Revision: Optics CUET (UG) Optics

The bouncing of light by any smooth or polished surface is called.

The phenomenon due to which a parallel beam of light traveling through a certain medium, on striking some polished surface, bounces off from it, as a parallel beam, in some other direction, is called regular reflection.

The principal axis is the straight line passing through the pole and the centre of curvature.

The phenomenon of bouncing back of light rays in the same medium on striking a surface is called reflection of light.

The distance of the image from the pole of the mirror is called the image distance (v).

In a spherical mirror, the distance of the object from its pole is called the object distance (u).

The distance of the principal focus from the pole is called the focal length (f).

Refracted light is the part of light enters into the other medium and travels in a straight path but in a direction different from its initial direction and is called the refracted light.

Light rays that are parallel to the principal axis of a concave mirror converge at a specific point on its principal axis after reflecting from the mirror. This point is known as the principal focus of the concave mirror.

When travelling obliquely from one medium to another, the direction of propagation of light in the second medium changes. This phenomenon is known as refraction of light.

OR

Light changes its direction when going from one transparent medium to another transparent medium. This is called the refraction of light.

OR

The bending of the light ray from its path in passing from one medium to the other medium is called 'refraction' of light.

OR

When a ray of light impinges on a polished, smooth, shiny surface, the rebounding of light within the same medium is called reflection of light.

The change in the direction of the path of light when it passes from one transparent medium to another transparent medium is called refraction. The refraction of light is essentially a surface phenomenon.

or

When light passes from one transparent medium to another, its speed and direction change. This is called refraction.

When a ray of light propagates from a denser medium to a rarer medium, the angle of incidence for which the angle of refraction is 90° is called the critical angle.

The phenomenon where light rays are completely reflected back into a medium instead of being refracted into another medium is called total internal reflection.

or

Complete reflection of a ray of light at the interface of an optically denser medium and a rarer medium, back into the denser medium.

The angle of incidence in the denser medium corresponding to an angle of refraction of 90° in the rarer medium is called the critical angle.

The aberration caused by the spherical shape of the lens, where light rays at the edges focus at a different point than those near the centre, leading to a blurred image, is called spherical aberration.

The aberration that occurs due to the lens refracting different wavelengths of light at different angles, resulting in an image consisting of different colours without a single focussed image, is called chromatic aberration.

When rays of light parallel to the principal axis of a mirror are incident on it, the rays after reflection either converge at a point or appear to diverge from a point. The distance of that point from the pole of the mirror is known as the focal length of the mirror.

The deviation of the incident light rays produced by a lens on refraction through it, is a measure of its power.

or

The power of a lens is defined as the reciprocal of its focal length. It is represented by the letter P.

OR

The power (P) of a thin lens is equal to the reciprocal of its focal length (f) measured in metres.

Power of a lens is defined as the ability of a lens to bend the rays of light. It is given by the reciprocal of focal length in metre.

The power of a lens is a measure of the deviation produced by it in the path of rays refracted through it.

The angular separation between the two extreme colours (violet and red) in the spectrum (which is obtained by passing a beam of white light through a prism) is known as angular dispersion.

Angular magnification or magnifying power of an optical instrument is defined as the ratio of the visual angle made by the image formed by that optical instrument (β) to the visual angle subtended by the object when kept at the least distance of distinct vision (α).

An optical instrument that uses a single convex lens to magnify small objects is called a simple microscope.

An optical instrument that uses objective and eye piece lenses to magnify tiny objects in detail is called a compound microscope.

An optical instrument that uses objective and eye piece lenses to magnify distant terrestrial or celestial objects is called a telescope.

The resolving power of an astronomical telescope is defined as the reciprocal of the smallest angular separation between two point objects whose images can just be resolved by the telescope.

R.P = `(1.22 lambda)/D`

Resolving power is the ability of the telescope to distinguish clearly between two points whose angular separation is less than the smallest angle that the observer’s eye can resolve.

Wavefront is defined as the locus of all the points in space that reach a particular distance by a propagating wave at the same instant.

A wave front is defined as a surface of constant phase.

The points of minimum intensity in the regions of superposition of waves are said to be in destructive interference.

The phenomenon of redistribution of energy on account of superposition of light waves from two coherent sources is called interference of light.

The phenomenon that occurs when two waves meet while travelling along the same medium is called wave interference.

The points of maximum intensity in the regions of superposition of waves are said to be in constructive interference.

Alignment of dipole moments (permanent or induced) in the direction of an applied electric field is called polarisation.

n = \[\frac {360°}{θ}\]

• If n is even → N = n − 1
• If n is odd → N = n (object not on bisector); N = n − 1 (object on bisector)
• If n is a fraction → N = integral part of n

\[\frac {1}{v}\] + \[\frac {1}{u}\] = \[\frac {1}{f}\]

Magnification (m) = \[\frac{\text{Height of the image (}h'\text{)}}{\text{Height of the object (}h\text{)}}\] = \[\frac {h'}{h}\]

Magnification in terms of object and image distances:

Magnification (m) = \[\frac {h'}{h}\] = -\[\frac {v}{u}\]

d = t - \[\frac {t}{μ}\] = t\[\left(1-\frac{1}{\mu}\right)\]

n = \[\frac {\text {sin i}}{\text {sin r}}\] = \[\frac {c}{v}\] = \[\frac {\text {Real depth}}{\text {Apparent depth}}\]

Power of lens (in D) = \[\frac{1}{\text{focal length (in metre)}}\]

or

P = \[\frac {1}{f}\]

or

P = \[\frac {1}{f (m)}\]

Power of a Lens in a Medium:

P = (n₂ - n₁)\[\left(\frac{1}{R_{1}}-\frac{1}{R_{2}}\right)\] = \[\frac {n_1}{f}\]

1. M_Max = 1 + \[\frac {D}{f}\]
2. M_Min = \[\frac {D}{f}\]

M = m_o × M_e

1. \[\mathrm{M_{D.D.V}=\frac{f_{o}}{f_{e}}\left(1+\frac{f_{e}}{D}\right)}\]
2. M = \[\frac{\mathrm{f}_{0}}{\mathrm{f}_{0}}\]

When two waves of amplitudes a₁ and a_2​ interfere at a point where phase difference is ϕ, the resultant amplitude is:

\[A^2=a_1^2+a_2^2+2a_1a_2\cos\phi\]

I = I₁ ​+ I₂​ + 2\[\sqrt {I_1​I_2}\] ​​⋅ cos ϕ

When I₁ = I₂ = I₀:

I = \[2I_0(1+\cos\phi)=4I_0\cos^2\left(\frac{\phi}{2}\right)\]

\[\frac{I_{\max}}{I_{\min}}=\left(\frac{a_1+a_2}{a_1-a_2}\right)^2=\left(\frac{\sqrt{I_1}+\sqrt{I_2}}{\sqrt{I_1}-\sqrt{I_2}}\right)^2\]

\[\begin{array} {c}x_n=\frac{n\lambda D}{d}=n\beta \end{array}\]

\[\begin{array} {cc} & x_m=\frac{(2m-1)\lambda D}{2d} \end{array}\]

\[\beta=\frac{\lambda D}{d}\]

\[\alpha=\frac{\beta}{D}=\frac{\lambda}{d}\]

\[x_n-x_m=\left[n-\frac{(2m-1)}{2}\right]\beta\]

Defined as dipole moment per unit volume:

\[P=\frac{\text{dipole moment}}{\mathrm{volume}}=np\]

• The angle of incidence ∠i = angle of reflection ∠r.
• The incident ray, reflected ray, and normal lie in one plane; both rays are on either side of the normal.

Thomas Young first demonstrated the phenomenon of interference of light with the help of a slit, using a monochromatic source and two slits S₁ and S₂​, producing alternating bright fringes (constructive interference) and dark fringes (destructive interference) on a screen.

Statement:

When unpolarised light is incident at polarising angle i_B on an interface separating air from a medium of refractive index μ, then the reflected light is plane polarised (perpendicular to the plane of incidence), provided:

Additional condition at polarising angle:

i.e., the reflected plane polarised light is at right angles to the refracted light.

OR

Statement:

• When the angle of incidence equals the polarising angle (θ_B), the reflected and refracted rays are perpendicular to each other.
• "The refractive index of a medium is equal to the tangent of the polarising angle θ_B."

• Reflection occurs when light bounces off a smooth surface like a mirror, following fixed laws.
• Plane mirrors always form virtual, erect, and same-sized images that are laterally inverted.
• Curved surfaces (like a spoon) act as spherical mirrors, changing the image size and orientation depending on the object's position.

Convex lens: Thicker at the middle, thinner at the edges, converges parallel rays.

• Types: Double-convex, Plano-convex, Concavo-convex.

Concave lens: Thinner at the middle, thicker at the edges, diverges parallel rays.

• Types: Double-concave, Plano-concave, Convexo-concave.

Important Terms:

• Principal Axis: Straight line through the centres of curvature of two surfaces.
• Optical Centre (O): Ray passing through it goes undeviated with no lateral displacement.
• Principal Focus (F): Point where parallel rays converge (convex) or appear to diverge (concave) after refraction.
• Aperture: Effective diameter of light-transmitting area; Intensity ∝ (Aperture)²

Lens Maker's Formula:

\[\frac{1}{f}=(\mu-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)\]

Where:

• f = focal length, μ = refractive index of lens material
• R₁, R₂ = radii of curvature of the two surfaces

New Cartesian Sign Conventions for Lens:

• Distances measured from optical centre.
• Along incident light → positive; against incident light → negative.
• For convex lens: f is positive; for concave lens: f is negative.

Rules for Image Formation:

1. Ray through optical centre → passes undeviated.
2. Ray parallel to principal axis → passes through F₂ (convex) or appears to come from F₁ (concave).
3. Ray towards/through first focus F₁ → emerges parallel to principal axis.

Lens Formula:

\[\frac{1}{v}-\frac{1}{u}=\frac{1}{f}\]

• A prism is a transparent medium bounded by two polished plane surfaces inclined at an angle (< 90°).
• The angle between the refracting faces = Refracting angle (A).
• A ray of light undergoes two refractions passing through a prism.

Angle of Deviation:

Minimum Deviation (δ_min):

• Occurs when i₁ = i₂ = i → ray passes symmetrically.
• At minimum deviation: \[r_1=r_2=r=\frac{A}{2}\]
• Minimum deviation: \[\delta_{min}=2i-A\]
• Angle of incidence at minimum deviation: \[i=\frac{A+\delta_{min}}{2}\]

Refractive Index of Prism:

• Red colour of the Sun at sunrise and sunset is due to maximum scattering of blue light and least scattering of red light in the atmosphere.
• The blue colour of the sky is due to the greater scattering of blue (or violet) light by air molecules because of its short wavelength.
• The black colour of the sky in the absence of atmosphere occurs because there is no scattering of sunlight.
• Red light is used for danger signals because it has the longest wavelength and is scattered the least, so it can be seen from a far distance.

• An optical instrument uses the principles of optics to enhance, modify, or analyse light for specific purposes.
• They manipulate light through reflection, refraction, diffraction, or interference.
• Common instruments: simple microscope, compound microscope, telescope.

Astronomical (Refracting) Telescope:

• Used to view distant objects.
• Objective lens (large focal length) forms image A'B' at its focus → acts as object for eyepiece.
• Eyepiece forms the final virtual, magnified image A''B''.

Magnifying Power: \[MP=\frac{\text{Visual angle with instrument}(\beta)}{\text{Visual angle for unaided eye}(\alpha)}\]

Resolving Power of Telescope:

• Ability to produce distinct images of two closely spaced objects.
• Angular separation between two resolvable objects: \[\sin\theta=1.22\frac{\lambda}{d}\]​ (Rayleigh's criterion), where d = aperture.
• Resolving power is inverse of angular separation.
• Larger aperture → better resolving power.

Types of Telescope:

• Refracting telescope: Uses two convex lenses; large objective + small eyepiece.
• Reflecting telescope: Uses concave mirror to reflect light internally; secondary mirror directs to eyepiece.
• Keplerian telescope: Converging lens as eyepiece → inverted image.
• Galilean telescope: Diverging lens as eyepiece → erect image.
• Magnification of Refracting Telescope: \[M=\frac{f_{o}}{f_{e}}\]

• Myopia is a vision defect in which distant objects appear blurry, while near objects are seen clearly.
• This occurs because the image of distant objects forms on the retina.
• The far point is not at infinity but is shifted closer to the eye.
• Causes include increased curvature of the cornea/lens or elongation of the eyeball.
• Corrected using a concave lens of negative power, which diverges light rays to focus the image on the retina.

Wave Optics (Physical Optics) treats light as a wave, explaining phenomena like interference, diffraction, and polarisation, which Ray Optics cannot explain.

• A wavefront is an imaginary surface where all points of a wave have the same phase (constant phase surface with maximum or minimum value).
• The direction of propagation of a wave is always perpendicular to the wavefront.
• Wavefront of a point source is a sphere; it propagates radially outward.

Types of Wavefronts:

S.No.	Wavefront	Shape of Light Source	Diagram or shape of wavefront	Variation of amplitude (A) with distance	Variation of intensity (I) with distance
1	Spherical	Point source		\[A\propto\frac{1}{r}\]	\[I\propto\frac{1}{r^2}\]
2	Cylindrical	Linear source / Slit		\[A\propto\frac{1}{\sqrt{r}}\]	\[I\propto\frac{1}{r}\]
3	Plane	Extended large source/ Point source at very large distinct		A = constant A ∝ r°	I = constant I ∝ r°

• Every point on a wavefront acts as a secondary source (point source) that emits new spherical wavelets in all directions with the same speed as the original wave.
• The new (forward) wavefront at any later time is the common tangential envelope (tangent surface) to all these secondary wavelets.
• The wavefront in a medium is always perpendicular to the direction of wave propagation.
• Secondary wavelets travel only in the forward direction — backward wavelets are ignored (zero amplitude in backward direction).

Memory: Every point → new source → envelope = new wavefront.

• Interference = redistribution of energy when two coherent waves superpose.
• Based on energy conservation, total energy remains constant, only redistributed.
• Constructive: I > (I₁ + I₂) → bright fringe
• Destructive: I < (I₁ + I₂) → dark fringe

Conditions for Sustained Interference:

• Sources must be coherent.
• Separation between sources must be small.
• Distance of screen from sources must be large.
• For good contrast: amplitudes of the two waves should be nearly equal.
• Two sources must propagate along same line.

Young's Double Slit Experiment (YDSE):

Setup: Light source → single slit → double slit (S₁ and S₂, separation d) → screen (distance D).

Path difference at point P: \[\delta=S_2P-S_1P=\frac{x_n\cdot d}{D}\]

Bright Fringe (Constructive): Path difference = even multiple of λ/2

δ = nλ, n = 0,1,2,3...

Dark Fringe (Destructive): Path difference = odd multiple of λ/2