Unit 12: Optics, Modern Physics and Semiconductors

This is a giant consolidated unit yielding 5–7 MCQs in NEET — combining Ray Optics, Wave Optics, Modern Physics, Atomic & Nuclear Physics, and Semiconductors. Each sub-unit is small individually but together they form the biggest contributor to the Physics section.

Concept Map

Ray optics: mirror & lens formulas, refraction, TIR, prism, dispersion, eye defects, microscope, telescope.

Wave optics: Huygens' principle, YDSE, single-slit diffraction, polarization.

Modern physics: photoelectric effect, de Broglie waves, Bohr atom, X-rays.

Nuclei: composition, mass defect, BE, radioactive decay, fission/fusion.

Semiconductors: bands, doping, p-n junction, rectifiers, Zener, LED, photodiode, logic gates.

Topic 1: Ray Optics

Sub-topic A: Reflection — Mirrors

Sign convention (NCERT, Cartesian): distances measured from pole, $+x$ in direction of incident light. Heights above principal axis positive.

Mirror formula:

Magnification:

Concave mirror: $f < 0$ (real focus). Convex: $f > 0$ (virtual focus).

Sub-topic B: Refraction

Snell's law: $n_1\sin\theta_1 = n_2\sin\theta_2$.

Refractive index: $n = c/v$.

Apparent depth: A pool of depth $d$ filled with liquid of index $n$ appears shifted by $d(1 - 1/n)$ upward when viewed from above.

Sub-topic C: Total Internal Reflection

Critical angle when going from denser ($n_1$) to rarer ($n_2$) medium:

For glass-air: $\theta_c \approx 42°$. Applied in optical fibre, prism reflectors, mirage, sparkle of diamond ($\theta_c \approx 24°$).

Sub-topic D: Refraction Through Spherical Surface

For light passing from $n_1$ to $n_2$ through a spherical interface of radius $R$:

Sub-topic E: Lens Maker's Formula

For a thin lens of index $n$ in air:

Sign convention: $R$ positive if centre is on the side of outgoing light.

Lens formula (same as mirror):

Magnification: $m = v/u$.

Power: $P = 1/f$ (in metres) → unit dioptre (D).

Two thin lenses in contact: $1/F = 1/f_1 + 1/f_2$, $P = P_1 + P_2$.

Sub-topic F: Prism

For a prism of refracting angle $A$ and angle of minimum deviation $\delta_m$:

For a thin prism ($A$ small): $\delta = (n - 1) A$.

Sub-topic G: Dispersion

White light through prism splits into spectrum. Angular dispersion $\theta = (n_v - n_r)A$. Dispersive power:

Achromatic combination: two prisms (different materials) such that net dispersion = 0 but net deviation ≠ 0.

Rayleigh scattering ($I \propto 1/\lambda^4$) explains the blue sky and red sunset.

Sub-topic H: Optical Instruments

Simple microscope (magnifier): angular magnification $m = 1 + D/f$ (image at near point), or $m = D/f$ (image at infinity), with $D = 25$ cm.

Compound microscope: $m = m_o \cdot m_e \approx (L/f_o)(1 + D/f_e)$, where $L$ is tube length, $f_o$ objective focal length, $f_e$ eyepiece focal length.

Astronomical telescope (refracting): $m = f_o/f_e$ (image at infinity), tube length = $f_o + f_e$.

Reflecting telescope (Cassegrain): uses parabolic mirror — free from chromatic aberration; large aperture possible.

Sub-topic I: Eye Defects

Topic 2: Wave Optics

Sub-topic A: Huygens' Principle

Every point on a wavefront acts as a source of secondary wavelets; the new wavefront is their forward envelope. Explains reflection, refraction.

Sub-topic B: Young's Double-Slit Experiment (YDSE)

Two coherent slits separated by $d$, screen at distance $D$:

• Fringe width: $\beta = \lambda D / d$.
• Path difference: $\Delta = d\sin\theta \approx d y/D$.
• Bright at $\Delta = n\lambda$; dark at $\Delta = (n + 1/2)\lambda$.

Intensity at point with phase difference $\phi$: $I = 4 I_0\cos^2(\phi/2)$, where $I_0$ is single-slit intensity. Max $I = 4I_0$; min $I = 0$.

For different intensities $I_1, I_2$: $I_\text{max} = (\sqrt{I_1} + \sqrt{I_2})^2$, $I_\text{min} = (\sqrt{I_1} - \sqrt{I_2})^2$.

When YDSE is immersed in liquid of index $n$: $\lambda$ becomes $\lambda/n$, so fringe width reduces by $n$.

Sub-topic C: Single-Slit Diffraction

Pattern from a slit of width $a$:

• First minimum at angle $\sin\theta = \lambda/a$.
• Central maximum width on screen: $2\lambda D/a$.
• Side maxima less intense.

Sub-topic D: Resolving Power

Rayleigh's criterion: two point sources just resolvable if the central maximum of one coincides with first minimum of other.

• Microscope (numerical aperture $\mu\sin\beta$): $d_\text{min} = 1.22\lambda/(2\mu\sin\beta)$.
• Telescope: $\theta_\text{min} = 1.22\lambda/D$, with $D$ aperture diameter.

Sub-topic E: Polarization

Transverse waves can be polarized. Only EM waves and string waves can — sound (longitudinal) cannot.

Malus's law: intensity through polarizer when angle between polarizer axis and incident polarization is $\theta$:

For unpolarised light passing through polarizer: $I = I_0/2$.

Brewster's law: at angle $\theta_B$, reflected ray is fully polarized perpendicular to plane of incidence:

At $\theta_B$, reflected and refracted rays are perpendicular.

Topic 3: Photoelectric Effect

Sub-topic A: Observations

When light hits a metal, electrons are ejected if frequency $\nu \ge \nu_0$ (threshold). Key facts:

• Existence of threshold frequency $\nu_0$ (cutoff).
• Max KE of ejected electrons depends on $\nu$, not on intensity.
• Photocurrent depends on intensity (not $\nu$).
• Effect is instantaneous (no time lag) — disproves classical wave theory.

Sub-topic B: Einstein's Equation

$\phi = h\nu_0$ is work function (energy needed to free an electron). Stopping potential $V_0$: $eV_0 = K_\text{max}$.

Note: $K_\text{max}$ vs $\nu$ is a straight line with slope $h/e$ and x-intercept $\nu_0$. Independent of metal beyond work function.

Sub-topic C: Photon Energy and Momentum

For $\lambda$ in nm: $E\ (\text{eV}) = 1240/\lambda(\text{nm})$.

Topic 4: Matter Waves

Sub-topic A: de Broglie Wavelength

For an electron accelerated through potential $V$:

Sub-topic B: Davisson-Germer

Confirmed wave nature of electron by observing diffraction maxima from Ni crystal — agreed with de Broglie's prediction.

Topic 5: Atomic Structure

Sub-topic A: Rutherford Model

Most of the atom is empty; nucleus is tiny and contains nearly all the mass. Failed to explain stability (accelerating electron should radiate).

Sub-topic B: Bohr Model

Bohr postulates:

1. Electrons orbit in stationary states without radiating.
2. Angular momentum is quantized: $L = n\hbar$, $n = 1, 2, 3, \dots$
3. Photons of energy $h\nu = E_i - E_f$ emitted on transition.

For hydrogen-like atoms (atomic number Z):

For hydrogen ($Z = 1$): $E_1 = -13.6$ eV (ground state), $E_2 = -3.4$ eV, etc.

Sub-topic C: Spectral Series of Hydrogen

Sub-topic D: X-Rays

Produced when fast electrons strike a target. Two components:

• Continuous (bremsstrahlung): cutoff wavelength $\lambda_\text{min} = hc/(eV)$ depends only on accelerating voltage.
• Characteristic: discrete lines characteristic of target material ($K_\alpha, K_\beta, L_\alpha$, etc.). Moseley's law: $\sqrt{\nu} = a(Z - b)$.

Topic 6: Nuclear Physics

Sub-topic A: Nuclear Composition

Nucleus contains Z protons and N = A − Z neutrons. Notation: $^A_Z X$.

Atomic mass unit: 1 u $= 1.66 \times 10^{-27}$ kg $= 931.5$ MeV/$c^2$.

Nuclear radius: $R = R_0 A^{1/3}$ with $R_0 = 1.2$ fm.

Sub-topic B: Mass Defect and Binding Energy

BE per nucleon is maximum around $A = 56$ (Fe-56), about 8.8 MeV. This explains why fission of heavy nuclei (e.g. U-235) and fusion of light nuclei (e.g. H + H) both release energy.

Sub-topic C: Radioactive Decay

• Half-life: $T_{1/2} = \ln 2/\lambda = 0.693/\lambda$.
• Mean life: $\tau = 1/\lambda = T_{1/2}/\ln 2$.
• Activity: $A = \lambda N$. Unit: becquerel (Bq, 1 decay/s). Older: curie (Ci, $3.7 \times 10^{10}$ Bq).

After $n$ half-lives, $N = N_0/2^n$.

Sub-topic D: Decay Modes

Sub-topic E: Fission and Fusion

Fission: U-235 + n → Ba-141 + Kr-92 + 3n + ~200 MeV. Used in reactors. Critical mass needed for chain reaction.

Fusion: H-1 + H-1 → He-2 (via D, T) + 17 MeV per reaction. Source of stellar energy (p-p chain, CNO cycle in stars). Requires extreme T ( $10^7$ K) to overcome Coulomb barrier.

Topic 7: Semiconductors

Sub-topic A: Energy Bands

In solids, allowed energies form bands:

• Valence band: occupied by valence electrons.
• Conduction band: empty or partially filled — carriers free to move.
• Band gap: forbidden region.

Sub-topic B: Intrinsic vs Extrinsic

Intrinsic: pure Si, Ge. At T > 0 K, some electrons thermally excite to conduction band, leaving holes. $n_e = n_h = n_i$.

Doping:

• n-type: pentavalent dopant (P, As). Donates electron. Majority: electrons. Minority: holes.
• p-type: trivalent dopant (B, Al). Creates hole. Majority: holes. Minority: electrons.

Mass-action law: $n_e \cdot n_h = n_i^2$.

Sub-topic C: p-n Junction

When p and n are joined, electrons diffuse from n to p (and holes vice versa), creating a depletion region with built-in potential (~0.7 V for Si, 0.3 V for Ge).

Forward bias ($p$ to $+$): low resistance, current flows.

Reverse bias ($n$ to $+$): high resistance, only tiny reverse saturation current. At a critical voltage, breakdown occurs (Zener / avalanche).

Sub-topic D: Rectifiers

Half-wave rectifier (one diode): output only during one half cycle. Ripple frequency = input frequency. Efficiency ~40.6%.

Full-wave rectifier (centre-tap or bridge): output in both halves. Ripple frequency = 2 × input. Efficiency ~81.2%.

Sub-topic E: Zener Diode

Heavily doped diode designed to operate in reverse breakdown with constant voltage. Used as voltage regulator.

Sub-topic F: Optoelectronic Devices

• LED: emits light when forward biased. Energy emitted $\approx E_g$.
• Photodiode: reverse biased; current increases under illumination.
• Solar cell: p-n junction with no bias; light generates EMF.

Sub-topic G: Logic Gates

Boolean operations on binary inputs:

NAND and NOR are universal — any logic function can be built using only NAND (or only NOR).

NEET Pattern MCQ Tips

• Mirror/lens formula: substitute $u, v, f$ with sign convention.
• TIR: critical angle from $\sin\theta_c = 1/n$.
• Prism min deviation: $n = \sin[(A+\delta_m)/2]/\sin(A/2)$.
• YDSE: fringe width $\beta = \lambda D/d$.
• Single-slit diffraction: first min at $a\sin\theta = \lambda$.
• Malus & Brewster (polarization).
• Photoelectric effect: $K_\text{max} = h(\nu - \nu_0)$; stopping potential.
• de Broglie: $\lambda = h/p$; for accelerated electron $\lambda = 12.27/\sqrt{V}$ Å.
• Bohr radius, energy, transitions: $E_n = -13.6/n^2$ eV; Lyman/Balmer formulas.
• Half-life: $N = N_0/2^n$ after $n$ half-lives.
• Mass defect → BE: BE/nucleon curve.
• p-n junction: forward vs reverse.
• Logic gates: truth tables.

Common Confusions and Traps

• For a convex lens immersed in a denser medium (water), the focal length increases. If immersed in a medium of refractive index equal to the lens, the lens behaves as a glass plate (no focusing).
• The eye's near point is 25 cm (for normal vision), far point at infinity.
• Lyman series is in UV, Balmer in visible — frequent factual check.
• Photoelectric effect: max KE depends on $\nu$, not intensity; photocurrent depends on intensity.
• Beyond the cutoff frequency, intensity does not affect $K_\text{max}$.
• Bohr's model works only for hydrogen-like (single electron) atoms.
• Mass number $A$ is conserved in alpha/beta decay; charge is conserved.
• $\beta^-$ emission: nucleon converts $n \to p + e^- + \bar\nu_e$.
• In p-type doping, the dopant is trivalent.
• LED emits photons of $E \approx E_g$, so wider band gap → bluer light.
• The bridge rectifier is a full-wave rectifier without a centre-tapped transformer.
• The NAND gate is universal — same for NOR.

Quick Revision Card

• Mirror: $1/v + 1/u = 1/f$, $f = R/2$, $m = -v/u$.
• Lens: $1/v - 1/u = 1/f$, $m = v/u$, Power $= 1/f$ in m.
• Lens-maker: $1/f = (n-1)(1/R_1 - 1/R_2)$.
• Snell: $n_1\sin\theta_1 = n_2\sin\theta_2$. TIR: $\sin\theta_c = 1/n$.
• Thin prism: $\delta = (n-1)A$.
• YDSE fringe width: $\beta = \lambda D/d$.
• Single slit first min: $a\sin\theta = \lambda$.
• Malus: $I = I_0\cos^2\theta$; Brewster: $\tan\theta_B = n$.
• Photoelectric: $K_\text{max} = h\nu - \phi$; $eV_0 = K_\text{max}$.
• de Broglie: $\lambda = h/p$; electron at $V$: $\lambda = 12.27/\sqrt V$ Å.
• Bohr: $r_n = n^2 a_0/Z$, $E_n = -13.6 Z^2/n^2$ eV.
• Rydberg: $1/\lambda = RZ^2(1/n_f^2 - 1/n_i^2)$.
• Half-life: $T_{1/2} = 0.693/\lambda$; mean life $\tau = 1/\lambda$.
• BE/nucleon peaks at Fe-56 (~8.8 MeV).
• Forward biased diode conducts; LED emits with $E \approx E_g$.
• NAND, NOR universal.

Worked NEET Examples

Example 1: Mirror Image Distance

Object at 30 cm from concave mirror of focal length 20 cm. Image:

$1/v + 1/(-30) = 1/(-20)$. So $1/v = -1/20 + 1/30 = (-3 + 2)/60 = -1/60$. $v = -60$ cm. Image is real, 60 cm in front of mirror, inverted, magnified by $m = -v/u = -(-60)/(-30) = -2$.

Example 2: Lens with Object at 2f

Object at 40 cm from convex lens of $f = 20$ cm. $1/v - 1/(-40) = 1/20$, so $1/v = 1/20 - 1/40 = 1/40$. $v = +40$ cm. Image: real, same size, inverted, on opposite side.

Example 3: YDSE Fringe Width

In YDSE, slit separation 0.5 mm, screen 1 m away, light $\lambda = 600$ nm.

Example 4: Photoelectric Effect

Light of $\lambda = 400$ nm on metal with work function $\phi = 2$ eV.

Photon energy: $E = hc/\lambda = 1240/400 = 3.1$ eV.

$K_\text{max} = 3.1 - 2 = 1.1$ eV. Stopping potential: $V_0 = 1.1$ V.

Example 5: Bohr Radius and Energy

For hydrogen $n = 3$: $r_3 = 9 a_0 = 4.76$ Å. Energy: $E_3 = -13.6/9 = -1.51$ eV.

Transition $n=3 \to n=2$: $\Delta E = -1.51 - (-3.4) = 1.89$ eV. Wavelength: $\lambda = 1240/1.89 \approx 656$ nm (visible red — H-alpha of Balmer series).

Derivations

Lens-Maker's Formula

For a thin lens of refractive index $n$ in air, with two spherical surfaces of radii $R_1$ and $R_2$:

Apply refraction at first surface ($n_1 = 1, n_2 = n$): $n/v_1 - 1/u = (n - 1)/R_1$.

Apply refraction at second surface ($n_1 = n, n_2 = 1$): $1/v - n/v_1 = (1 - n)/R_2$.

Adding: $1/v - 1/u = (n - 1)(1/R_1 - 1/R_2) = 1/f$.

Power of Lenses in Contact

Two thin lenses in contact: the image of first acts as object for second. Net focal length:

In terms of power (in dioptres): $P_\text{net} = P_1 + P_2$.

YDSE Fringe Width

Path difference at point P on screen at height y from central maximum: $\Delta \approx d y/D$.

Bright fringe: $\Delta = n\lambda$, so $y_n = n\lambda D/d$. Fringe width: $\beta = y_{n+1} - y_n = \lambda D/d$.

Single-Slit Diffraction Minimum

Slit of width $a$. Light from the two edges of the slit travels paths differing by $a\sin\theta$. First minimum when this equals $\lambda$ (the slit is divided into two halves, each pair cancels):

Photoelectric Effect — Einstein Equation

Photon of energy $h\nu$ ejects electron. Work function $\phi$ is the minimum energy to free an electron. Max KE of ejected electron:

For $\nu < \nu_0 = \phi/h$, no ejection. Stopping potential $V_0 = K_\text{max}/e$.

Bohr Model Quantitative

Quantisation condition: $mvr = n\hbar$.

Coulomb attraction provides centripetal force: $kZe^2/r^2 = mv^2/r$.

Combining gives $r_n = n^2\hbar^2/(mk Ze^2)$. For hydrogen ($Z = 1, n = 1$): $r_1 = a_0 = 0.529$ Å.

Energy: $E_n = -kZe^2/(2 r_n) = -k^2 Z^2 e^4 m/(2 n^2 \hbar^2) = -13.6 Z^2/n^2$ eV.

Rydberg Formula

Photon emitted in transition $n_i \to n_f$ has energy $E = E_i - E_f$. Using Bohr energies:

so $1/\lambda = R Z^2(1/n_f^2 - 1/n_i^2)$ with $R = 1.097 \times 10^7$ m⁻¹.

Half-Life from Decay Law

From $N(t) = N_0 e^{-\lambda t}$, half-life is when $N = N_0/2$:

Mean life $\tau = 1/\lambda$.

Optical Instruments — Magnifications

Simple Microscope

Magnifying glass: angular magnification when image at near point ($D = 25$ cm):

When image at infinity: $m = D/f$.

Compound Microscope

Two lenses: objective $f_o$ (small), eyepiece $f_e$. Object close to focal point of objective, image at length $L$ (tube length). Total magnification (image at near point):

Refracting Telescope

For distant objects, parallel rays focus at $f_o$. Eyepiece magnifies that image. For final image at infinity:

Tube length $L = f_o + f_e$.

For large magnification, use large $f_o$ and small $f_e$. Large objective also collects more light (aperture).

Wave Optics — Polarization

Malus's Law

Polarized light of intensity $I_0$ passing through analyzer at angle $\theta$ to its axis transmits

For unpolarized light passing first polarizer: intensity halves to $I_0/2$.

Brewster's Law

At $\theta_B$, reflected ray is fully polarized (perpendicular to plane of incidence):

For glass-air: $\theta_B \approx 56.3°$. At this angle, reflected and refracted rays are perpendicular.

Nuclear Physics — More Details

Mass-Energy Equivalence

1 u $= 931.5$ MeV/$c^2$.

For a nucleus with $Z$ protons and $A - Z$ neutrons:

BE per Nucleon Curve

• Low A: BE/A increases (binding stronger as nucleus grows).
• Peak at A ≈ 56 (Fe-56): ~8.8 MeV/nucleon.
• High A: BE/A decreases (Coulomb repulsion grows).

Energy released:

• Fission of heavy: BE/A higher in products → energy released.
• Fusion of light: same reason.

Radioactive Decay Activity

$A(t) = \lambda N(t) = \lambda N_0 e^{-\lambda t} = A_0 e^{-\lambda t}$.

Unit: becquerel (Bq) = 1 decay/s. 1 Ci = $3.7 \times 10^{10}$ Bq.

Successive Decays

If A decays to B (with rate $\lambda_A$) and B decays to C (with rate $\lambda_B$), in equilibrium $\lambda_A N_A = \lambda_B N_B$.

Semiconductors — Diode Details

Diode Equation

• Forward bias ($V > 0$): exponential current increase.
• Reverse bias ($V < 0$): $I \to -I_0$ (saturation).

Zener Diode

Heavily doped, designed to operate in breakdown region. Reverse voltage is nearly constant (Zener voltage $V_Z$), providing voltage regulation.

Use as regulator: input voltage $V_\text{in}$ through resistor $R_s$ to Zener in reverse. Output across Zener = $V_Z$, regardless of variations in $V_\text{in}$ or load.

LED

Forward-biased p-n junction releases photons of energy ~$E_g$ on recombination:

For GaAs ($E_g = 1.4$ eV): $\lambda \approx 887$ nm (near-IR). For GaP ($E_g = 2.3$ eV): $\lambda \approx 540$ nm (green). For GaN ($E_g = 3.4$ eV): $\lambda \approx 365$ nm (UV).

Solar Cell

A p-n junction without external bias. Photons with $h\nu > E_g$ create electron-hole pairs in the depletion region. The built-in field separates them, creating a current.

Open-circuit voltage close to $V_\text{oc} = (E_g - kT\ln(\dots))/e$, max ~0.5–0.7 V per cell of Si.

Logic Gates — Implementation

NAND realizations:

So all logic can be built from NAND alone. This makes NAND a universal gate. Similarly NOR.

Formula Sheet