Unit 12: Optics, Modern Physics & Electronics

This is the broadest unit of JEE, combining ray optics, wave optics, photoelectric effect, atomic & nuclear physics, and semiconductor electronics — together worth 15–20 % of the paper. Expect:

• JEE-Main: 1–2 ray-optics (lens, prism, total internal reflection), 1 wave-optics (Young's slits, single-slit diffraction), 1 photoelectric, 1 Bohr-model, 1 nuclear/radioactivity, 1 semiconductor/diode/logic gate.
• JEE-Advanced: Multi-step optics (lens + mirror combo); photoelectric + de Broglie crossover; Bohr model with isotope/recoil correction.

This unit splits neatly into five blocks:

1. Ray optics (geometric, classical).
2. Wave optics (interference, diffraction, polarization).
3. Photoelectric + de Broglie (the wave–particle bridge).
4. Atoms, nuclei, radioactivity (Bohr → modern).
5. Semiconductors and basic logic (the JEE electronics core).

Concept Map

RAY OPTICS
├── Mirrors: 1/v + 1/u = 1/f, m = -v/u
├── Refraction at plane / spherical surface
├── Lenses: lens-maker, 1/v − 1/u = 1/f
├── Combinations of thin lenses, power P = 1/f
├── Prism: A + D = i₁ + i₂; thin prism D = (n−1)A
├── TIR: sin θ_c = 1/n
└── Dispersion, scattering

WAVE OPTICS
├── Huygens construction
├── Young's: β = λD/d; intensity I = I_max cos²(πd sinθ/λ)
├── Single-slit diffraction: a sinθ = mλ
├── Resolving power
├── Thin films
└── Polarization: Malus I = I₀cos²θ, Brewster tan θ_B = n

MODERN
├── Photoelectric: hν = φ + KE_max
├── de Broglie: λ = h/p
├── Bohr model: E_n = −13.6 Z²/n² eV
├── Spectral series (Lyman, Balmer, …)
├── X-rays (Moseley): √f = a(Z − b)
├── Nuclear: ΔE = Δm c²
├── BE/A curve, fission, fusion
└── Radioactivity: N = N₀e^(−λt), t½, mean life

ELECTRONICS
├── Band theory: insulator / conductor / semiconductor
├── Intrinsic vs extrinsic (n, p)
├── pn-junction, forward / reverse I–V
├── Half-wave, full-wave rectifier
├── Zener (regulator)
└── Logic gates (NOT, AND, OR, NAND, NOR, XOR)

Topic 1: Ray Optics

Sub-topic A: Reflection at a Spherical Mirror — Mirror Formula

Sign convention (Cartesian): take pole as origin, optic axis along $x$, light incident from the left as the positive direction. Distances measured from pole; positive in the direction of incident light.

Mirror formula:

where $f = R/2$ (concave mirror $f > 0$ in many texts; sign depends on convention).

Lateral magnification: $m = -v/u$ (negative when image is inverted).

Sub-topic B: Refraction at a Plane Surface (Snell's Law)

Apparent depth (object in medium 1, viewer in medium 2):

Sub-topic C: Refraction at a Spherical Surface

Convention same as above. For a single refracting surface of radius $R$:

Derivation sketch: Use paraxial geometry. Consider an object at distance $u$ in medium $n_1$ in front of a spherical surface of radius $R$ separating media $n_1, n_2$. A small-angle ray traces from object to a point on the axis; applying Snell's law in linear approximation and using sign convention yields the result.

Sub-topic D: Thin Lens — Lens-maker Formula and Thin-Lens Equation

Apply the spherical-surface formula twice, once at each surface (radii $R_1, R_2$). For the first surface light goes $n_1\to n_2$; for the second $n_2\to n_1$. Adding the two:

For $n_1 = 1$ (air), $n_2 = n$:

This defines the focal length $f$ via the lens-maker formula:

For a biconvex lens both $R$'s contribute the same sign; for biconcave, the opposite.

Thin-lens equation:

Magnification: $m = v/u$.

Sub-topic E: Lens Combinations and Power

For two thin lenses in contact:

Power: $P = 1/f$ (in diopters when $f$ in metres). For combinations:

Sub-topic F: Prism

A ray entering and leaving a prism of angle $A$ at angles $i_1, i_2$ (on each face) gives deviation $\delta = i_1 + i_2 - A$.

Minimum deviation condition: $i_1 = i_2 = i$. Then symmetric:

Thin prism ($A$ small): deviation

Sub-topic G: Total Internal Reflection

Going from denser to lighter medium, beyond a critical angle there is no refraction — only reflection.

For water-air: $\theta_c = \arcsin(1/1.33)\approx 48.6^\circ$.

Applications: optical fibres, sparkle of diamond ($n = 2.4$, $\theta_c \approx 24.4^\circ$).

Sub-topic H: Dispersion and Scattering

Dispersion: $n(\lambda)$, blue refracted more than red. Angular dispersion $= (n_v - n_r)A$ for thin prism. Dispersive power $\omega = (n_v - n_r)/(n_y - 1)$.

Rayleigh scattering: intensity $\propto 1/\lambda^4$. Sky is blue (preferentially scatters blue out of sunlight); sunset is red (transmitted long-wavelength light).

Sub-topic I: Optical Instruments

Simple microscope (magnifying glass): $M = 1 + D/f$ where $D = 25$ cm is least distance of distinct vision.

Compound microscope: objective + eyepiece. $M = (L/f_o)(1 + D/f_e)$ where $L \approx$ tube length.

Refracting astronomical telescope (final image at infinity, normal adjustment): $M = -f_o/f_e$, tube length $= f_o + f_e$.

Reflecting telescope (Cassegrain/Newtonian): uses a concave mirror as objective — no chromatic aberration.

Resolving power of microscope: $1.22\lambda/(2n\sin\theta)$; telescope: $1.22\lambda/D$.

Eye defects:

• Myopia (short sight): far image forms before retina. Correct with diverging (concave) lens, $P < 0$.
• Hypermetropia (long sight): near image forms behind retina. Correct with converging (convex) lens, $P > 0$.
• Astigmatism: cylindrical lens.
• Presbyopia: bifocal.

Worked Problem 1

A convex lens of focal length $20$ cm is in contact with a concave lens of focal length $30$ cm. Find the equivalent focal length and power.

Solution. $1/f = 1/20 - 1/30 = (3 - 2)/60 = 1/60$. So $f = 60$ cm (converging). $P = 1/0.6 = +1.67$ D.

Worked Problem 2

A ray of light enters a prism of angle $60^\circ$ at angle $45^\circ$ and exits at $45^\circ$ on the other side (symmetric, $\delta_m$). Find $n$.

Solution. $\delta_m = 45 + 45 - 60 = 30^\circ$. $n = \sin((60+30)/2)/\sin(30) = \sin 45/\sin 30 = (\sqrt 2/2)/(1/2) = \sqrt 2$.

Topic 2: Wave Optics

Sub-topic A: Huygens' Principle

Each point on a wavefront acts as a secondary source. The envelope of secondary wavelets at a later time is the new wavefront. From this:

Reflection: incident plane wavefront hits a flat mirror, secondary wavelets form a reflected plane wavefront; angle of reflection = angle of incidence.

Refraction: at the interface, secondary waves travel with different speeds in the two media. Construction shows Snell's law: $n_1\sin\theta_1 = n_2\sin\theta_2$.

Sub-topic B: Young's Double-Slit Experiment

Two narrow slits separated by $d$ illuminated by monochromatic light of wavelength $\lambda$. Observation screen at distance $D \gg d$.

Path difference at a point at distance $y$ from the central axis:

Constructive (bright fringe): $\Delta x = m\lambda$. Destructive (dark fringe): $\Delta x = (m + \tfrac{1}{2})\lambda$.

Fringe width:

Intensity distribution:

where $I_0$ is the intensity from one slit alone. Max intensity $= 4I_0$.

Sub-topic C: Missing Fringes (Slits with Finite Width)

For real slits of width $a$, each slit produces a single-slit diffraction pattern (envelope). The double-slit pattern is modulated by this envelope. A fringe is missing when an interference maximum coincides with a diffraction minimum:

Sub-topic D: Thin Film Interference

For a soap film of thickness $t$, index $n$, in air, viewed at angle $\theta$ (in film):

Path difference between top-reflected and bottom-reflected ray: $2nt\cos\theta$. Plus phase shift of $\pi$ on reflection from the upper surface (going from lower to higher $n$): equivalent to $\lambda/2$ extra.

Constructive: $2nt\cos\theta = (m + \tfrac{1}{2})\lambda$. Destructive: $2nt\cos\theta = m\lambda$.

(Why colourful soap bubbles: different colours satisfy condition at different points of varying $t$.)

Sub-topic E: Single-Slit Diffraction

A slit of width $a$ illuminated by parallel light of wavelength $\lambda$. Diffraction minima at

Width of central maximum (between $m = +1$ and $m = -1$):

or linear width on a screen at distance $D$: $2\lambda D/a$.

Intensity distribution:

Sub-topic F: Resolving Power

Microscope (Abbe): $d_{\min} = 1.22\lambda/(2n\sin\theta)$. Resolving power $= 1/d_{\min}$.

Telescope (Rayleigh): $\theta_{\min} = 1.22\lambda/D$.

Sub-topic G: Polarization

Malus's Law: light of intensity $I_0$ polarized along an axis, then passed through a polarizer at angle $\theta$ to that axis:

(For unpolarized light passing through the first polarizer, $I_0 \to I_{\text{incident}}/2$.)

Brewster's Angle: reflected light is completely polarized when reflected ray is perpendicular to refracted ray:

At $\theta_B$, the reflected ray has $\vec E$ entirely perpendicular to plane of incidence (s-polarized).

Worked Problem 3

A YDSE has slits $0.5$ mm apart, screen $1$ m away. With $\lambda = 600$ nm find the fringe width.

Solution. $\beta = \lambda D/d = 6\times 10^{-7}\cdot 1/(5\times 10^{-4}) = 1.2\times 10^{-3}$ m $= 1.2$ mm.

Worked Problem 4

A $0.1$ mm wide slit is illuminated by $600$ nm light. Find the width of the central diffraction maximum on a screen $1$ m away.

Solution. Half-width angular = $\lambda/a = 6\times 10^{-3}$ rad. Full width on screen $= 2\lambda D/a = 1.2\times 10^{-2}$ m $= 1.2$ cm.

Topic 3: Photoelectric Effect

Sub-topic A: Phenomenology and Einstein's Explanation

When monochromatic light of frequency $\nu$ strikes a metal:

• Below a threshold frequency $\nu_0$, no electrons are emitted, regardless of intensity.
• Above $\nu_0$, electrons are emitted instantaneously (no time lag).
• Maximum KE of emitted electrons depends on $\nu$, not on intensity.
• Number of photoelectrons per second is proportional to intensity (for $\nu > \nu_0$).

Classical wave theory cannot explain these. Einstein (1905): light is composed of photons, each of energy $h\nu$. A single photon either ejects an electron (if $h\nu > \phi$, the work function) or doesn't.

Stopping potential $V_s$: applied retarding voltage that just stops the most energetic electron: $\text{KE}_{\max} = eV_s$, so

A plot of $V_s$ vs $\nu$ is a straight line of slope $h/e$ and $x$-intercept $\nu_0$.

Sub-topic B: Photon Properties

• Energy $E = h\nu = hc/\lambda$.
• Momentum $p = E/c = h/\lambda$ (massless particle).
• Mass equivalence $m = E/c^2 = h\nu/c^2$ (relativistic concept; rest mass is zero).

Sub-topic C: de Broglie Hypothesis

Particles also have wave nature:

For an electron accelerated through potential $V$: $\tfrac{1}{2}m v^2 = eV \Rightarrow p = \sqrt{2meV}$. So

Sub-topic D: Davisson–Germer Experiment

Electrons scattering off a Ni crystal showed Bragg-like diffraction peaks, confirming the wave nature of matter. The measured $\lambda$ matched de Broglie's $h/p$ exactly.

Worked Problem 5

Light of wavelength $300$ nm falls on a metal of work function $2$ eV. Find KE of the most energetic photoelectron and stopping potential.

Solution. $E_\gamma = 1240/300 = 4.13$ eV. KE = $4.13 - 2 = 2.13$ eV. $V_s =$ KE $/e = 2.13$ V.

Worked Problem 6

An electron is accelerated through $100$ V. Find its de Broglie wavelength.

Solution. $\lambda = 12.27/\sqrt{100} = 1.227$ Å.

Topic 4: Bohr Model of the Atom

Sub-topic A: Postulates

1. Electrons revolve in circular orbits around the nucleus under Coulomb attraction.
2. Only orbits with angular momentum $L = n\hbar = nh/2\pi$ are allowed (quantization).
3. While in an allowed orbit, the electron does not radiate. When it transitions, it emits/absorbs a photon of energy $h\nu = E_i - E_f$.

Sub-topic B: Derivation of Orbital Parameters

Centripetal balance for an electron (charge $-e$) orbiting a nucleus of charge $+Ze$:

Angular momentum quantization: $mvr = n\hbar$ ⇒ $v = n\hbar/(mr)$.

Substitute:

where $a_0 = \hbar^2/(kme^2) = 0.529$ Å is the Bohr radius for hydrogen ($Z = 1, n = 1$).

Velocity:

Energy (kinetic + potential):

For hydrogen: $E_1 = -13.6$ eV (ground state), $E_2 = -3.4$ eV, $E_3 = -1.51$ eV, …, $E_\infty = 0$.

Sub-topic C: Spectral Series — Rydberg Formula

A transition from $n_i$ to $n_f$ ($n_f < n_i$) emits a photon with

Spectral series of hydrogen:

• Lyman ($n_f = 1$): UV. First line (Lyman-α): $n = 2 \to 1$, $\lambda = 121.6$ nm.
• Balmer ($n_f = 2$): visible. H-α ($3\to 2$): $\lambda = 656.3$ nm (red).
• Paschen ($n_f = 3$): IR.
• Brackett ($n_f = 4$), Pfund ($n_f = 5$): far IR.

Sub-topic D: X-rays and Moseley's Law

X-rays are emitted when high-energy electrons strike a metal target:

• Continuous (Bremsstrahlung): slowing-down of electrons in the target produces a broad spectrum, with sharp short-wavelength cutoff at $\lambda_{\min} = hc/(eV)$.
• Characteristic spectrum: discrete lines from electron transitions to inner shells. K-α: $L\to K$; K-β: $M\to K$.

Moseley's law (semi-empirical):

For K-α, $b = 1$ (one inner electron shields the nuclear charge by $-1e$). Established the central role of atomic number $Z$.

Worked Problem 7

The wavelength of the first line in the Lyman series of hydrogen is approximately:

Solution. $1/\lambda = R(1 - 1/4) = 3R/4$. $\lambda = 4/(3R) = 4/(3\cdot 1.097\times 10^7)\approx 1.22\times 10^{-7}$ m $= 122$ nm.

Topic 5: Nuclei and Radioactivity

Sub-topic A: Nuclear Composition and Binding Energy

A nucleus $^A_Z X$ has $Z$ protons and $N = A-Z$ neutrons. Mass of the bound nucleus $M$ is less than the sum of free masses: the mass defect $\Delta m = (Z m_p + (A-Z)m_n) - M$ shows up as binding energy:

In practice, masses are expressed in u (atomic mass units), where $1$ u $= 931.5$ MeV/c².

Binding energy per nucleon (B/A) curve:

• Rises sharply for light nuclei.
• Peaks near $A = 56$ (iron) at $\approx 8.8$ MeV/nucleon.
• Falls slowly for heavier nuclei.

This shape explains:

• Fission of heavy nuclei (e.g. ²³⁵U) into medium nuclei releases energy.
• Fusion of light nuclei (e.g. H → He in stars) also releases energy.

Sub-topic B: Radioactive Decay Law

The rate of decay of a sample is proportional to the number of unstable nuclei:

where $\lambda$ is the decay constant (probability per nucleus per unit time).

Half-life:

Mean lifetime:

Activity: $A = -dN/dt = \lambda N = \lambda N_0 e^{-\lambda t}$. Unit: becquerel (Bq, $1$ disintegration/s); curie (Ci) $= 3.7\times 10^{10}$ Bq.

Sub-topic C: Types of Decay

• α-decay: nucleus emits ⁴He. $A\to A-4$, $Z\to Z-2$. Range in air few cm; stopped by paper.
• β⁻-decay: $n\to p + e^- + \bar\nu_e$. $A$ same, $Z\to Z+1$. Range ~m in air.
• β⁺-decay: $p\to n + e^+ + \nu_e$.
• γ-emission: excited nucleus releases photon; $A, Z$ unchanged. Highly penetrating; needs lead.

Sub-topic D: Fission and Fusion

Fission (²³⁵U + slow neutron): releases ≈ 200 MeV per fission; harnessed in reactors and bombs.

Fusion (in stars):

• pp-chain: 4¹H → ⁴He + 2e⁺ + 2ν + 26.7 MeV.
• CNO cycle: catalysed by C, N, O.
• Earth-based fusion: deuterium-tritium D + T → ⁴He + n + 17.6 MeV.

Worked Problem 8

A radioactive sample has half-life of 5 years. What fraction remains after 15 years?

Solution. $15$ y $= 3$ half-lives. Fraction $= (1/2)^3 = 1/8 = 12.5\%$.

Worked Problem 9

The activity of a sample at $t = 0$ is $8\times 10^4$ Bq and falls to $2\times 10^4$ Bq in 6 hours. Find half-life and decay constant.

Solution. $A/A_0 = 1/4 = (1/2)^{t/T_{1/2}}\Rightarrow t/T_{1/2} = 2 \Rightarrow T_{1/2} = 3$ h. $\lambda = \ln 2/T_{1/2} = 0.231/$h.

Topic 6: Semiconductors and Electronics

Sub-topic A: Band Theory

In solids, atomic energy levels broaden into bands (valence and conduction) separated by a forbidden gap $E_g$.

Sub-topic B: Intrinsic Semiconductors

Pure Si or Ge. At $T > 0$, some electrons thermally excited from valence band to conduction band, leaving behind holes.

$n = p = n_i$ (intrinsic carrier density). At room T, $n_i(\text{Si})\approx 10^{16}$/m³ and $n_i(\text{Ge})\approx 2.4\times 10^{19}$/m³.

Conductivity rises rapidly with T (exponentially): $n_i\propto T^{3/2}e^{-E_g/(2k_BT)}$.

Sub-topic C: Extrinsic Semiconductors (Doping)

Add traces of impurity to control carrier type and density.

n-type: dope Si with group V (P, As). Each donor adds a loosely-bound electron — promoted easily into conduction band. Majority carriers: electrons; minority: holes.

p-type: dope Si with group III (B, Al). Each acceptor needs an extra electron to bond — creates a hole. Majority: holes; minority: electrons.

For both: $np = n_i^2$ (mass-action law).

Sub-topic D: pn-junction

Junction of p- and n-type semiconductors. Initially diffusion creates depletion region with built-in potential $V_0 \approx 0.7$ V for Si.

Forward bias (p more positive): depletion region shrinks, large current flows. I-V characteristic:

Reverse bias (n more positive): depletion widens, only tiny saturation current $I_0$. At a critical reverse voltage (breakdown), the junction conducts (avalanche or Zener).

Sub-topic E: Diode Applications

Half-wave rectifier: a single diode in series with a load and AC source. Passes only positive half-cycles. DC component $V_{dc} = V_m/\pi$ where $V_m$ is peak.

Full-wave rectifier:

• Centre-tap: two diodes and centre-tapped transformer.
• Bridge: four diodes, no centre tap needed.

In both, $V_{dc} = 2V_m/\pi$, ripple factor much lower than half-wave.

Zener diode: operates in reverse breakdown at a sharp voltage. Used as a voltage regulator — output stays at $V_z$ regardless of input variation, provided the current is within rated range.

LED, photodiode, solar cell — all variants of the pn-junction with light coupling.

Sub-topic F: Logic Gates

A logic gate takes binary inputs and produces a binary output.

NAND and NOR are universal (can build any logic from them).

Worked Problem 10

A common-emitter NPN transistor amplifier has $\beta = 100$. If $I_B = 50$ μA, find $I_C$ and $I_E$.

Solution. $I_C = \beta I_B = 5$ mA. $I_E = I_C + I_B = 5.05$ mA.

(Note: detailed transistor analysis is mostly out of the JEE-Advanced syllabus 2025 onwards, but β and the I_C/I_B ratio still appear in JEE-Main.)

Problem-Solving Heuristics

1. Sign convention is everything in geometric optics — mark distances from the pole/lens, sign per direction.
2. For a lens combination, add powers; for mirror–lens combinations, treat each surface in turn.
3. At a flat surface, apparent depth scales by $n_2/n_1$.
4. Total internal reflection requires going from denser to lighter and $\theta > \theta_c$.
5. For minimum deviation in a prism, light goes symmetrically; $i_1 = i_2$.
6. In YDSE, fringe width is $\beta = \lambda D/d$ — independent of which fringe.
7. For single-slit diffraction, central max is twice the width of side maxima.
8. Photoelectric: stopping potential depends only on $\nu$, not intensity. Plot $V_s$ vs $\nu$ for $h/e$.
9. de Broglie: $\lambda = 12.27/\sqrt V$ Å for electrons in V volts.
10. Bohr levels: $E_n = -13.6 Z^2/n^2$ eV, $r_n = n^2 a_0/Z$.
11. For X-rays, short-wavelength cutoff is $\lambda_{\min} = hc/(eV)$, independent of target.
12. Half-life vs mean lifetime: $T_{1/2} = \tau\ln 2 \approx 0.693\tau$.
13. Universal gates: any circuit reducible to NAND-only or NOR-only.
14. For a Zener regulator, the load voltage equals $V_z$ as long as the Zener is in reverse breakdown and current is within rating.

Common Traps & Mistakes

• Sign of focal length and image distance. Be consistent — and check which side the image is on physically.
• Lens formula vs mirror formula: signs of $u, v$ differ between conventions; pick one (Cartesian recommended) and stick with it.
• Critical angle is from denser to lighter, not the other way.
• In YDSE, increasing slit separation $d$ shrinks the fringe width — common reversed answer.
• Single-slit diffraction: minimum (not maximum) at $a\sin\theta = m\lambda$.
• Malus law uses $\cos^2$, not $\cos$.
• Brewster's angle gives 100% polarized reflected light, but the refracted light is still partially polarized.
• Photoelectric: stopping potential is $V_s = (h\nu - \phi)/e$ — not $(h\nu)/e$.
• Bohr orbits don't apply to multi-electron atoms (only one-electron systems and hydrogen-like ions like He⁺, Li²⁺).
• Half-life and mean life are different: $T_{1/2} = 0.693\tau$, not equal.
• Activity decays at the same rate as $N$: $A(t) = A_0 e^{-\lambda t}$.
• Doping doesn't change resistivity by a constant factor — depends on dopant concentration and temperature.
• Zener diode in forward bias behaves like a normal diode; only in reverse breakdown does it regulate.

Quick Revision Card

Ray Optics:

• Mirror: $1/v + 1/u = 1/f$, $m = -v/u$.
• Lens: $1/v - 1/u = 1/f$, $m = v/u$.
• Lens-maker: $1/f = (n-1)(1/R_1 - 1/R_2)$.
• Refraction at sphere: $n_2/v - n_1/u = (n_2 - n_1)/R$.
• Prism: thin $\delta = (n-1)A$; min $n = \sin((A+\delta_m)/2)/\sin(A/2)$.
• TIR: $\sin\theta_c = 1/n$.

Wave Optics:

• YDSE: $\beta = \lambda D/d$, intensity $4I_0\cos^2(\pi d\sin\theta/\lambda)$.
• Single-slit: minima $a\sin\theta = m\lambda$; central width $2\lambda D/a$.
• Malus: $I = I_0\cos^2\theta$. Brewster: $\tan\theta_B = n$.

Modern:

• Photoelectric: $h\nu = \phi + KE$; $eV_s = h(\nu - \nu_0)$.
• de Broglie: $\lambda = h/p$; electron, $V$ volts: $\lambda = 12.27/\sqrt V$ Å.
• Bohr: $E_n = -13.6 Z^2/n^2$ eV, $r_n = n^2 a_0/Z$, $a_0 = 0.529$ Å.
• Lyman ($n_f=1$, UV), Balmer ($n_f=2$, visible), Paschen ($n_f=3$, IR).
• Moseley: $\sqrt\nu \propto (Z - b)$.
• Decay: $N = N_0 e^{-\lambda t}$, $T_{1/2} = \ln 2/\lambda$, $\tau = 1/\lambda$.

Electronics:

• Semiconductors: $E_g$(Si) = 1.1 eV, $E_g$(Ge) = 0.7 eV.
• $np = n_i^2$ (mass-action).
• Half-wave rectifier: $V_{dc} = V_m/\pi$. Full-wave: $V_{dc} = 2V_m/\pi$.
• Zener: voltage regulator.
• NAND, NOR are universal gates.

Formula Sheet