Ray Optics

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Ray Optics (also called Geometrical Optics) deals with the propagation of light in terms of rays, ignoring wave effects. The ray model assumes light travels in straight-line paths called rays when passing through a uniform medium. This topic forms a crucial foundation for understanding mirrors, lenses, optical instruments, and the behaviour of light at boundaries between media.

Fundamental Laws:

1. Law of Rectilinear Propagation: Light travels in straight lines in a homogeneous medium
2. Law of Independence: Light rays from different sources do not interfere with each other
3. Law of Reversibility: A light ray retraces its path when the direction of propagation is reversed

Reflection Laws: $$\angle i = \angle r$$ The angle of incidence equals the angle of reflection. Both angles are measured from the normal (perpendicular to the reflecting surface).

Refraction Laws (Snell’s Law): $$n_1 \sin \theta_1 = n_2 \sin \theta_2$$ Where $n_1$ and $n_2$ are the refractive indices of medium 1 and 2 respectively. When light enters a denser medium, it bends toward the normal; entering a rarer medium, it bends away from the normal.

Critical Angle and Total Internal Reflection (TIR): When light travels from denser to rarer medium: $$\sin \theta_c = \frac{n_2}{n_1} \text{ (where } n_2 < n_1\text{)}$$ If $\theta_i > \theta_c$, total internal reflection occurs — no refraction, all light reflects back into the denser medium.

Mirror Formula (for spherical mirrors): $$\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$$ $$f = \frac{R}{2}$$ Where $f$ = focal length, $R$ = radius of curvature, $u$ = object distance, $v$ = image distance.

For concave mirror: f is negative; for convex mirror: f is positive.

⚡ Exam Tip: For a concave mirror, real image forms when object is beyond focus (u > f); virtual image forms when object is between pole and focus (u < f). For convex mirror, always virtual, upright, diminished image behind the mirror.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding and problem-solving practice.

Refractive Index:

Absolute refractive index of a medium: $n = \frac{c}{v}$ where $c = 3 \times 10^8$ m/s (speed in vacuum), $v$ = speed in medium.

Relative refractive index: $n_{12} = \frac{n_2}{n_1} = \frac{v_1}{v_2} = \frac{\sin \theta_1}{\sin \theta_2}$

For water: $n_w = 4/3 ≈ 1.33$; For glass: $n_g ≈ 1.5$

Lens Formula: $$\frac{1}{f} = \frac{1}{v} - \frac{1}{u} \text{ (sign convention careful)}$$

Lens Maker’s Formula (for thin lens in air): $$\frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)$$ Where $n$ = refractive index of lens material, $R_1$ and $R_2$ = radii of curvature of the two surfaces.

Magnification:

For mirrors: $m = \frac{-v}{u} = \frac{h’}{h}$ For lenses: $m = \frac{v}{u} = \frac{h’}{h}$

Linear magnification $m$ can be:

• |m| > 1: Magnified image
• |m| < 1: Diminished image
• m > 0: Virtual and upright
• m < 0: Real and inverted

Power of a Lens: $$P = \frac{1}{f} \text{ (in metres)}$$ Unit: Dioptre (D); $f$ must be in metres.

For converging lens: P > 0; For diverging lens: P < 0.

Combined Lens System: $$P_{eq} = P_1 + P_2 + P_3 + …$$ $$frac{1}{f_{eq}} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2}$$ Where $d$ = distance between lenses (for separated lenses).

Dispersion of Light:

Refractive index varies with wavelength: $n_{violet} > n_{red}$ This causes white light to split into constituent colours when passing through a prism.

Angular dispersion: $\theta_v - \theta_r = (\mu_v - \mu_r)A$ where $A$ = prism angle.

Deviation by a Prism: $$\delta = (\mu - 1)A \text{ (for thin prism, small angle approximation)}$$ Minimum deviation: $\delta_m = (\mu - 1)A$ when the ray passes through the prism symmetrically ($i = e$).

Optical Phenomena:

• Mirage: Refraction causing image of sky on road (hot air layers near ground act as rarer medium)
• Looming: Inverted image seen in cold regions (cold air acts as denser medium)
• Rainbow: Dispersion + internal reflection in water droplets
• Optical fibres: TIR used to transmit light through flexible fibres

⚡ CUET-Specific Tip: For lens problems, ALWAYS use the sign convention consistently. The Cartesian sign convention: object distance (u) is always negative; distances measured opposite to incident light direction are positive; distances along incident light direction are negative. Real is positive for image distance.

Common Student Mistakes:

• Not converting cm to metres before using lens maker’s formula
• Confusing focal length sign for concave vs convex mirrors
• Forgetting that magnification can be negative (real inverted images)
• Using wrong sign convention in lens formula (there are multiple conventions — stick to one)

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Spherical Mirrors — Detailed Analysis:

Concave mirror properties:

• Reflects from inner (concave) surface
• Can form real or virtual images
• Focal length f = R/2 (negative by convention)
• Used in torches, headlights, shaving mirrors, ophthalmoscopes

Convex mirror properties:

• Reflects from outer (convex) surface
• Always forms virtual, upright, diminished image
• f is positive by convention
• Used as rear-view mirrors, security mirrors in shops

Image formation by concave mirror:

Object Position	Image Position	Image Nature	Size
At infinity	At focus F	Real	Point (highly diminished)
Beyond C (C is 2F)	Between F and C	Real	Diminished
At C (2F)	At C	Real	Same size
Between C and F	Beyond C	Real	Magnified
At F	At infinity	Real	Highly magnified
Between F and P	Behind mirror	Virtual	Magnified

Image formation by convex mirror: Always virtual, upright, diminished image between pole and focus.

Thin Lenses — Detailed Analysis:

Types of thin lenses:

• Converging (convex): Thicker at centre, converges rays; biconvex, plano-convex, concavo-convex
• Diverging (concave): Thinner at centre, diverges rays; biconcave, plano-concave, convexo-concave

Lens types and image formation:

Lens Type	Object Position	Image Position	Nature	Size
Convex	At infinity	At F₂	Real	Point
Convex	Beyond 2F₁	Between F₂ and 2F₂	Real	Diminished
Convex	At 2F₁	At 2F₂	Real	Same
Convex	Between 2F₁ and F₁	Beyond 2F₂	Real	Magnified
Convex	At F₁	At infinity	Real	Highly magnified
Convex	Between F₁ and O	Same side as object	Virtual	Magnified
Concave	Any position	Between F and O (same side)	Virtual	Diminished

Optical Instruments:

Simple Microscope (Magnifying Glass):

• Angular magnification $M = \frac{D}{f} + 1$ (for relaxed eye, normal adjustment)
• $M = \frac{D}{f}$ (for near point adjustment, final image at near point)
• $D$ = least distance of distinct vision = 25 cm

Compound Microscope: $$M = \frac{v_o}{u_o} \times \frac{D}{f_e} = m_o \times M_e$$ Where $m_o$ = magnification by objective, $M_e$ = angular magnification by eyepiece.

Normal adjustment: $m_o = \frac{L}{f_o}$ (where $L$ = tube length, $f_o$ = objective focal length)

Astronomical Telescope (Refracting):

• Astronomical: $M = -\frac{f_o}{f_e}$ (final image at infinity)
• Terrestrial: $M = -\frac{f_o}{f_e}$ (with erecting lens)
• $f_o$ = objective focal length, $f_e$ = eyepiece focal length
• Length of tube: $L = f_o + f_e$ (when final image at infinity for astronomical)

Reflecting Telescopes:

• Newtonian: Concave mirror as objective + convex secondary mirror
• Cassegrain: Concave mirror + convex secondary (light reflected through hole in primary)

Aberrations:

Spherical Aberration:

• Paraxial rays and marginal rays focus at different points
• Caused by spherical surfaces
• Minimised by: using parabolic mirrors, stopping down aperture, using appropriate glass

Chromatic Aberration:

• Different wavelengths focus at different points
• Violet focuses nearer lens than red
• Minimised by: achromatic doublet (combination of converging and diverging lenses of different glasses)

For achromatic doublet: $\frac{\omega_1}{f_1} + \frac{\omega_2}{f_2} = 0$ Where $\omega$ = dispersive power.

Coma: Oblique rays produce comet-shaped images Astigmatism: Different magnifications in different planes Field Curvature: Image of flat object forms on curved surface Distortion: Barrel (wide edges) or pincushion (wide centre) distortion

Young’s Double Slit Experiment (Interference): $$y_n = \frac{n\lambda D}{d} \text{ (position of nth bright fringe)}$$ $$y_n = \frac{(2n-1)\lambda D}{2d} \text{ (position of nth dark fringe)}$$ Fringe width $\beta = \frac{\lambda D}{d}$

Where $D$ = distance slit to screen, $d$ = distance between slits.

Thin Film Interference:

For reflected light:

• Constructive: $2\mu t \cos r = (m + \frac{1}{2})\lambda$ (bright)
• Destructive: $2\mu t \cos r = m\lambda$ (dark)

For transmitted light (opposite):

• Constructive: $2\mu t \cos r = m\lambda$
• Destructive: $2\mu t \cos r = (m + \frac{1}{2})\lambda$

Phase change of $\pi$ occurs upon reflection from denser medium.

Newton’s Rings: $$r_m^2 = m\lambda R \text{ (bright rings)}$$ $$r_m^2 = (2m-1)\frac{\lambda R}{2} \text{ (dark rings)}$$

Used to measure wavelength of light or radius of curvature of lens.

Diffraction:

Single slit: $a \sin \theta = m\lambda$ (minima, $m = \pm 1, \pm 2, …$)

Resolving power of telescope: $R = \frac{D}{1.22\lambda}$ Where $D$ = diameter of objective (circular aperture).

CUET Previous Year Patterns (2023-2025):

• 2024: Lens maker’s formula and combined lens system calculations
• 2023: Prism deviation minimum deviation conditions and calculations
• 2024: Young’s double slit experiment — fringe width calculation with given wavelengths
• 2023: TIR conditions and critical angle for glass-air interface
• 2024: Magnification by spherical mirrors and sign convention applications

⚡ Advanced Tip: For compound microscope/telescope problems, distinguish between magnifying power ($M$) and magnification ($m$). Angular magnification is ratio of angle subtended by image at eye to angle subtended by object at eye. In telescopes, separation of objective and eyepiece matters: for final image at infinity, tube length $L = f_o + f_e$. For relaxed eye, $L = f_o + f_e$ if final image at D, or $L = f_o + f_e$ for other cases depending on the specific problem.

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