Engineering Physics

1. Units, Dimensions, and Errors

Physical Quantities and SI Units

Dimensional Analysis

Examples: \([\text{Velocity}]=[LT^{-1}]\), \([\text{Force}]=[MLT^{-2}]\), \([\text{Energy}]=[ML^2T^{-2}]\), \([\text{Pressure}]=[ML^{-1}T^{-2}]\), \([\text{Charge}]=[IT]\).

Dimensional analysis is used to check equation homogeneity, derive relations between quantities, and convert units. The principle of homogeneity requires every term in a valid equation to share the same dimensions.

Limitations: it cannot determine dimensionless constants (e.g., the \(\tfrac{1}{2}\) in \(\tfrac{1}{2}mv^2\)), fails for transcendental functions, and cannot distinguish quantities with identical dimensions such as work and torque.

Errors in Measurement

For \(n\) repeated measurements \(a_1, a_2, \ldots, a_n\):

Propagation of Errors

2. Waves and Oscillations

Simple Harmonic Motion (SHM)

Newton's second law gives \(m\ddot{x} = -kx\), i.e., \(\ddot{x} + \omega_0^2 x = 0\) with \(\omega_0 = \sqrt{k/m}\). The general solution is

Energy in SHM

Damped Harmonic Oscillator

With a velocity-dependent damping force \(-b\dot{x}\):

Three regimes determined by the trial solution \(x \propto e^{\alpha t}\):

• Under-damped (\(\gamma < \omega_0\)): oscillatory decay, \(x(t) = A_0 e^{-\gamma t}\cos(\omega' t + \phi)\), \(\omega' = \sqrt{\omega_0^2 - \gamma^2}\).
• Critically damped (\(\gamma = \omega_0\)): fastest non-oscillatory return to equilibrium.
• Over-damped (\(\gamma > \omega_0\)): slow exponential return.

Quality Factor and Logarithmic Decrement

High \(Q\) implies low damping and a sharp resonance peak; \(\delta\) measures the decay per cycle.

Forced Oscillations and Resonance

With a driving force \(F_0\cos\omega t\), the steady-state amplitude is

Resonance occurs at \(\omega_r = \sqrt{\omega_0^2 - 2\gamma^2} \approx \omega_0\) for small damping.

Travelling Waves

A harmonic transverse wave on a string: \(y = A\sin(kx - \omega t)\), satisfying

Standing Waves

Superposition of two counter-propagating waves gives

Nodes at \(kx = n\pi\); antinodes at \(kx = (n + \tfrac{1}{2})\pi\). Normal modes of a string of length \(L\) fixed at both ends:

Sound Waves and Doppler Effect

Sound in air consists of longitudinal pressure waves with speed \(v = \sqrt{B/\rho}\) (\(B\) = bulk modulus). Intensity \(I = \tfrac{1}{2}\rho v\omega^2 A^2\). Sound level in decibels:

For a source moving at velocity \(v_s\) and observer at \(v_o\) along the same line:

Applications include radar speed guns, Doppler ultrasound, and astronomical redshift.

Phase Velocity and Group Velocity

A monochromatic wave travels at the phase velocity \(v_p = \omega/k\). A wave packet (real signal) has its envelope moving at the group velocity

3. Interference of Light

Wave Theory and Coherence

Light is a transverse electromagnetic wave with visible wavelengths from 400 to 700 nm. Two beams from coherent sources combine by linear superposition, producing intensity variations called interference.

Intensity Distribution

For fields \(E_1 = E_0\cos\omega t\) and \(E_2 = E_0\cos(\omega t + \delta)\), the resultant time-averaged intensity is

Maxima when \(\delta = 2n\pi\); minima when \(\delta = (2n+1)\pi\).

Young's Double-Slit Experiment (YDSE)

For slit separation \(d\) and screen distance \(D\), the path difference for a point at height \(y\) is \(\Delta = yd/D\). Bright fringes at \(\Delta = n\lambda\), dark at \(\Delta = (n + \tfrac{1}{2})\lambda\). The fringe width (spacing between successive bright or dark fringes) is

Thin-Film Interference

For a film of refractive index \(\mu\) and thickness \(t\), with light incident at angle \(i\) refracting at \(r\):

The \(\lambda/2\) term arises from the phase reversal on reflection at the denser medium. For reflected light: maxima when \(2\mu t\cos r = (n + \tfrac{1}{2})\lambda\); minima when \(2\mu t\cos r = n\lambda\).

Newton's Rings

A plano-convex lens of radius \(R\) resting on a flat glass plate creates an air wedge. In reflected monochromatic light the radii of dark and bright rings are

Wavelength determination using two ring orders \(n\) and \(n+p\):

4. Diffraction

Definition and Types

Single-Slit Fraunhofer Diffraction

For slit width \(a\) and wavelength \(\lambda\), the intensity at angle \(\theta\) is

Minima (nodes) occur when \(a\sin\theta = m\lambda\), \(m = \pm 1, \pm 2, \ldots\) The width of the central maximum on a screen at focal length \(f\) is \(w_0 = 2\lambda f/a\).

Double-Slit Diffraction

The pattern combines the single-slit envelope with two-source interference:

Missing orders occur when \(d/a\) is an integer, causing an interference maximum to coincide with a diffraction minimum.

Diffraction Grating

For \(N\) slits with grating element \(d = a + b\), the principal maxima satisfy the grating equation \(d\sin\theta = n\lambda\). The resolving and dispersive powers are

Applications include spectrometers, wavelength-division multiplexing (WDM) in optical fiber systems, and astronomical spectroscopy.

Resolving Power of Optical Instruments

For a telescope with circular aperture \(D\): \(\theta_{\min} = 1.22\lambda/D\). For a microscope with numerical aperture \(NA\): \(d_{\min} = 0.61\lambda/NA\).

5. Polarization

Methods of producing polarized light: reflection (Brewster's law), refraction in birefringent crystals, selective absorption (Polaroid films), and Rayleigh scattering.

Brewster's Law

At the polarizing angle \(\theta_p\), the reflected beam is completely plane-polarized:

For glass (\(\mu = 1.5\)), \(\theta_p \approx 56.3°\). The reflected and refracted rays are perpendicular. Applications: polarizing sunglasses, laser Brewster windows.

Malus's Law

Linearly polarized light of intensity \(I_0\) transmitted through an analyzer at angle \(\theta\):

Double Refraction and Wave Plates

Calcite and quartz are birefringent: the incident ray splits into an ordinary ray (obeys Snell's law) and an extraordinary ray (direction-dependent index \(n_e\)). The optic axis is the direction along which \(n_o = n_e\). A quarter-wave plate has thickness such that \((n_o - n_e)t = \lambda/4\), converting linear to circular polarization; a half-wave plate satisfies \((n_o - n_e)t = \lambda/2\) and rotates the plane of polarization by 90°.

6. Lasers

Principle of Laser Action

LASER stands for Light Amplification by Stimulated Emission of Radiation. Three processes operate between atomic levels \(E_1\) and \(E_2\) (\(E_2 > E_1\)): stimulated absorption (\(E_1 \to E_2\)), spontaneous emission (\(E_2 \to E_1\), random phase), and stimulated emission (incident photon induces an identical coherent photon).

Einstein Coefficients

In thermal equilibrium with radiation density \(\rho(\nu)\):

Population Inversion and Pumping

Three- and Four-Level Laser Schemes

An optical resonator (high-reflectance mirror + output coupler) provides feedback to sustain and amplify the stimulated emission.

Properties of Laser Light

• Monochromaticity: extreme spectral purity, \(\Delta\lambda/\lambda \sim 10^{-8}\).
• Coherence: high spatial and temporal coherence.
• Directionality: divergence \(\theta \sim \lambda/D\).
• High intensity/brightness.

Specific Laser Types

Ruby Laser

Active medium: Cr³⁺-doped Al₂O₃. Pump: xenon flash lamp. Wavelength: 694.3 nm (red). Pulsed 3-level operation. Efficiency ~1%. Applications: holography, ranging, tattoo removal.

He–Ne Laser

Active medium: He–Ne gas (10:1) at low pressure. Pump: DC discharge. Principal wavelength: 632.8 nm (red). Continuous wave, 4-level operation. Applications: interferometry, alignment, barcode readers.

Semiconductor (Diode) Laser

Stimulated emission across the band gap in a heavily doped p–n junction. Wavelength: \(\lambda = hc/E_g\) (e.g., GaAs at ~840 nm, InGaAsP at 1.55 μm). Electrically pumped, compact, efficiency >50%. Applications: optical communication, DVD/Blu-ray, LiDAR.

Holography

Recording intensity: \(I(\vec{r}) = |E_R + E_O|^2 = |E_R|^2 + |E_O|^2 + 2\,\text{Re}(E_R^* E_O)\). The cross term carries the object phase. Reconstruction with \(E_R\) produces the term \(\propto E_O\). Applications: security holograms, holographic data storage, head-up displays, vibration analysis.

7. Fiber Optics

Basic Principle

An optical fiber is a glass or plastic cylinder guiding light by repeated total internal reflection (TIR) at the core–cladding interface. The core has refractive index \(n_1\), the cladding \(n_2 < n_1\).

Acceptance Angle and Numerical Aperture

TIR condition at the core–cladding interface: \(\sin\theta_c = n_2/n_1\). The acceptance (half-)cone angle in air:

Classification of Optical Fibers

The V-number \(V = (2\pi a/\lambda)\,\text{NA}\) determines how many modes propagate. Single-mode operation requires \(V < 2.405\).

Attenuation and Dispersion

Attenuation: \(\alpha = \frac{10}{L}\log_{10}(P_{\text{in}}/P_{\text{out}})\) [dB/km]. Typical silica fiber: ~0.2 dB/km at 1.55 μm. Sources of loss: absorption (OH ions, IR lattice), Rayleigh scattering (\(\propto 1/\lambda^4\)), and bending losses. Dispersion (modal, material, waveguide) broadens pulses and limits data rate.

8. Origins of Quantum Mechanics

Failure of Classical Physics

By 1900, several experimental results could not be explained by classical theory: the blackbody radiation spectrum (UV catastrophe), photoelectric effect, Compton scattering, atomic emission line spectra, and the specific heats of solids at low temperature. These resolutions gave birth to quantum mechanics.

Planck's Radiation Law (1900)

Energy of a harmonic oscillator is quantized: \(E_n = nh\nu\). Planck's radiation law:

Low-frequency limit recovers the Rayleigh–Jeans law; high-frequency limit gives Wien's law. The Stefan–Boltzmann result \(E = \sigma T^4\) with \(\sigma = 5.67 \times 10^{-8}\ \text{W m}^{-2} \text{K}^{-4}\) follows by integration.

Photoelectric Effect (Einstein, 1905)

Einstein's equation:

Compton Scattering (1923)

An X-ray photon of wavelength \(\lambda\) scattered through angle \(\theta\) by a free electron shifts to

This establishes the photon as a relativistic particle with energy \(h\nu\) and momentum \(h\nu/c\).

de Broglie Hypothesis (1924)

Every particle has an associated matter wave:

For an electron accelerated through potential \(V\): \(\lambda = 12.27/\sqrt{V}\ \text{Å}\). The Davisson–Germer experiment (1927) confirmed this via electron diffraction from a nickel crystal (\(\lambda_{\text{de Broglie}} \approx 1.67\ \text{Å}\), \(\lambda_{\text{exp}} \approx 1.65\ \text{Å}\) from Bragg's law on Ni (110) planes).

Heisenberg Uncertainty Principle

For conjugate variables:

Wave Function — Born Interpretation

A particle is described by a complex wave function \(\psi(\vec{r}, t)\) such that \(|\psi(\vec{r}, t)|^2\,dV\) is the probability of finding the particle in volume \(dV\). Normalization: \(\int|\psi|^2\,dV = 1\). The wave function must be single-valued, finite, continuous, with continuous first derivative wherever the potential is finite.

9. Schrödinger Equation

Time-Dependent Schrödinger Equation

For a non-relativistic particle of mass \(m\) in potential \(V(\vec{r}, t)\):

For a free particle (\(V = 0\)), the plane wave \(\psi = e^{i(\vec{k}\cdot\vec{r} - \omega t)}\) requires \(\hbar\omega = \hbar^2 k^2/2m\), recovering the de Broglie relation.

Time-Independent Schrödinger Equation (TISE)

If \(V = V(\vec{r})\), writing \(\psi = u(\vec{r})e^{-iEt/\hbar}\) leads to the eigenvalue equation:

For a stationary state, \(|\psi|^2\) is independent of time.

Probability Current

\(\vec{J} = \frac{\hbar}{2mi}\left[\psi^*\nabla\psi - \psi\nabla\psi^*\right]\) satisfies the continuity equation \(\partial|\psi|^2/\partial t + \nabla\cdot\vec{J} = 0\). Probability is conserved. This describes tunneling currents in scanning tunneling microscopes.

Postulates of Quantum Mechanics

1. Physical state described by \(\psi\), a vector in Hilbert space.
2. Every observable is represented by a Hermitian operator.
3. The only possible measurement results are eigenvalues of that operator.
4. Probability of obtaining eigenvalue \(a_i\) is \(|\langle\phi_i|\psi\rangle|^2\).
5. After measurement of \(a_i\), \(\psi\) collapses to \(\phi_i\).
6. Time evolution between measurements governed by the Schrödinger equation.

10. Particle in a Box

One-Dimensional Infinite Well

Potential: \(V(x) = 0\) for \(0 < x < L\), \(V = \infty\) elsewhere. Inside the box TISE gives \(u'' + k^2 u = 0\) with \(k = \sqrt{2mE}/\hbar\). Boundary conditions \(u(0) = u(L) = 0\) quantize \(k\) to \(k_n = n\pi/L\):

Key features: energies scale as \(n^2\); there is a non-zero zero-point energy \(E_1\); nodes increase with \(n\); spacing widens as \(L\) shrinks — the basis of quantum confinement.

Three-Dimensional Infinite Box

Inside a cube of side \(L\):

Different triples \((n_x, n_y, n_z)\) with the same sum of squares are degenerate.

11. Free Electron Theory and Band Theory

Classical (Drude) Free Electron Theory

Valence electrons are treated as a classical ideal gas colliding with positive ions (relaxation time \(\tau\)). Electrical conductivity: \(\sigma = ne^2\tau/m\). Wiedemann–Franz law: \(\kappa/\sigma = LT\) with \(L \approx 2.45 \times 10^{-8}\ \text{W}\Omega\,\text{K}^{-2}\). The model fails to predict the correct electronic specific heat and has no explanation for insulators.

Sommerfeld Quantum Free-Electron Theory

Electrons are non-interacting fermions filling states up to the Fermi energy \(E_F\):

At \(T > 0\), electrons obey the Fermi–Dirac distribution:

This gives the correct \(T\)-linear electronic specific heat \(C_v^{\text{el}} = \frac{\pi^2}{2}nk_B(T/T_F)\) and recovers the Wiedemann–Franz law.

Bloch's Theorem and Band Formation

A periodic potential \(V(\vec{r}) = V(\vec{r} + \vec{R})\) leads to Bloch eigenfunctions:

Allowed energies form continuous bands separated by forbidden gaps.

Kronig–Penney Model

A 1-D periodic square-well potential (barrier strength \(P\)) gives the condition

Forbidden gaps appear at Brillouin-zone boundaries \(k = n\pi/a\). As \(P \to 0\) the free-electron parabola is recovered; as \(P \to \infty\) discrete atomic levels emerge.

Effective Mass

The curvature of the dispersion relation \(E(k)\) defines the effective mass:

Near band minima \(m^* > 0\) (electron-like); near band maxima \(m^* < 0\) leading to hole behavior. The effective mass controls carrier mobility and density of states.

Classification by Band Filling

Conductors have overlapping valence and conduction bands. Semiconductors have a small gap (~1 eV); insulators have a large gap (>5 eV).

12. Semiconductor Physics

Intrinsic Semiconductors

In a pure semiconductor, thermal generation across the band gap governs carrier concentration:

Typical band gaps: Si (1.12 eV), Ge (0.67 eV), GaAs (1.43 eV).

Extrinsic Semiconductors — Doping

n-type: pentavalent donors (P, As, Sb) donate electrons; donor level \(E_d\) just below \(E_c\). For fully ionized donors: \(n \approx N_D\), \(p = n_i^2/N_D\). p-type: trivalent acceptors (B, Al) create holes; acceptor level \(E_a\) just above \(E_v\). Law of mass action: \(np = n_i^2\).

Carrier Transport: Drift and Diffusion

Drift: \(J_n^{\text{drift}} = en\mu_n E\), \(J_p^{\text{drift}} = ep\mu_p E\). Diffusion: \(J_n^{\text{diff}} = eD_n(dn/dx)\), \(J_p^{\text{diff}} = -eD_p(dp/dx)\). Einstein relation: \(D/\mu = k_BT/e\).

Hall Effect

A current-carrying conductor in a transverse magnetic field develops a transverse Hall voltage \(V_H = IB/(net)\). Hall coefficient \(R_H = 1/(nq)\); its sign identifies the carrier type and its magnitude gives carrier density.

p–n Junction

Diffusion of carriers creates a depletion region with a built-in potential:

Shockley ideal diode equation: \(I = I_S(e^{eV/k_BT} - 1)\). Reverse breakdown: Zener (tunneling, narrow depletion layer, \(|V_z| \lesssim 5\) V, negative temperature coefficient) and avalanche (impact ionization, positive \(T\)-coefficient).

13. Diodes and Transistors

Diode Types

Rectifier diodes (power conversion), Zener diodes (voltage regulation), LEDs (electroluminescence), photodiodes (optical detection), solar cells (photovoltaics), Schottky diodes (fast switching), tunnel diodes (negative resistance), and varactors (voltage-controlled capacitance).

Rectifier Circuits

Half-wave rectifier: \(V_{\text{dc}} = V_m/\pi\), efficiency \(\eta = 40.6\%\), ripple \(\gamma = 1.21\). Full-wave bridge rectifier: \(V_{\text{dc}} = 2V_m/\pi\), \(\eta = 81.2\%\), \(\gamma = 0.48\). LC or capacitor filters reduce ripple.

Bipolar Junction Transistor (BJT)

Two back-to-back p–n junctions (NPN or PNP). In active mode (EB forward, CB reverse): \(I_E = I_B + I_C\), \(\alpha = I_C/I_E < 1\), \(\beta = I_C/I_B = \alpha/(1-\alpha) \gg 1\).

Field-Effect Transistors (FETs)

JFET: current controlled by reverse-biased gate; very high input impedance (>10⁹ Ω). MOSFET: oxide-isolated gate; enhancement or depletion mode. Saturation drain current (enhancement MOSFET):

CMOS (complementary MOSFET) is the foundation of modern digital electronics.

14. Superconductivity

Critical Parameters

The superconducting state is bounded by three quantities: critical temperature \(T_c\), critical field \(H_c(T) = H_0[1-(T/T_c)^2]\), and critical current density \(J_c\). The Meissner effect (\(\vec{B} = 0\) inside the bulk below \(T_c\)) distinguishes a superconductor from a perfect conductor.

Type-I vs Type-II Superconductors

BCS Theory

Bardeen–Cooper–Schrieffer (1957): phonon-mediated attraction causes pairs of electrons with opposite momenta and spins (Cooper pairs) to condense into a coherent macroscopic quantum state below \(T_c\). An energy gap \(2\Delta\) forms; excitation requires breaking pairs, hence zero resistance. BCS prediction: \(2\Delta(0) \approx 3.52\,k_BT_c\).

London Equations and Penetration Depth

\(\lambda_L\) is the London penetration depth, the skin depth for magnetic flux.

Josephson Effects

DC Josephson: Cooper-pair tunneling through a thin insulating barrier gives \(I = I_c\sin\phi\) at zero voltage. AC Josephson: a constant voltage \(V\) produces oscillating current at \(\omega = 2eV/\hbar\). Applications: SQUIDs (world's most sensitive magnetometers), voltage standards, superconducting qubits.

15. Nanotechnology

At the nanoscale: quantum confinement modifies electronic properties, the surface-to-volume ratio is enormously enhanced, mean-free-path effects govern transport, and optical resonances (plasmons) appear at characteristic sizes.

Dimensional Classification of Nanostructures

Synthesis Methods

Top-down (start large, carve to nanoscale): lithography (photo, e-beam, EUV), etching (RIE, wet), ball milling. Bottom-up (atom-by-atom assembly): CVD/PVD, MBE, ALD, sol-gel, self-assembly, colloidal chemistry.

Characterization Techniques

Special Nanomaterials

Carbon nanotubes (CNTs): rolled graphene, ballistic transport, Young's modulus ~TPa. Graphene: monolayer carbon, massless Dirac fermions, extremely high conductivity. Quantum dots: size-tunable photoluminescence. Nano-Ag, Nano-Au: plasmonic absorption, sensing, antibacterial. TiO₂, ZnO nanoparticles: photocatalysis, UV protection.

16. Electromagnetism

Vector Calculus Toolkit

Gradient \(\nabla\phi\) (steepest rise), divergence \(\nabla\cdot\vec{F}\) (source density), curl \(\nabla\times\vec{F}\) (rotation), Laplacian \(\nabla^2\phi = \nabla\cdot(\nabla\phi)\). Gauss's theorem: \(\oint_S \vec{F}\cdot d\vec{A} = \int_V (\nabla\cdot\vec{F})\,dV\). Stokes' theorem: \(\oint_C \vec{F}\cdot d\vec{\ell} = \int_S (\nabla\times\vec{F})\cdot d\vec{A}\).

Electrostatics

Coulomb force: \(\vec{F}_{12} = \frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{r_{12}^2}\hat{r}_{12}\). Gauss's law: \(\oint\vec{E}\cdot d\vec{A} = Q_{\text{enc}}/\varepsilon_0\), equivalently \(\nabla\cdot\vec{E} = \rho/\varepsilon_0\). Potential: \(\vec{E} = -\nabla V\). Energy stored in a field: \(U = \frac{\varepsilon_0}{2}\int E^2\,dV\). Parallel-plate capacitance: \(C = \varepsilon_0\varepsilon_r A/d\).

Magnetostatics

Biot–Savart law: \(d\vec{B} = \frac{\mu_0}{4\pi}\frac{I\,d\vec{\ell}\times\hat{r}}{r^2}\). Ampère's law: \(\oint\vec{B}\cdot d\vec{\ell} = \mu_0 I_{\text{enc}}\), \(\nabla\times\vec{B} = \mu_0\vec{J}\). Lorentz force: \(\vec{F} = q(\vec{E} + \vec{v}\times\vec{B})\).

Electromagnetic Induction

Faraday's law: \(\varepsilon = -d\Phi_B/dt\), \(\nabla\times\vec{E} = -\partial\vec{B}/\partial t\). Lenz's law: induced current opposes change. Self-inductance: \(\varepsilon = -L\,dI/dt\), energy \(U = \tfrac{1}{2}LI^2\). Mutual inductance: \(M = \Phi_{21}/I_1\) — basis of transformers.

Series LCR Circuit and Resonance

For a series LCR driven at angular frequency \(\omega\):

Quality factor and bandwidth: \(Q = \omega_0 L/R = (1/R)\sqrt{L/C}\), \(\Delta\omega = \omega_0/Q\). Engineering uses: radio/TV tuning, band-pass filters, power-factor correction, MRI coils, wireless charging.

17. Maxwell's Equations and EM Waves

Displacement Current

Ampère's law fails for a charging capacitor. Maxwell introduced the displacement current density \(\vec{J}_d = \varepsilon_0\,\partial\vec{E}/\partial t\), modifying Ampère's law to

Maxwell's Equations in Vacuum

EM Wave Equation

In free space (\(\rho = 0\), \(\vec{J} = 0\)), taking the curl of Faraday's law:

The same equation holds for \(\vec{B}\). EM waves are transverse (\(\vec{E} \perp \vec{B} \perp \hat{k}\)), in phase, with \(E = cB\).

Energy and Poynting Vector

Energy density: \(u = \varepsilon_0 E^2\). Poynting vector: \(\vec{S} = (1/\mu_0)\vec{E}\times\vec{B}\). Time-averaged intensity: \(\langle S\rangle = \frac{1}{2}c\varepsilon_0 E_0^2\).

Electromagnetic Spectrum

18. Special Theory of Relativity

Postulates

1. The laws of physics are identical in every inertial frame.
2. The speed of light in vacuum, \(c\), is the same in all inertial frames, independent of source motion.

Lorentz Transformations

For frame \(S'\) moving at velocity \(v\) along \(x\):

Maxwell's equations are invariant under Lorentz transformations (unlike Galilean ones).

Time Dilation and Length Contraction

Moving clocks run slow: \(\Delta t = \gamma\,\Delta t_0\). Moving lengths contract: \(L = L_0/\gamma\). Example: muons at \(0.99c\) (\(\gamma \approx 7.09\)) survive to reach detectors on Earth — confirming time dilation.

Relativistic Velocity Addition

Two photons approaching each other still give a relative speed \(c\), never \(2c\).

Relativistic Energy and Momentum

Rest energy \(E_0 = m_0 c^2\). Kinetic energy \(K = (\gamma - 1)m_0 c^2\).

19. Crystal Structure and X-Ray Diffraction

Crystalline vs Amorphous Solids

Crystalline solids have long-range periodic order, a sharp melting point, and are anisotropic (e.g., NaCl, metals). Amorphous solids have short-range order only, soften gradually, and are isotropic (e.g., glass, polymers). Most metals and ceramics are polycrystalline.

Fourteen Bravais Lattices and Seven Crystal Systems

Common Cubic Structures

Miller Indices

To find the \((hkl)\) indices: (1) find intercepts on axes in units of lattice constants, (2) take reciprocals, (3) reduce to smallest integers. Interplanar spacing in cubic: \(d_{hkl} = a/\sqrt{h^2 + k^2 + l^2}\).

Bragg's Law of X-Ray Diffraction

XRD Methods

Laue method (white X-rays, single crystal, spot pattern — orientation determination). Powder (Debye-Scherrer) method (monochromatic X-rays, powder sample, rings — phase identification, lattice constant). Rotating-crystal method (3-D structure information).

Crystal Defects

Point defects (vacancies, interstitials — Schottky, Frenkel), line defects (edge and screw dislocations → plastic deformation), planar defects (grain boundaries, stacking faults), and volume defects (voids, precipitates). Defects often govern mechanical, electrical, and optical properties of real materials.

20. Acoustics and Ultrasonics

Sound and Frequency Ranges

Audible: 20 Hz – 20 kHz. Infrasonic: <20 Hz (earthquakes, large animals). Ultrasonic: >20 kHz (medical, NDT). Speed of sound: \(v = \sqrt{B/\rho}\) (fluids), \(v = \sqrt{E/\rho}\) (solids, longitudinal).

Architectural Acoustics

Reverberation time \(T_{60}\) is the time for the sound level to drop 60 dB. Sabine's formula:

Common acoustic defects and remedies: echo (use absorbers), excessive reverberation (acoustic panels), focusing from concave surfaces (use diffusers), dead spots (diffusers), outside noise (soundproofing).

Production of Ultrasonic Waves

Magnetostriction oscillator: ferromagnetic rod (Ni) elongates and contracts under alternating magnetic field; works up to ~100 kHz. Piezoelectric oscillator: quartz or PZT crystal in a tank circuit; high frequencies up to MHz, high power.

21. Dielectrics and Magnetic Materials

Dielectric Materials

Polarization Mechanisms

• Electronic: distortion of electron cloud relative to nucleus.
• Ionic: relative displacement of positive and negative ions.
• Orientational (dipolar): alignment of permanent dipoles.
• Space-charge: at interfaces between media of different conductivity (Maxwell–Wagner).

Total: \(\alpha_{\text{tot}} = \alpha_e + \alpha_i + \alpha_o + \alpha_s\). As frequency increases, polarization mechanisms drop out successively.

Clausius–Mossotti relation:

Dielectric loss is quantified by the loss tangent \(\tan\delta = \varepsilon''/\varepsilon'\). Dielectric strength is the maximum field before breakdown.

Magnetic Materials

\(\vec{B} = \mu_0(\vec{H} + \vec{M})\), \(\vec{M} = \chi_m\vec{H}\), \(\mu_r = 1 + \chi_m\).

Ferromagnetism and Weiss Domains

Below the Curie temperature \(T_C\), exchange interaction aligns spins into magnetic domains. Above \(T_C\), the Curie–Weiss law governs: \(\chi_m = C/(T - T_C)\). The hysteresis loop is characterized by retentivity \(B_r\) (residual flux density) and coercivity \(H_c\) (field needed to demagnetize); its area equals energy lost per cycle per unit volume.

Soft vs Hard Magnetic Materials

22. Modern Physics — Atom and Hydrogen Spectra

Hydrogen Atomic Spectrum

Empirical Rydberg formula: \(1/\lambda = R(1/n_1^2 - 1/n_2^2)\), \(R = 1.097 \times 10^7\ \text{m}^{-1}\). Spectral series: Lyman (\(n_1 = 1\), UV), Balmer (\(n_1 = 2\), visible), Paschen/Brackett/Pfund (IR).

Bohr Model

Postulates: (1) angular momentum is quantized \(L = n\hbar\); (2) radiation emitted or absorbed when transitioning between orbits \(h\nu = E_i - E_f\). For hydrogen-like atoms:

Quantum Numbers and Pauli Exclusion Principle

The quantum-mechanical hydrogen atom is described by \(\psi_{nlm} = R_{nl}(r)Y_l^m(\theta,\phi)\) with: principal \(n = 1, 2, \ldots\); orbital \(\ell = 0, \ldots, n-1\) (s, p, d, f); magnetic \(m = -\ell, \ldots, +\ell\); spin \(s = \pm\tfrac{1}{2}\). The Pauli exclusion principle (no two fermions can share all four quantum numbers) determines the structure of the periodic table. Selection rules: \(\Delta\ell = \pm 1\), \(\Delta m = 0, \pm 1\), \(\Delta s = 0\).

23. Nuclear Physics

Nuclear Properties

Nucleus contains \(Z\) protons and \(N\) neutrons (\(A = Z + N\)). Nuclear radius: \(R = R_0 A^{1/3}\), \(R_0 \approx 1.2\ \text{fm}\). Nuclear density is nearly constant, ~\(2.3 \times 10^{17}\ \text{kg m}^{-3}\).

Binding Energy and the Semi-Empirical Mass Formula

Mass defect: \(\Delta m = Zm_p + Nm_n - M_{\text{nucleus}}\). Binding energy: \(B = (\Delta m)c^2\). Binding energy per nucleon \(B/A\) peaks near \(A \approx 56\) (iron). Weizsäcker's formula:

Terms: volume, surface, Coulomb, asymmetry, pairing.

Radioactive Decay

Activity \(A(t) = \lambda N(t)\), measured in becquerels (1 Bq = 1 decay/s). Decay modes: alpha (emission of \({}^4_2\text{He}\)), beta-minus (\(n \to p + e^- + \bar{\nu}_e\)), beta-plus, electron capture, and gamma (nuclear de-excitation).

Nuclear Fission and Fusion

Fission: \(n + {}^{235}\text{U} \to {}^{141}\text{Ba} + {}^{92}\text{Kr} + 3n + 200\ \text{MeV}\). Fusion: \({}^{2}\text{H} + {}^{3}\text{H} \to {}^{4}\text{He} + n + 17.6\ \text{MeV}\). Fusion powers stars and is pursued in ITER and NIF.

24. Renewable Energy Physics

Solar Photovoltaics

A p–n junction absorbs photons with \(h\nu > E_g\), generating electron–hole pairs separated by the built-in field. The equivalent circuit gives the solar cell I-V equation:

Key parameters: short-circuit current \(I_{sc}\), open-circuit voltage \(V_{oc}\), fill factor \(\text{FF} = V_{mp}I_{mp}/(V_{oc}I_{sc})\), efficiency \(\eta = V_{oc}I_{sc}\cdot\text{FF}/P_{\text{in}}\). The Shockley–Queisser limit for a single junction is ~33%.

Wind Energy

Power available in wind crossing area \(A\) at speed \(v\): \(P = \tfrac{1}{2}\rho Av^3\). The Betz limit: only \(C_{P,\max} = 16/27 \approx 59.3\%\) of wind power can be extracted by an ideal turbine. Practical wind turbines achieve \(C_P \sim 0.4\text{–}0.5\).

Hydro, Tidal, and Wave Energy

Hydropower: \(P = \rho g Q H \eta\) (world's largest renewable contribution). Tidal range potential: \(E = \tfrac{1}{2}\rho Agh^2\). Wave energy flux proportional to \(H^2 T\).

Geothermal, Biomass, and Hydrogen

Geothermal uses Earth's internal heat (capacity factor often >90%). Biomass is carbon-neutral if sustainably sourced. Green hydrogen, produced by electrolysis from surplus renewables, is a promising energy carrier for fuel cells.

Energy Storage Technologies

Batteries (Li-ion, flow, Na-ion), pumped hydro (95% of global grid storage), compressed air (CAES), flywheels, supercapacitors, thermal storage (molten salts in CSP), and hydrogen as a chemical carrier. Key engineering challenges: round-trip efficiency, lifecycle, and cost.

25. Engineering Applications of Physics

Engineering physics bridges fundamental principles and practical hardware across every engineering domain.

MEMS and NEMS

Micro/Nano-Electro-Mechanical Systems are miniaturized machines fabricated by IC-compatible processes, exploiting SHM, beam mechanics, and capacitive/piezoelectric transduction. Applications: accelerometers (airbag deployment, gaming), gyroscopes (drones), pressure sensors, microphones, lab-on-a-chip biosensors, RF MEMS filters and switches.

Display Technologies

• LCD: birefringence and polarizers, requires backlight.
• LED/mini-LED: direct emission from semiconductor junctions.
• OLED: organic semiconductors, self-emissive, flexible.
• Micro-LED: inorganic LED arrays <50 μm.
• Quantum-dot LED (QLED): size-tunable color via quantum confinement.

Medical Imaging and Sensing

X-ray CT (absorption tomography), MRI (nuclear spin precession in \(\vec{B}\)), ultrasound (piezoelectric transduction), OCT (low-coherence interferometry), PET (positron annihilation), and hyperspectral imaging.

Communications and Computing

Optical fiber networks (EDFAs, DWDM), 5G mm-wave and MIMO antennas, quantum communication and QKD (BB84), photonic integrated circuits, spintronics (GMR/TMR in hard-disk read heads), and quantum computing (superconducting and trapped-ion qubits).

Appendix A: Fundamental Physical Constants (CODATA)

Useful Conversion Factors

Appendix B: Formula Cheat Sheet

Mechanics and Oscillations

• SHM: \(x(t) = A\cos(\omega t + \phi)\), \(\omega = \sqrt{k/m}\)
• Energy: \(E = \tfrac{1}{2}kA^2 = \tfrac{1}{2}m\omega^2 A^2\)
• Damped: \(\ddot{x} + 2\gamma\dot{x} + \omega_0^2 x = 0\); quality factor \(Q = \omega_0/2\gamma\)
• Wave equation: \(\partial^2 y/\partial t^2 = v^2\,\partial^2 y/\partial x^2\)
• Group / phase velocity: \(v_g = d\omega/dk\), \(v_p = \omega/k\)

Optics

• YDSE fringe width: \(\beta = \lambda D/d\)
• Thin film (constructive, reflected): \(2\mu t\cos r = (n + \tfrac{1}{2})\lambda\)
• Newton's rings (dark, reflected): \(r_n^2 = n\lambda R\)
• Single-slit minima: \(a\sin\theta = m\lambda\)
• Grating principal maxima: \((a + b)\sin\theta = m\lambda\)
• Rayleigh criterion: \(\sin\theta_{\min} = 1.22\lambda/D\)
• Brewster: \(\tan\theta_p = n_2/n_1\); Malus: \(I = I_0\cos^2\theta\)
• Fiber NA: \(\text{NA} = \sqrt{n_1^2 - n_2^2} = n_0\sin\theta_a\)
• V-number: \(V = (2\pi a/\lambda)\,\text{NA}\); single mode if \(V < 2.405\)

Quantum and Atomic

• de Broglie: \(\lambda = h/p\)
• Uncertainty: \(\Delta x\,\Delta p \ge \hbar/2\)
• TISE: \(-(\hbar^2/2m)\nabla^2\psi + V\psi = E\psi\)
• Particle in 1-D box: \(E_n = n^2\pi^2\hbar^2/(2mL^2)\), \(\psi_n = \sqrt{2/L}\sin(n\pi x/L)\)
• Hydrogen: \(E_n = -13.6/n^2\ \text{eV}\); \(1/\lambda = R_H(1/n_1^2 - 1/n_2^2)\)
• Bohr radius: \(a_0 = 4\pi\varepsilon_0\hbar^2/(m_e e^2)\)
• Compton shift: \(\Delta\lambda = (h/m_e c)(1 - \cos\theta)\)
• Planck radiation: \(u(\nu, T) = \frac{8\pi h\nu^3}{c^3}\frac{1}{e^{h\nu/k_BT}-1}\)

Solid State, Semiconductors, Superconductivity

• Bragg: \(2d\sin\theta = n\lambda\); interplanar: \(d = a/\sqrt{h^2+k^2+l^2}\) (cubic)
• Fermi–Dirac: \(f(E) = 1/[1 + \exp((E - E_F)/k_BT)]\)
• Intrinsic carriers: \(n_i = \sqrt{N_cN_v}\exp(-E_g/2k_BT)\)
• Mass action: \(np = n_i^2\)
• Hall coefficient: \(R_H = 1/(nq)\)
• Shockley diode: \(I = I_0(e^{qV/k_BT} - 1)\)
• BCS gap (\(T=0\)): \(2\Delta(0) \approx 3.52\,k_BT_c\)
• London penetration: \(\lambda_L = \sqrt{m/(\mu_0 n_s e^2)}\)
• Flux quantum: \(\Phi_0 = h/(2e) = 2.07 \times 10^{-15}\ \text{Wb}\)

Electromagnetism and Relativity

Maxwell's equations (vacuum, differential form):

EM waves: \(c = 1/\sqrt{\mu_0\varepsilon_0}\); intensity \(I = \tfrac{1}{2}\varepsilon_0 c E_0^2\).

Special relativity: \(\gamma = 1/\sqrt{1-v^2/c^2}\); time dilation \(\Delta t = \gamma\,\Delta t_0\); length contraction \(L = L_0/\gamma\); energy-momentum \(E^2 = (pc)^2 + (m_0c^2)^2\); velocity addition \(u = (u'+v)/(1+u'v/c^2)\).

Appendix C: Greek Letters and Mathematical Operators in Physics

Common Mathematical Operators

• Gradient: \(\nabla f = (\partial_x f, \partial_y f, \partial_z f)\)
• Divergence: \(\nabla\cdot\vec{A} = \partial_x A_x + \partial_y A_y + \partial_z A_z\)
• Curl: \(\nabla\times\vec{A}\)
• Laplacian: \(\nabla^2 f = \nabla\cdot\nabla f\)
• D'Alembertian: \(\Box = \nabla^2 - (1/c^2)\partial_t^2\)
• Commutator: \([\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}\)
• Expectation value: \(\langle\hat{O}\rangle = \int\psi^*\hat{O}\psi\,d\tau\)
• Inner product: \(\langle\phi|\psi\rangle = \int\phi^*\psi\,d\tau\)
• Dirac delta: \(\int\delta(x-a)f(x)\,dx = f(a)\)

References and Bibliography

1. D. Halliday, R. Resnick, J. Walker, Fundamentals of Physics, 10th ed., Wiley, 2014.
2. A. Beiser, Concepts of Modern Physics, 6th ed., McGraw-Hill, 2003.
3. D. J. Griffiths, Introduction to Electrodynamics, 4th ed., Pearson, 2013.
4. D. J. Griffiths, Introduction to Quantum Mechanics, 3rd ed., Cambridge, 2018.
5. A. Ghatak, Optics, 6th ed., McGraw-Hill, 2017.
6. C. Kittel, Introduction to Solid State Physics, 8th ed., Wiley, 2005.
7. N. W. Ashcroft, N. D. Mermin, Solid State Physics, Cengage, 1976.
8. S. O. Pillai, Solid State Physics, 8th ed., New Age International, 2018.
9. B. K. Pandey, S. Chaturvedi, Engineering Physics, Cengage, 2012.
10. H. K. Malik, A. K. Singh, Engineering Physics, McGraw-Hill, 2010.
11. P. A. Tipler, R. Llewellyn, Modern Physics, 6th ed., W. H. Freeman, 2012.
12. S. M. Sze, K. K. Ng, Physics of Semiconductor Devices, 3rd ed., Wiley, 2007.
13. B. G. Streetman, S. K. Banerjee, Solid State Electronic Devices, 7th ed., Pearson, 2016.
14. K. S. Krane, Introductory Nuclear Physics, Wiley, 1987.
15. M. Tinkham, Introduction to Superconductivity, 2nd ed., Dover, 2004.
16. C. P. Poole et al., Introduction to Nanotechnology, Wiley, 2003.
17. J. D. Jackson, Classical Electrodynamics, 3rd ed., Wiley, 1998.
18. A. P. French, Vibrations and Waves, Nelson Thornes, 1971.
19. B. D. Cullity, Elements of X-Ray Diffraction, 3rd ed., Prentice Hall, 2001.
20. B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, 2nd ed., Wiley, 2007.