is reflected back into the denser medium. This is total internal reflection (TIR).
Conditions for TIR:
1. Light must travel from an optically denser medium towards an optically rarer medium.
2. The angle of incidence must be greater than the critical angle for the pair of media.
For crown glass (n = 1.5), C ≈ 41.8°; for water (n ≈ 1.33), C ≈ 48.8°.

Applications of total internal reflection:
(i) Totally reflecting prisms: A right-angled isosceles glass prism (angles 90°, 45°, 45°) reflects light totally at its hypotenuse face because the angle of incidence there is 45°, which is greater than the critical angle of glass (about 41.8°). Such prisms bend a ray through 90° or 180°, or invert an image without changing its size, and are used in periscopes and binoculars.
(ii) Optical fibres: An optical fibre has a thin glass or quartz core surrounded by a cladding of lower refractive index. Light entering one end strikes the core–cladding boundary at angles greater than the critical angle and undergoes repeated total internal reflections, so it travels along the fibre with very little loss even when the fibre is bent. Uses: transmitting telephone and internet signals, endoscopy (viewing inside the body) and decorative lamps.

5. Refraction at a Spherical Surface

For a spherical surface of radius of curvature R separating a medium of refractive index n₁ (where the object lies) from a medium of refractive index n₂, with paraxial rays,
\( \frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R} \)
Here u, v and R follow the Cartesian sign convention, measured from the pole of the surface. R is positive when the centre of curvature lies on the side of the outgoing light (a surface bulging towards the object).

6. Thin Lenses and the Lens Maker's Formula

A lens is a transparent medium bounded by two surfaces, at least one of which is curved. A convex (converging) lens is thicker at the middle; a concave (diverging) lens is thinner at the middle. The optical centre O is the point through which a ray passes undeviated.
Applying the result for refraction at a spherical surface to both surfaces of a thin lens of refractive index n placed in air gives the lens maker's formula:
\( \frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \)
R₁ and R₂ are the radii of curvature of the first and second surfaces met by the light.
1. Double convex lens: R₁ positive, R₂ negative, so f is positive. Double concave lens: f is negative.
2. For a lens of refractive index n₂ placed in a medium of refractive index n₁, replace (n − 1) by \( \left(\frac{n_2}{n_1} - 1\right) \). So the focal length of a glass lens increases in water.
3. If the surrounding medium has a greater refractive index than the lens, \( \left(\frac{n_2}{n_1} - 1\right) \) becomes negative and a convex lens behaves as a diverging lens.
4. If the lens and the medium have the same refractive index, f becomes infinite and the lens produces no bending at all.

7. Thin Lens Formula and Magnification

For an object at distance u and its image at distance v, both measured from the optical centre,
\( \frac{1}{v} - \frac{1}{u} = \frac{1}{f} \)
Linear magnification: m = \( \frac{h'}{h} = \frac{v}{u} \)
For a convex lens, a real object beyond F gives a real, inverted image on the other side; an object between O and F gives a virtual, erect, magnified image on the same side (magnifying glass). A concave lens always gives a virtual, erect, diminished image between the object and the lens.

Ray diagram rules: (i) a ray parallel to the principal axis passes through the second focus F₂ after refraction (a concave lens makes it appear to come from F₁); (ii) a ray through the optical centre goes straight on; (iii) a ray through the first focus F₁ emerges parallel to the axis. The figure, drawn to scale, shows an object between F₁ and 2F₁ of a convex lens: the image is real, inverted and twice the size, beyond 2F₂.

If part of a lens is covered, a complete image is still formed, because every part of the lens forms the whole image; only the brightness of the image decreases.

8. Power of a Lens

The power of a lens is a measure of how strongly it converges or diverges light. It is the reciprocal of the focal length in metres.
P = \( \frac{1}{f\,(\text{in m})} \) = \( \frac{100}{f\,(\text{in cm})} \)
SI unit: dioptre (D); 1 D = 1 m⁻¹. A convex lens has positive power and a concave lens has negative power. For example, a lens of focal length +25 cm has P = +4 D.

9. Combination of Thin Lenses in Contact

For two thin lenses of focal lengths f₁ and f₂ placed in contact, the combination acts as a single lens of focal length F given by
\( \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} \), and in terms of power, P = P₁ + P₂ + P₃ + …
For lenses placed one after another (not necessarily in contact), the total magnification is the product of the individual magnifications: m = m₁ × m₂ × m₃ × …
Combining lenses helps to obtain a desired power or magnification and to make sharper images, which is why camera and microscope objectives use several lenses.

10. Refraction through a Prism

A prism is a transparent medium bounded by two plane refracting faces inclined at an angle A, called the angle of the prism. A ray passing through the prism is bent towards its base. The angle between the incident ray and the emergent ray is the angle of deviation δ.
If r₁ and r₂ are the angles of refraction at the first face and of incidence at the second face (inside the prism), and e is the angle of emergence:
A = r₁ + r₂ and δ = i + e − A

Minimum deviation: As the angle of incidence i is increased from a small value, the deviation first decreases, reaches a minimum value δ_m, and then increases. At minimum deviation:
1. i = e and r₁ = r₂ = \( \frac{A}{2} \); the ray passes symmetrically through the prism (parallel to the base in an isosceles prism).
2. i = \( \frac{A + \delta_m}{2} \), and the refractive index of the prism is
n = \( \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)} \)
Thin prism (A small, a few degrees): δ = (n − 1)A, so the deviation does not depend on the angle of incidence.

11. Simple Microscope

A simple microscope (magnifier) is a convex lens of short focal length f held close to the eye. The object is placed between the lens and its focus, so an erect, virtual, magnified image is seen. D = 25 cm is the least distance of distinct vision (near point of a normal eye).
(i) Image at the near point (v = −D): m = 1 + \( \frac{D}{f} \)
(ii) Image at infinity (object at F, relaxed eye): m = \( \frac{D}{f} \)
The magnifying power is the ratio of the angle subtended by the image at the eye to the angle subtended by the object when kept at D. A shorter focal length gives a larger magnification.

12. Compound Microscope

A compound microscope has two convex lenses: an objective of very short focal length and small aperture near the object, and an eyepiece of somewhat larger focal length near the eye. The objective forms a real, inverted, magnified image, which acts as the object for the eyepiece; the eyepiece works as a simple microscope and gives a large virtual image. The final image is inverted with respect to the object.
Magnifying power m = m_o × m_e
With the final image at infinity: m = \( \frac{L}{f_o} \times \frac{D}{f_e} \), where L (tube length) is the distance between the second focal point of the objective and the first focal point of the eyepiece.
With the final image at the near point: m = \( \frac{L}{f_o}\left(1 + \frac{D}{f_e}\right) \) (approximately).
To get a large magnifying power, both f_o and f_e should be small.

13. Telescopes

Refracting (astronomical) telescope: It has an objective of large focal length and large aperture, and an eyepiece of small focal length. Light from a distant object enters the objective as a parallel beam, and the objective forms a real, inverted image at its focal plane. The eyepiece magnifies this image. In normal adjustment the final image is at infinity, and the eyepiece focus coincides with the objective focus.

1. Magnifying power (normal adjustment): m = \( \frac{\beta}{\alpha} = \frac{f_o}{f_e} \)
2. Length of the telescope tube (normal adjustment): L = f_o + f_e
3. Final image at the near point: m = \( \frac{f_o}{f_e}\left(1 + \frac{f_e}{D}\right) \)
4. A large objective aperture collects more light, so faint objects such as distant stars appear brighter.

Reflecting telescope (Cassegrain type): The objective is a large concave (paraboloidal) mirror. Light reflected from it is reflected again by a small convex secondary mirror back through a hole in the primary mirror to the eyepiece. Advantages over a refracting telescope:
1. A mirror produces no chromatic aberration (no colour fringes), and a paraboloidal mirror removes spherical aberration.
2. A mirror can be supported over its whole back, so it can be made much larger and lighter than a lens of the same size.

14. Quick Formula Recap