Ray Optics and Optical Instrument

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Ray Optics and Optical Instruments

1.0Introduction to Ray Optics

Ray Optics, also known as Geometrical Optics, is the branch of optics that describes the propagation of light in terms of rays. A ray is an imaginary line that represents the direction of energy flow of light. This approach simplifies the study of light by ignoring its wave nature and is highly effective for explaining phenomena like reflection and refraction, which are crucial for understanding optical instruments.

2.0Reflection of Light

Reflection is the phenomenon of light bouncing back from a surface. When a light ray strikes a boundary between two media, a part of it returns into the same medium.

Laws of Reflection

The angle of incidence ( $θ_{i} $ ) is equal to the angle of reflection ( $θ_{r} $ ).
The incident ray, the reflected ray, and the normal to the surface at the point of incidence all lie in the same plane.

(a) The incident ray, the reflected ray and the normal at the point of incidence lie in the same plane. This plane is called the plane of incidence (or plane of reflection). This condition can be expressed mathematically as

$\overset{e}{^} \cdot (\overset{n}{^} \times \overset{r}{^}) = 0$

(b) The angle of incidence (the angle between normal and the incident ray) and the angle of reflection (the angle between the reflected ray and the normal) are equal, i.e.

$∠ i = ∠ r$

In vector form both laws can be expressed as

In Vector form

$\overset{r}{^} = \overset{e}{^} - 2 (\overset{e}{^} \cdot \overset{n}{^}) \overset{n}{^}$

Plane Mirrors

A plane mirror is a flat reflecting surface. The(a) The incident ray, the reflected ray and the normal at the point of incidence lie in the same plane. This plane is called the plane of incidence (or plane of reflection). This condition can be expressed mathematically as

$\overset{e}{^} \cdot (\overset{n}{^} \times \overset{r}{^}) = 0$

(b) The angle of incidence (the angle between normal and the incident ray) and the angle of reflection (the angle between the reflected ray and the normal) are equal, i.e.

$∠ i = ∠ r$

The image formed by a plane mirror is always virtual, erect, and of the same size as the object. The distance of the image from the mirror is equal to the distance of the object from the mirror.

Spherical Mirrors (Concave and Convex)

Spherical mirrors are part of a hollow sphere.

Concave Mirror: The reflecting surface is curved inwards. It converges incident light rays. These mirrors are used in headlights, telescopes, and shaving mirrors.
Convex Mirror: The reflecting surface is curved outwards. It diverges incident light rays. These mirrors are used as rear-view mirrors in vehicles due to their wide field of view.

Mirror Formula

The relationship between the object distance (u), image distance (v), and focal length (f) for spherical mirrors is given by the mirror formula:

$\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$

The sign convention is critical for solving problems. Usually, all distances are measured from the pole (P) of the mirror. Distances measured in the direction of the incident light are taken as positive, and those in the opposite direction are negative.

3.0Refraction of Light

Laws of Refraction:

(a) The incident ray, the normal to any refracting surface at the point of incidence and the refracted ray all lie in the same plane called the plane of incidence or plane of refraction.

(b) $\frac{s i n i}{s i n r} = Constant for any pair of media and for light of a given wave length. This is known as Snell’s Law. A l so, \frac{s i n i}{s i n r} = \frac{n _{2}}{n _{1}} = \frac{v _{1}}{v _{2}} = \frac{λ _{1}}{λ _{2}} For applying in problems remember n_{1} sin i = n_{2} sin r \frac{n _{2}}{n _{1}} =^{1} n_{2} = Refractive Index of the second medium with respect to the first medium. c = speed of light in air (or vacuum) = 3 \times 1 0^{8} m/s .$

Refraction through a Glass Slab

When light passes through a parallel-sided glass slab, it emerges parallel to the incident ray, but with a lateral shift or displacement. The angle of emergence is equal to the angle of incidence.

Total Internal Reflection (TIR)

Critical Angle and Total Internal Reflection (T. I. R.)

Critical angle is the angle made in denser medium for which the angle of refraction in rarer medium is 90°. When angle in denser medium is more than critical angle, then the light ray reflects back in denser medium following the laws of reflection and the interface behaves like a perfectly reflecting mirror.

In the figure

Critical Angle and Total Internal Reflection (T. I. R.)

$In the figure O = Object N N^{'} = Normal to the interface I I^{'} = Interface θ_{c} = Critical angle; A B = reflected ray due to T. I. R. c When i = θ_{c} then r = 9 0^{\circ} ∴ θ_{c} = sin^{- 1} (\frac{n _{r}}{n _{d}})$

Conditions of T.I.R.

(a) Light is incident on the interface from denser medium.

(b) Angle of incidence should be greater than the critical angle (i > θ_c). Figure shows a luminous object placed in denser medium at a distance h from an interface separating two media of refractive indices μ_r and μ_d. Subscript r and d stand for rarer and denser medium respectively.

In the figure, ray 1 strikes the surface at an angle less than critical angle θc and gets refracted in rarer medium. Ray 2 strikes the surface at a critical angle and grazes the interface. Ray 3 strikes the surface making an angle more than critical angle and gets internally reflected. The locus of points where ray strikes at critical angle is a circle, called circle of illuminance. All light rays striking inside the circle of illuminance get refracted in a rarer medium. If an observer is in rarer medium, he/she will see light coming out only from within the circle of illuminance. If a circular opaque plate covers the circle of illuminance, no light will get refracted in rarer medium and then the object can not be seen from the rarer medium. Radius of C.O.I can be easily found.

4.0Refraction at Spherical Surfaces and Lenses

Lens Maker's Formula

Lens-Maker's formula:

It relates the focal length of the lens to the relative refractive index μ of the lens material and the radii of curvature of the two surfaces

$\frac{1}{f} = (μ - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}}) Where, μ = \frac{μ _{2}}{μ _{1}} = \frac{Refractive index of lens}{Refractive index of surrounding}$

$R_{1}$ is the radius of curvature of first surface and $R_{2}$ is the radius of curvature of the second surface from where light emerges out in the first medium.

Thin Lens Formula

This formula is the lens equivalent of the mirror formula:

$\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$

The same Cartesian sign convention applies here.

Power of a Lens

The power (P) of a lens is its ability to converge or diverge light. It is defined as the reciprocal of the focal length in meters.

$P = \frac{1}{f ( inm e t ers )}$

The unit of power is the dioptre (D).
A convex lens has a positive power, and a concave lens has a negative power.

Combination of Lenses

When multiple thin lenses are placed in contact, their powers simply add up.

$P_{eq} = P_{1} + P_{2} + P_{3} + \dots$

The equivalent focal length is given by:

$\frac{1}{f _{eq}} = \frac{1}{f _{1}} + \frac{1}{f _{2}} + \dots$

5.0Prisms

A prism is a transparent optical element with flat, polished surfaces that refract light.

Angle of Deviation (δ):

It is the angle between the emergent and the incident ray. In other words, it is the angle through which incident ray turns while passing through a prism.

δ = (i − r₁) + (e − r₂)

= i + e − [r₁ + r₂]

= i + e − A

Condition for minimum Deviation:

The minimum deviation occurs when the angle of incidence is equal to the angle of emergence.

$i . e ., i = e A s δ = i + e - A δ_{m i n} = 2 i - A Using Snell’s law 1 \times sin i = μ sin r_{1} an d μ sin r_{2} = 1 \times sin i ∴ sin r_{1} = sin r_{2} \Rightarrow r_{1} = r_{2} = r ∴ δ_{m i n} = (2 i - A) w h ere r = \frac{A}{2} μ = \frac{s i n i}{s i n r} = \frac{s i n ( \frac{δ _{m i n} + A}{2} )}{s i n ( \frac{A}{2} )}$

Dispersion of Light:

The angular splitting of a ray of white light into a number of components and spreading in different directions is called Dispersion of Light. [It is for whole Electro Magnetic Wave in totality]. This phenomenon is because waves of different wavelength move with same speed in vacuum but with different speeds in a medium.

Therefore, the refractive index of a medium depends slightly on wavelength also. This variation of refractive index with wavelength is given by Cauchy’s formula.

Cauchy’s formula $μ (λ) = a + (\frac{b}{λ ^{2}})$ where a and b are positive constants of a medium.

6.0Optical Instruments

Optical instruments use the principles of reflection and refraction to aid vision or analyze light.

The Human Eye:

A natural optical instrument. It has a convex lens that focuses light onto the retina. Common defects include myopia (nearsightedness, corrected by a concave lens) and hypermetropia (farsightedness, corrected by a convex lens).

Simple Microscope:

Also known as a magnifying glass, it is a single convex lens used to view magnified virtual images of small objects. The maximum magnifying power is $M = 1 + \frac{D}{f}$ where D is the least distance of distinct vision.

Compound Microscope:

Uses two convex lenses—an objective lens and an eyepiece—to achieve a much higher magnification than a simple microscope. The total magnification is the product of the magnifications of the two lenses: $M_{t o t a l} = m_{o} \times m_{e}$

Astronomical Telescope:

Used to view distant celestial objects. It also uses an objective lens and an eyepiece, but the objective lens has a large focal length and a large aperture to gather more light. The magnifying power is $M = - \frac{f _{o}}{f _{e}}$