Light and Optics

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Light is a form of electromagnetic radiation that is visible to the human eye, with wavelengths ranging approximately from 400 nm (violet) to 700 nm (red). Geometrical optics treats light as travelling in straight-line rays and governs how mirrors and lenses form images. Wave optics (physical optics) deals with interference, diffraction, and polarisation, treating light as a transverse wave.

Reflection of Light:

The law of reflection states that the angle of incidence $i$ equals the angle of reflection $r$, measured from the normal (perpendicular) to the reflecting surface. For a plane mirror:

• Image is as far behind the mirror as the object is in front: $d_i = -d_o$ (virtual image, same size)
• Laterally inverted (left-right reversal)

Spherical Mirrors:

For concave and convex mirrors, the mirror formula is: $$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$$

Where $f$ is the focal length (positive for concave, negative for convex), $d_o$ is object distance (positive if object is in front of the mirror), $d_i$ is image distance (positive for real images, negative for virtual).

Linear magnification: $m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}$

Sign conventions for spherical mirrors:

Mirror Type	f	Real Image	Virtual Image
Concave	+	$d_i > 0$ (in front)	$d_i < 0$ (behind mirror)
Convex	−	$d_i > 0$ (in front)	$d_i < 0$ (behind mirror)

Refraction of Light:

When light passes from one medium to another, its speed changes, causing it to bend. Snell’s Law: $$n_1 \sin \theta_1 = n_2 \sin \theta_2$$ where $n$ is the absolute refractive index ($n = c/v$, speed of light in vacuum / speed of light in medium).

Critical Angle and Total Internal Reflection: When light travels from a denser to a rarer medium: $\sin \theta_c = \frac{n_2}{n_1}$ (for $n_1 > n_2$) If $\theta_i > \theta_c$, total internal reflection occurs — 100% of light is reflected back into the denser medium. This is the principle behind optical fibres and diamond sparkles.

Thin Lenses:

Lensmaker’s formula: $\frac{1}{f} = (n-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$ Same sign convention as mirrors: $\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$; $m = \frac{h_i}{h_o} = \frac{d_i}{d_o}$

Power of lens: $P = \frac{1}{f}$ (in metres), unit = dioptre (D) For lenses in contact: $P_{total} = P_1 + P_2 + …$

⚡ Exam Tip (MDCAT): For a prism, the angle of deviation $\delta = (i_1 + i_2) - A$, where $A$ is the prism angle. Minimum deviation occurs when $i_1 = i_2$, and the refractive index is $n = \frac{\sin(A + \delta_m)/2}{\sin(A/2)}$. MDCAT frequently asks for the emergent angle when light passes through a glass slab — remember that the emergent ray is parallel to the incident ray (lateral shift but no deviation).

⚡ MDCAT Trap: Don’t confuse the focal length sign conventions between mirrors and lenses. For lenses, a converging lens has positive $f$ (like a concave mirror), but the sign convention for object and image distances follows the same Cartesian sign convention.

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🟡 Standard — Regular Study (2d–2mo)

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

Refraction through a Glass Slab:

A rectangular glass slab shifts the emergent ray laterally without changing its direction. The lateral shift $x$ is: $$x = t \frac{\sin(\theta_1 - \theta_2)}{\cos \theta_2}$$ where $t$ is the thickness of the slab. The emergent ray is always parallel to the incident ray.

Prism Spectrometer:

A prism disperses white light into its constituent colours (spectrum) because the refractive index of a material is slightly different for each wavelength — this is dispersion. Violet light bends more than red light because $n_{violet} > n_{red}$.

Optical Phenomena in Nature:

• Rainbow: Caused by dispersion + total internal reflection in water droplets. Primary rainbow: red on top, violet inside; secondary rainbow (double rainbow): reversed order, weaker intensity.
• Mirage: In very hot conditions (desert roads), light from the sky refracts upward beyond the critical angle, creating the illusion of water ahead — an inferior mirage.
• Sparkling of diamond: Diamond has a very high refractive index ($n \approx 2.42$) and strong dispersion, giving many internal reflections before light emerges.

Human Eye — Defects and Corrections:

Defect	Cause	Near Point / Far Point	Correction
Myopia (near-sightedness)	Eyeball too long or lens too convex	Far point less than infinity	Concave (diverging) lens
Hypermetropia (far-sightedness)	Eyeball too short or lens too flat	Near point > 25 cm	Convex (converging) lens
Presbyopia	Loss of accommodation with age	Both near and far vision affected	Bifocal lens
Astigmatism	Unequal curvature of cornea	Objects at all distances blurred in one meridian	Cylindrical lens

The power of accommodation of the human eye decreases with age. The near point recedes from about 25 cm in youth to several metres in old age — this condition is called presbyopia.

Magnifying Glass (Simple Microscope):

Angular magnification $M = 1 + \frac{D}{f}$ (for relaxed eye, image at near point) Or $M = \frac{D}{f}$ (for image at infinity, normal adjustment)

Astronomical Telescope (Refracting): $M = \frac{f_o}{f_e}$ (normal adjustment, final image at infinity) $f_o$ = focal length of objective, $f_e$ = focal length of eyepiece

⚡ Common MDCAT Error: Students often forget that the lens formula $\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$ requires sign conventions consistently applied. When an object is placed between the focal point and optical centre of a convex lens ($d_o < f$), the image is virtual and upright — this is how a magnifying glass works.

Optical Instruments — Compound Microscope:

Total magnification $M = m_o \times m_e = \left(\frac{L}{f_o}\right) \times \left(\frac{D}{f_e}\right)$ Where $L$ is the tube length (distance between objective and eyepiece focal points), $D = 25$ cm (near point distance), $f_o < f_e$ typically.

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🔴 Extended — Deep Study (3mo+)

Comprehensive theory, derivations, and exam pattern analysis for serious MDCAT aspirants.

Derivation of Snell’s Law from Wave Theory:

When light passes from medium 1 (speed $v_1$, wavelength $\lambda_1$) to medium 2 (speed $v_2$, wavelength $\lambda_2$), the frequency $\nu$ remains constant (since the source doesn’t change). Therefore: $$v_1 = \nu \lambda_1, \quad v_2 = \nu \lambda_2$$ The wavefront theory shows that the incident ray, normal, and refracted ray all lie in the same plane. Applying the geometry of equal time paths along the wavefront: $$\frac{\sin \theta_1}{\sin \theta_2} = \frac{v_1}{v_2} = \frac{n_2}{n_1}$$ This is Snell’s Law: $n_1 \sin \theta_1 = n_2 \sin \theta_2$.

Derivation of the Mirror Formula:

For a spherical concave mirror with an object at distance $u$ from the pole, the image forms at distance $v$. From geometry of similar triangles formed by the object, image, and focal point: $$\frac{h_i}{h_o} = \frac{v - f}{f} = \frac{f}{u - f}$$ Rearranging gives: $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$

Thin Lens Combination:

When two thin lenses are placed in contact, the equivalent focal length $F$ satisfies: $$\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2}$$ If separated by distance $d$: $\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2}$

Optical Fibre Communication:

Total internal reflection is the operating principle behind fibre optic cables. A step-index fibre has a core of higher refractive index $n_1$ surrounded by cladding of lower refractive index $n_2$. Light entering within the acceptance angle $\theta_a$ undergoes total internal reflection at the core-cladding boundary: $$\sin \theta_a = \sqrt{n_1^2 - n_2^2} = NA \text{ (numerical aperture)}$$ The critical angle at the core-cladding boundary: $\theta_c = \sin^{-1}\left(\frac{n_2}{n_1}\right)$

Fibre optics is preferred for telecommunications because: (1) much higher bandwidth than copper wire, (2) no electromagnetic interference, (3) signals travel with minimal loss over long distances.

Young’s Double Slit Experiment (Wave Optics):

When two coherent light sources interfere: $$d \sin \theta = n\lambda \text{ (constructive, bright fringe)}$$ $$d \sin \theta = \left(n + \frac{1}{2}\right)\lambda \text{ (destructive, dark fringe)}$$

Fringe width $\beta = \frac{\lambda D}{d}$ where $D$ is the distance to screen, $d$ is slit separation.

Diffraction Grating:

For a diffraction grating with $N$ lines per metre, slit separation $d = 1/N$. Maxima occur at: $d \sin \theta = n\lambda$ Resolving power $R = \frac{\lambda}{\Delta\lambda} = nN$ where $n$ is the order and $N$ is total number of illuminated slits.

Polarisation:

Light is polarised when its electric field vectors oscillate in a single plane. Unpolarised light becomes partially or fully polarised by:

• Reflection at Brewster’s angle: $\tan \theta_B = n_2/n_1$ (reflected ray is completely polarised perpendicular to the plane of incidence)
• Transmission through a Polaroid filter (selective absorption)
• Scattering (sky appears blue due to scattering intensity $\propto 1/\lambda^4$)

MDCAT Exam Pattern for Light and Optics:

Approximately 2–3 questions per MDCAT paper. Common question types:

• Mirror/lens formula problems (find image distance, magnification, or focal length)
• Refractive index calculations (Snell’s Law applications)
• Total internal reflection (critical angle, optical fibre acceptance angle)
• Prism deviation problems
• Magnifying glass/telescope magnification

⚡ Advanced Tip: For minimum deviation in a prism, the condition $i_1 = i_2 = i$ and $r_1 = r_2 = A/2$ applies, giving $\delta_m = 2i - A$, so $n = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin(A/2)}$. This formula is frequently asked in MDCAT. Also remember that a convex mirror always produces a virtual, diminished, upright image — it can never form a real image regardless of object position.

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