§ 7.1Vector Calculus & Electrostatics
The three operators
- Gradient ∇V (scalar → vector): direction and rate of maximum increase; \( \mathbf{E} = -\nabla V \).
- Divergence ∇·A (vector → scalar): net outflow per unit volume — source strength; Divergence theorem: \( \oint_S \mathbf{A}\cdot d\mathbf{S} = \int_V (\nabla\cdot\mathbf{A})\,dv \).
- Curl ∇×A (vector → vector): circulation per unit area; Stokes' theorem: \( \oint_L \mathbf{A}\cdot d\mathbf{l} = \int_S (\nabla\times\mathbf{A})\cdot d\mathbf{S} \).
Identities used constantly
$$ \nabla\times(\nabla V) = 0, \qquad \nabla\cdot(\nabla\times\mathbf{A}) = 0, \qquad \nabla\times(\nabla\times\mathbf{A}) = \nabla(\nabla\cdot\mathbf{A}) - \nabla^{2}\mathbf{A} $$
Conservative field test: ∇×E = 0 ⟺ E = −∇V ⟺ ∮E·dl = 0. Coordinate systems: rectangular, cylindrical (ρ, φ, z), spherical (r, θ, φ) — exam expects the Laplacian/divergence forms in all three to be recognizable.
Electrostatics fundamentals
Coulomb's law & field
$$ \mathbf{F} = \frac{Q_1Q_2}{4\pi\varepsilon_0 r^{2}}\hat{r}, \qquad \mathbf{E} = \frac{Q}{4\pi\varepsilon_0 r^{2}}\hat{r}, \qquad \varepsilon_0 = 8.854\times10^{-12}\ \text{F/m} $$
- Field of standard distributions: infinite line \( E = \lambda/2\pi\varepsilon_0\rho \); infinite sheet \( E = \sigma_s/2\varepsilon_0 \) (uniform, independent of distance); ring/disc on axis by superposition.
- Potential \( V = \frac{Q}{4\pi\varepsilon_0 r} \) (zero at ∞); \( V_{AB} = -\int_B^A \mathbf{E}\cdot d\mathbf{l} \); equipotentials ⊥ field lines.
- Dipole (moment p = Qd): \( V = \dfrac{p\cos\theta}{4\pi\varepsilon_0 r^{2}} \), E ∝ 1/r³; torque τ = p × E.
- Energy density \( w_E = \tfrac12\varepsilon E^2 \); system energy \( W = \tfrac12\sum Q_iV_i = \tfrac12\int \varepsilon E^2\,dv \).
UGC NET focus
curl(grad) = 0 and div(curl) = 0 identities; sheet field σ/2ε₀ distance-independence; dipole 1/r² potential vs 1/r³ field; divergence/Stokes theorem statement matching; conservative-field tests.
§ 7.2Gauss's Law & Applications
Gauss's law — integral & point forms
$$ \oint_S \mathbf{D}\cdot d\mathbf{S} = Q_{enc} \qquad\Longleftrightarrow\qquad \nabla\cdot\mathbf{D} = \rho_v, \qquad \mathbf{D} = \varepsilon\mathbf{E} $$
- Useful only with symmetry (Gaussian surface where D is constant and normal): spherical, cylindrical, planar.
- Standard results (compute on sight):
- Point/outside sphere: \( E = Q/4\pi\varepsilon_0r^2 \); inside a uniformly charged sphere: \( E = \rho r/3\varepsilon_0 \propto r \); inside a charged conducting shell: E = 0.
- Infinite line: \( E = \lambda/2\pi\varepsilon_0\rho \); coaxial cable between conductors: same form → capacitance \( C/L = 2\pi\varepsilon/\ln(b/a) \).
- Infinite sheet: \( E = \sigma_s/2\varepsilon_0 \); parallel-plate region: \( E = \sigma_s/\varepsilon_0 \), \( C = \varepsilon A/d \).
- Conductor properties (electrostatic): E_inside = 0, ρ_inside = 0; charge on surface; E just outside = σ_s/ε₀, normal; conductor is an equipotential; shielding (Faraday cage).
- Boundary conditions (memorize as a block):
Dielectric–dielectric boundary $$ E_{t1} = E_{t2}\ \ (\text{tangential E continuous}); \qquad D_{n1} - D_{n2} = \rho_s\ \ (\to D_n\ \text{continuous if } \rho_s = 0) $$Refraction of field lines: tan θ₁/tan θ₂ = ε₁/ε₂.
UGC NET focus
E ∝ r inside / 1/r² outside a charged sphere (graph identification); coax capacitance; boundary-condition statements (which component is continuous); flux through a surface from enclosed charge only.
§ 7.3Laplace & Poisson Equations
The governing equations
$$ \nabla^{2}V = -\frac{\rho_v}{\varepsilon} \ \ (\textbf{Poisson}); \qquad \rho_v = 0 \Rightarrow \nabla^{2}V = 0 \ \ (\textbf{Laplace}) $$
- Derivation: ∇·D = ρ with E = −∇V; the equations govern V in charge-containing and charge-free regions respectively.
- Uniqueness theorem: a solution satisfying the boundary conditions is the solution — licenses guessing, images, and numerical methods.
- 1-D workhorse solutions (asked as derivations):
- Parallel plates: V = V₀x/d (linear), E uniform = V₀/d → C = εA/d.
- Coaxial: V ∝ ln ρ → \( C/L = 2\pi\varepsilon/\ln(b/a) \); spherical shells: V ∝ 1/r → \( C = 4\pi\varepsilon\,\dfrac{ab}{b-a} \) (isolated sphere: 4πεa).
- Properties of harmonic (Laplace) solutions: no local maxima/minima inside the region (extrema on boundaries); mean-value property; superposition holds.
- Method of images: point charge above a grounded plane ≡ charge + image −Q at the mirror point — force \( = -Q^2/4\pi\varepsilon_0(2d)^2 \), induced surface charge integrates to −Q.
- Capacitance energy: \( W = \tfrac12CV^2 = Q^2/2C \); dielectrics raise C by ε_r (polarization P = ε₀χ_eE, D = ε₀E + P).
UGC NET focus
Poisson↔Laplace condition (ρ = 0); the three classic capacitances; uniqueness statement; image-charge force numericals; "V has no interior extremum" property.
§ 7.4Magnetostatics: Biot–Savart & Ampere
Biot–Savart law
$$ d\mathbf{H} = \frac{I\,d\mathbf{l}\times\hat{r}}{4\pi r^{2}} \qquad (\mathbf{B} = \mu\mathbf{H},\ \mu_0 = 4\pi\times10^{-7}\ \text{H/m}) $$
- Standard fields (instant recall): infinite straight wire \( H = I/2\pi\rho \); finite segment \( H = \frac{I}{4\pi\rho}(\sin\alpha_1 + \sin\alpha_2) \); circular-loop centre \( H = I/2a \); on axis \( H = \dfrac{Ia^2}{2(a^2+z^2)^{3/2}} \); solenoid interior \( H = nI \) (uniform, n = turns/length); toroid \( H = NI/2\pi\rho \).
Ampere's circuital law
$$ \oint \mathbf{H}\cdot d\mathbf{l} = I_{enc} \qquad\Longleftrightarrow\qquad \nabla\times\mathbf{H} = \mathbf{J} $$
- The magnetic analogue of Gauss — needs symmetry (Amperian loop); inside a thick wire H = Iρ/2πa² ∝ ρ.
- No monopoles: ∇·B = 0, ∮B·dS = 0 — flux lines close on themselves; vector potential B = ∇×A.
- Forces: Lorentz \( \mathbf{F} = q(\mathbf{E} + \mathbf{v}\times\mathbf{B}) \); on a current element dF = I dl × B; parallel wires \( F/L = \mu_0I_1I_2/2\pi d \) (attract if parallel currents — defines the ampere); torque on a loop τ = m × B with moment m = IA n̂.
- Magnetic boundary conditions: B_n continuous; H_t discontinuous by surface current K: \( H_{t1} - H_{t2} = K \) (continuous if K = 0).
- Materials: dia/para/ferromagnetic (χ_m sign/magnitude); B = µ₀(H + M); hysteresis vocabulary (retentivity, coercivity) for Unit-X transformer contexts.
UGC NET focus
H formulas for wire/loop/solenoid/toroid (most-asked plug-ins); inside-wire linear profile; parallel-wire force and the ampere definition; which components are continuous at magnetic boundaries; ∇·B = 0 meaning.
§ 7.5Electromagnetic Induction
Faraday's law
$$ e = -\frac{d\Phi}{dt} = -N\frac{d\phi}{dt} \qquad\Longleftrightarrow\qquad \nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t} $$
- Lenz's law (the minus sign): induced effects oppose the change — energy conservation in disguise.
- EMF varieties: transformer EMF (stationary circuit, time-varying B) and motional EMF \( e = \oint(\mathbf{v}\times\mathbf{B})\cdot d\mathbf{l} = Blv \) for a sliding rod; rotating loop → \( e = \omega NAB\sin\omega t \) (AC generator).
- Inductance: self \( L = N\phi/I \) — solenoid \( L = \mu N^2A/l \); toroid \( \mu N^2A/2\pi\rho \); coax \( L/L' = \frac{\mu}{2\pi}\ln(b/a) \) per length. Mutual \( M = k\sqrt{L_1L_2} \) (0 ≤ k ≤ 1); dot convention for series aiding/opposing \( L = L_1 + L_2 \pm 2M \).
- Energy \( W = \tfrac12LI^2 \); density \( w_m = \tfrac12\mu H^2 \) — symmetric partner of w_E.
- Eddy currents (lamination cure), skin effect preview (§7.7), induction applications: transformers, induction heating, generators.
UGC NET focus
Blv and ωNAB sin ωt plug-ins; Lenz-direction reasoning; solenoid L ∝ N²; series-aiding/opposing ±2M; transformer vs motional EMF classification.
§ 7.6Maxwell's Equations & the Wave Equation
| Law | Differential | Integral | Statement |
|---|---|---|---|
| Gauss (E) | \( \nabla\cdot\mathbf{D} = \rho_v \) | \( \oint\mathbf{D}\cdot d\mathbf{S} = Q_{enc} \) | charges source D |
| Gauss (M) | \( \nabla\cdot\mathbf{B} = 0 \) | \( \oint\mathbf{B}\cdot d\mathbf{S} = 0 \) | no monopoles |
| Faraday | \( \nabla\times\mathbf{E} = -\dfrac{\partial\mathbf{B}}{\partial t} \) | \( \oint\mathbf{E}\cdot d\mathbf{l} = -\dfrac{d\Phi}{dt} \) | changing B makes E |
| Ampere–Maxwell | \( \nabla\times\mathbf{H} = \mathbf{J} + \dfrac{\partial\mathbf{D}}{\partial t} \) | \( \oint\mathbf{H}\cdot d\mathbf{l} = I + \dfrac{d\Psi_D}{dt} \) | currents + changing D make H |
- Displacement current \( J_d = \partial D/\partial t \) — Maxwell's addition: rescues continuity \( \nabla\cdot\mathbf{J} = -\partial\rho/\partial t \) (capacitor-charging paradox) and predicts EM waves. Ratio J_c/J_d = σ/ωε classifies media (§7.7).
- Wave-equation derivation (source-free, linear medium): curl of Faraday + Ampere →
Helmholtz / wave equations $$ \nabla^{2}\mathbf{E} = \mu\varepsilon\frac{\partial^{2}\mathbf{E}}{\partial t^{2}}, \qquad v = \frac{1}{\sqrt{\mu\varepsilon}}, \qquad c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} = 3\times10^{8}\ \text{m/s} $$— light is electromagnetic (the historic punchline).
- Time-harmonic (phasor) forms: ∂/∂t → jω; \( \nabla^2\mathbf{E}_s + k^2\mathbf{E}_s = 0 \), \( k = \omega\sqrt{\mu\varepsilon} \).
- Constitutive set: D = εE, B = µH, J = σE (Ohm); potentials V, A with Lorenz gauge → retarded potentials (§7.13).
UGC NET focus
Write/identify all four equations (both forms) — guaranteed; what term Maxwell added and why (continuity + waves); v = 1/√µε numericals in dielectrics; phasor substitution ∂/∂t → jω.
§ 7.7Plane Waves: Free Space, Dielectrics, Conductors
Uniform plane wave anatomy
- E ⊥ H ⊥ propagation k̂ (TEM); E × H points along propagation; E and H in phase in lossless media; E/H = η.
Free-space parameters
$$ \eta_0 = \sqrt{\frac{\mu_0}{\varepsilon_0}} = 120\pi \approx 377\ \Omega; \qquad \beta = \frac{\omega}{c} = \frac{2\pi}{\lambda}; \qquad \lambda = \frac{c}{f} $$
General lossy medium
Propagation constant
$$ \gamma = \alpha + j\beta = \sqrt{j\omega\mu(\sigma + j\omega\varepsilon)}; \qquad \text{loss tangent } \tan\delta = \frac{\sigma}{\omega\varepsilon} = \frac{J_c}{J_d} $$
- Classification: lossless/perfect dielectric σ = 0; good dielectric σ/ωε ≪ 1: \( \alpha \approx \tfrac{\sigma}{2}\sqrt{\mu/\varepsilon} \), v ≈ 1/√µε, η real-ish; good conductor σ/ωε ≫ 1:
Good-conductor results (memorize the whole row) $$ \alpha = \beta = \sqrt{\pi f\mu\sigma}; \qquad \boxed{\delta = \frac{1}{\sqrt{\pi f\mu\sigma}}}\ (\text{skin depth}); \qquad \eta = \sqrt{\frac{\omega\mu}{\sigma}}\angle 45^{\circ} \Rightarrow \mathbf{H}\ \text{lags}\ \mathbf{E}\ \text{by } 45^{\circ} $$
- Skin-depth facts: field falls to 1/e (37%) in δ; copper at 1 MHz → δ ≈ 66 µm (∝ 1/√f) — basis of RF surface currents, plating, shielding; surface resistance \( R_s = 1/\sigma\delta = \sqrt{\pi f\mu/\sigma} \).
- In dielectrics v = c/√ε_r, λ shrinks by √ε_r, η = η₀/√ε_r (non-magnetic).
- Wave velocities vocabulary: phase v_p = ω/β; group v_g = dω/dβ (information/energy in dispersive systems — full use in §7.12).
UGC NET focus
377 Ω; skin-depth formula and ∝ 1/√f numericals; loss-tangent classification thresholds; H lagging E by 45° in good conductors; λ and η scaling by √ε_r.
§ 7.8Poynting Theorem
Poynting vector & theorem
$$ \mathbf{S} = \mathbf{E}\times\mathbf{H}\ \ (\text{W/m}^2); \qquad -\oint_S(\mathbf{E}\times\mathbf{H})\cdot d\mathbf{S} = \frac{\partial}{\partial t}\int_V\left(\frac{\varepsilon E^2}{2} + \frac{\mu H^2}{2}\right)dv + \int_V \sigma E^2\,dv $$
- Reading: power flowing in through a closed surface = rate of increase of stored (electric + magnetic) energy + ohmic dissipation — EM energy conservation.
- Time-average power density (the usable formula):
Average Poynting vector $$ \mathbf{S}_{avg} = \frac12\,\mathrm{Re}(\mathbf{E}_s\times\mathbf{H}_s^{*}) = \frac{E_0^{2}}{2\eta}\hat{k} \ \ (\text{lossless; } E_0 = \text{peak}) $$e.g., E₀ = 1 V/m in free space → S_avg = 1/(2×377) ≈ 1.33 mW/m².
- Applications: antenna radiated power = ∮S_avg·dS (§7.14); power entering a resistor's surface equals I²R (classic demonstration); solar constant ~1.37 kW/m² ↔ E_rms computations.
- In a good conductor S decays as e^{−2αz} — power dissipates within a few skin depths.
UGC NET focus
S = E×H direction checks; E₀²/2η numericals (peak vs RMS factor!); term-by-term meaning of the theorem; power ∝ e^{−2αz} (field e^{−αz}).
§ 7.9Reflection & Refraction
Normal incidence (impedance language)
Reflection & transmission coefficients
$$ \Gamma = \frac{\eta_2 - \eta_1}{\eta_2 + \eta_1}, \qquad \tau = \frac{2\eta_2}{\eta_2 + \eta_1} = 1 + \Gamma; \qquad \text{SWR} = \frac{1+|\Gamma|}{1-|\Gamma|} $$
- Perfect conductor (η₂ = 0): Γ = −1 — total reflection, standing wave with E-nulls at the surface and every λ/2; H maximum at the surface.
- Power bookkeeping: reflected fraction |Γ|², transmitted 1 − |Γ|².
Oblique incidence (optics language)
- Snell's laws: θ_r = θ_i; \( n_1\sin\theta_1 = n_2\sin\theta_2 \), n = √ε_r (non-magnetic) = c/v.
- Polarization split: perpendicular/s/TE (E ⊥ plane of incidence) and parallel/p/TM — Fresnel coefficients differ.
- Brewster angle (parallel pol. only, Γ_∥ = 0):
Brewster & critical angles $$ \tan\theta_B = \frac{n_2}{n_1} \quad (\text{reflected light fully s-polarized}); \qquad \sin\theta_c = \frac{n_2}{n_1}\ (n_1 > n_2) $$
- Total internal reflection beyond θ_c: |Γ| = 1, transmitted field becomes evanescent (exponentially decaying, carries no average power) — optical fibers (NA = √(n₁²−n₂²)), prisms, frustrated-TIR couplers.
- Glass n = 1.5: θ_B ≈ 56.3°, θ_c ≈ 41.8° (the two stock numericals).
UGC NET focus
Γ and τ = 1 + Γ plug-ins; Γ = −1 on conductors and the λ/2 null pattern; tan θ_B = n₂/n₁ vs sin θ_c = n₂/n₁ (don't swap!); Brewster polarizes; evanescent-wave property.
§ 7.10Polarization, Interference, Coherence & Diffraction
Polarization (tip of E vs time)
- Linear: components in phase (0/180°); circular: equal amplitudes, 90° phase (RHCP/LHCP by rotation sense vs propagation); elliptical: the general case (axial ratio; AR = 1 circular, ∞ linear).
- Devices: polarizers — Malus's law \( I = I_0\cos^2\theta \); unpolarized → I₀/2; quarter-wave plate linear↔circular (90° retardation), half-wave plate rotates linear by 2θ; polarization by reflection at Brewster, by scattering (sky), dichroism (Polaroid).
- Antenna link: polarization mismatch loss cos²ψ; circular polarization defeats Faraday rotation (satcom).
Interference & coherence
- Superposition of coherent waves: \( I = I_1 + I_2 + 2\sqrt{I_1I_2}\cos\delta \); equal beams → 4I₀ max, 0 min; fringe visibility \( V = (I_{max}-I_{min})/(I_{max}+I_{min}) \).
- Young's double slit: fringe width \( \beta = \lambda D/d \); bright at path difference mλ, dark (m+½)λ; thin films: 2nt cos θ = (m+½)λ for bright (reflection π-shift at denser medium) — Newton's rings, AR coatings (λ/4, n = √(n₁n₂), link §1.15).
- Coherence: temporal — coherence time τ_c ≈ 1/Δν, length \( l_c = c\tau_c = \lambda^2/\Delta\lambda \) (lasers: metres; white light: µm); spatial — across the wavefront (source size; stellar interferometry). Interference demands coherence; incoherent sources add intensities.
Diffraction
- Bending around obstacles — Huygens–Fresnel; regimes: Fresnel (near) vs Fraunhofer (far field, the exam case).
- Single slit (width a): minima at \( a\sin\theta = m\lambda \); central max twice as wide; intensity sinc². Grating: maxima \( d\sin\theta = m\lambda \), resolving power R = λ/Δλ = mN.
- Rayleigh criterion (circular aperture D): \( \theta_{min} = 1.22\lambda/D \) — telescope/antenna resolution (beamwidth ∝ λ/D bridges to §7.14).
- Double-slit with finite slits = interference pattern under a diffraction envelope; missing orders when d/a integer.
UGC NET focus
Malus cos²θ and the unpolarized ½ factor; circular-polarization conditions (equal + 90°); fringe width λD/d; coherence length λ²/Δλ; single-slit minima vs grating maxima (min/max swap trap); 1.22λ/D.
§ 7.11Transmission Lines: Equations, Impedance, VSWR
Distributed model & line equations
Per-unit-length R, L, G, C → telegrapher's equations; solutions are forward + backward waves with:
Primary results
$$ \gamma = \sqrt{(R + j\omega L)(G + j\omega C)} = \alpha + j\beta; \qquad Z_0 = \sqrt{\frac{R + j\omega L}{G + j\omega C}} \xrightarrow{\text{lossless}} \sqrt{\frac{L}{C}};\qquad v_p = \frac{1}{\sqrt{LC}} $$
- Lossless: α = 0, β = ω√(LC); distortionless (Heaviside): R/L = G/C → real Z₀, frequency-flat α (telephony loading coils). Typical Z₀: coax 50/75 Ω, twisted pair 100–120 Ω, twin-lead 300 Ω.
Reflection & standing waves
The cluster that decides most numericals
$$ \Gamma_L = \frac{Z_L - Z_0}{Z_L + Z_0}; \qquad \text{VSWR} = S = \frac{1 + |\Gamma|}{1 - |\Gamma|};\qquad |\Gamma| = \frac{S-1}{S+1};\qquad \frac{P_{refl}}{P_{inc}} = |\Gamma|^{2} $$
- Landmarks: matched Z_L = Z₀ → Γ = 0, S = 1; open → Γ = +1; short → Γ = −1 (both S = ∞). V_max = (1+|Γ|)V⁺, V_min = (1−|Γ|)V⁺; adjacent maxima λ/2 apart; R at V_max = SZ₀, at V_min = Z₀/S.
Input impedance & the standard tricks
Impedance transformation
$$ Z_{in}(l) = Z_0\,\frac{Z_L + jZ_0\tan\beta l}{Z_0 + jZ_L\tan\beta l} $$
- Shorted line: Z_in = jZ₀tan βl — inductive for l < λ/4, open at λ/4, repeats each λ/2 → stubs, resonators. Open line: −jZ₀cot βl (capacitive first).
- λ/4 transformer: \( Z_{in} = Z_0^2/Z_L \) — match real loads with \( Z_0' = \sqrt{Z_0Z_L} \); inverts short↔open. λ/2: Z_in = Z_L (repeats).
- Stub matching: single shorted stub in parallel where line admittance = Y₀ + jB; stub supplies −jB. Smith chart = Γ-plane map of normalized impedance: constant-r and constant-x circles; one full revolution = λ/2; VSWR circle radius |Γ|; admittance = point diametrically opposite.
UGC NET focus
Γ↔VSWR conversions; Z_in of λ/4 and λ/2 sections (Z₀²/Z_L is the most-fired single formula); shorted-stub reactance cycle; distortionless condition R/L = G/C; λ/2 spacing of maxima; Smith-chart facts (rotation = λ/2, opposite point = admittance).
§ 7.12Rectangular Waveguides
- Hollow conductor (a × b, a > b) guiding TE (E_z = 0) and TM (H_z = 0) modes — no TEM (single conductor). High-pass behaviour: each mode propagates only above its cutoff.
Cutoff & the dominant mode
$$ f_{c(mn)} = \frac{c}{2}\sqrt{\left(\frac{m}{a}\right)^{2} + \left(\frac{n}{b}\right)^{2}} \qquad \textbf{TE}_{10}: f_c = \frac{c}{2a}\ \ (\lambda_c = 2a) $$
- TE₁₀ dominant (lowest f_c, single-mode band ≈ 1.25–1.9 f_c); TM modes start at TM₁₁ (no TM₁₀/TM₀₁); X-band guide WR-90 (a = 22.86 mm) → f_c = 6.56 GHz, used 8.2–12.4 GHz.
Above-cutoff dispersion set (memorize as one block)
$$ \lambda_g = \frac{\lambda}{\sqrt{1 - (f_c/f)^2}}, \qquad v_p = \frac{c}{\sqrt{1-(f_c/f)^2}} > c, \qquad v_g = c\sqrt{1-(f_c/f)^2} < c, \qquad \boxed{v_pv_g = c^2} $$
$$ Z_{TE} = \frac{\eta}{\sqrt{1-(f_c/f)^2}}\ (>\eta); \qquad Z_{TM} = \eta\sqrt{1-(f_c/f)^2}\ (<\eta) $$
- v_p > c is phase only (no information faster than light — v_g carries energy); below cutoff: evanescent attenuation (waveguide-below-cutoff attenuators).
- Field picture TE₁₀: E_y half-sine across a, max at centre, zero at side walls; wall currents dictate slot placement (slotted-line measurement along the broad-wall centre).
- Excitation: probe (E-coupling) at E-max, λ_g/4 from the shorted end; loop (H-coupling) at H-max.
- Cavity resonator = shorted guide section: \( f_{mnp} = \frac{c}{2}\sqrt{(m/a)^2+(n/b)^2+(p/d)^2} \), dominant TE₁₀₁; very high Q (10³–10⁴) — wavemeters, klystron circuits.
- Why waveguides at microwaves: lower loss and higher power than coax (no dielectric/centre conductor); circular guides and modes (TE₁₁ dominant) exist but rectangular dominates practice.
UGC NET focus
f_c = c/2a and λ_c = 2a numericals; the λ_g/v_p/v_g/Z block with the v_pv_g = c² identity; "no TEM in hollow guides" and "no TM₁₀"; X-band WR-90 example; below-cutoff evanescence.
§ 7.13Antennas: Retarded Potential, Hertzian & Half-Wave Dipoles
Retarded potentials — why antennas radiate
Retarded vector potential
$$ \mathbf{A}(r, t) = \frac{\mu_0}{4\pi}\int \frac{\mathbf{J}\left(r',\, t - R/c\right)}{R}\,dv' $$
Fields at distance R respond to the source as it was a time R/c earlier — finite propagation speed converts accelerating charge into radiation; B = ∇×A, E from A and V (Lorenz gauge).
Hertzian (infinitesimal) dipole — length dl ≪ λ
- Field zones: near/induction terms ∝ 1/r², 1/r³ (reactive energy) vs far/radiation field ∝ 1/r dominating beyond r ≫ λ/2π:
Far field & radiation resistance $$ E_\theta = \frac{j\eta_0 \beta I\,dl}{4\pi r}\sin\theta\,e^{-j\beta r}; \qquad R_{rad} = 80\pi^{2}\left(\frac{dl}{\lambda}\right)^{2}\ \Omega $$
- Pattern sin θ (doughnut, null on axis); directivity D = 1.5 (1.76 dB); tiny R_rad → inefficient (why real antennas are λ-scale).
Half-wave dipole (l = λ/2)
- Standing-wave current I(z) = I₀cos(βz); far-field pattern \( F(\theta) = \dfrac{\cos\left(\tfrac{\pi}{2}\cos\theta\right)}{\sin\theta} \) — slightly sharper doughnut.
- The number set: R_rad ≈ 73 Ω (input Z ≈ 73 + j42.5 Ω; trim slightly short for resonance); D = 1.64 (2.15 dBi = 0 dBd); HPBW = 78°.
- Monopole (λ/4 over ground): image theory doubles the dipole → R_rad ≈ 36.5 Ω, D = 3.28 — broadcast towers, whips.
- Feeding: balanced dipole from unbalanced coax needs a balun; folded dipole ≈ 4×73 ≈ 292 Ω (matches 300 Ω twin-lead; Yagi driven element).
UGC NET focus
80π²(dl/λ)² and the 73 Ω / 1.64 / 2.15 dBi / 36.5 Ω number set; 1/r radiation vs 1/r²,³ near fields; retarded-time meaning; sin θ doughnut with axial nulls; monopole image doubling.
§ 7.14Antenna Parameters: Patterns, Gain, Effective Area, Friis
- Radiation pattern: field/power vs angle; main lobe, side lobes (SLL), back lobe, nulls; E-plane and H-plane cuts; HPBW (−3 dB width) and first-null beamwidth; front-to-back ratio.
- Radiation intensity U(θ, φ) = r²S_avg (W/sr); total P_rad = ∮U dΩ.
Directivity, gain, efficiency
$$ D = \frac{U_{max}}{U_{avg}} = \frac{4\pi U_{max}}{P_{rad}}; \qquad G = \eta_{ant}D,\ \ \eta_{ant} = \frac{R_{rad}}{R_{rad} + R_{loss}}; \qquad D \approx \frac{4\pi}{\Omega_A} \approx \frac{41253}{\theta_E^\circ\,\theta_H^\circ} $$
- Reference scales: dBi (vs isotropic) and dBd (vs half-wave dipole) — dBi = dBd + 2.15.
- Effective aperture A_e = P_received/S_incident; the universal relation
Gain–aperture relation $$ G = \frac{4\pi A_e}{\lambda^{2}} \qquad (\text{any antenna}); \quad \text{apertures: } A_e = e_{ap}A_{phys},\ e_{ap}\approx 0.5\text{–}0.7\ \text{for dishes} $$Isotropic A_e = λ²/4π; dish gain G = e_ap(πD/λ)² with beamwidth ≈ 70λ/D degrees (diffraction limit of §7.10).
Friis free-space transmission equation — the link formula
$$ \frac{P_r}{P_t} = G_tG_r\left(\frac{\lambda}{4\pi R}\right)^{2}; \qquad \text{Path loss (dB)} = 20\log\frac{4\pi R}{\lambda} = 32.44 + 20\log f_{MHz} + 20\log R_{km} $$
- Conditions: far field (\(R > 2D^2/\lambda\)), matched polarization and impedance; (λ/4πR)² is free-space path loss, not absorption — spreading of power.
- Other terms: bandwidth, input impedance/VSWR (ties §7.11), polarization (ties §7.10); reciprocity — transmit and receive properties identical.
UGC NET focus
G = 4πA_e/λ² and Friis numericals (link budgets in dB); D from beamwidths (41253 rule); η = R_rad/(R_rad + R_loss); dBi↔dBd offset 2.15; far-field 2D²/λ criterion; path loss grows 6 dB per doubling of R or f.
§ 7.15Microwave Tubes: Reflex Klystron, Magnetron, TWT
Conventional tubes/transistors fail at microwaves (transit time, lead L/C, gain–BW); microwave tubes exploit transit time via velocity modulation → bunching → energy extraction. Two families: linear-beam O-type (klystron, TWT — beam ∥ B-focus field) and crossed-field M-type (magnetron — E ⊥ B).
Reflex klystron (low-power oscillator)
- Single re-entrant cavity + repeller (negative electrode) returns the beam through the same gap. Gap voltage velocity-modulates electrons; drift in the repeller space bunches them; bunches re-cross the gap and deliver energy when arriving at the retarding phase.
- Oscillation condition: round-trip transit time \( T = (n + \tfrac34)\,T_{RF} \) → modes n = 0, 1, 2…; the 1¾ mode typically optimum power.
- Electronic tuning: repeller voltage shifts frequency slightly (~MHz) within a mode — easy FM; mechanical tuning via cavity. Output 10–500 mW, 1–25 GHz, efficiency 2–10% (low). Historic local oscillator / lab bench source (now Gunn-replaced).
Magnetron (high-power oscillator)
- Cylindrical cathode inside an anode block of N resonant cavities; radial E, axial B (crossed fields) → electrons follow cycloidal paths.
- Electrons synchronous with the rotating RF field form spokes ("favourable" electrons give energy and spiral outward; unfavourable return to the cathode — back-heating).
- π-mode operation (adjacent cavities 180° out of phase) stabilized by strapping or rising-sun anodes; frequency set by cavity resonance.
- The high-power champion: kW (CW, ovens at 2.45 GHz) to MW (pulsed radar transmitters); efficiency 40–70%; oscillator only (noisy, not phase-coherent — modern coherent radars prefer amplifier chains).
Travelling-Wave Tube (TWT — wideband amplifier)
- Electron beam interacts continuously with an RF wave slowed by a helix (slow-wave structure: v_RF,axial ≈ c·pitch/circumference ≈ beam velocity).
- Beam slightly faster than the wave → bunches form and surrender energy along the whole length → growing wave; attenuator/sever mid-helix suppresses backward oscillation.
- No resonant cavities → octave-plus bandwidth (the TWT's defining edge over the klystron); gain 30–60 dB; W–kW; satellite transponders (TWTA), ECM, radar drivers. Coupled-cavity TWTs trade bandwidth for power.
- (Two-cavity klystron = narrowband high-power amplifier sibling: buncher/catcher cavities, applegate diagram vocabulary.)
UGC NET focus
n + ¾ repeller modes and repeller-voltage electronic tuning; π-mode + strapping keywords; crossed-field vs linear-beam classification; helix as slow-wave structure; TWT = wide bandwidth (no cavities) — the discriminating fact; magnetron 2.45 GHz oven anchor.
§ 7.16Microwave Diodes: Gunn, IMPATT, Crystal Detector, PIN
Gunn diode (transferred-electron device)
- Bulk GaAs/InP effect — no junction. Above ~3.2 kV/cm electrons transfer from the light-mass Γ valley to heavy-mass L satellite valleys (Ridley–Watkins–Hilsum mechanism) → velocity falls as field rises → bulk negative differential mobility.
- High-field domains nucleate at the cathode and transit to the anode → current pulses at \( f \approx v_d/L \) (transit-time/Gunn mode; ~10⁷ m/s ÷ 10 µm → 10 GHz); LSA mode for higher power/frequency.
- Low power (mW–W), low noise, 1–100 GHz; the standard lab/Doppler/motion-sensor and LO source — direct successor to the reflex klystron. (Contrast with the tunnel diode's junction-tunneling NDR, §1.9.)
IMPATT (IMPact Avalanche Transit Time)
- Reverse-biased junction at avalanche breakdown (Read structure n⁺-p-i-p⁺): avalanche multiplication injects a carrier pulse with a built-in 90° delay; transit across the drift region adds the other ~90° → current lags voltage by ~180° → negative resistance.
- Optimum drift length: transit angle π → \( f \approx v_{sat}/2L \).
- Highest CW power of solid-state microwave diodes (W-level, to >100 GHz) but very noisy (avalanche statistics) and needs high bias — transmitters where noise tolerable; TRAPATT (higher efficiency, lower f) and BARITT (injection by barrier, low noise/low power) as siblings.
Crystal (point-contact Schottky) detector
- Metal whisker / Schottky junction on Si or GaAs — majority-carrier rectifier with femto-farad capacitance → detects/mixes to 100+ GHz (1N21/1N23 heritage).
- Square-law region at small signals: DC output ∝ RF power (V ∝ E²) — power meters and slotted-line/VSWR measurement; large-signal → linear (envelope) detection. As a mixer, nonlinearity produces the IF = |f_RF − f_LO| (superheterodyne front ends, conversion loss ~5–8 dB); figures of merit: tangential sensitivity, video resistance.
PIN diode
- P⁺ – intrinsic (wide i-layer) – N⁺. The i-layer gives high reverse breakdown, low constant capacitance, and slow stored-charge response.
- At microwave frequencies (f ≫ 1/τ_carrier) it cannot rectify — it behaves as a current-controlled RF resistor: forward bias floods the i-layer (R falls from kΩ to <1 Ω); reverse bias = small fixed C.
- Applications (each is a one-word answer in MCQs): RF switches (T/R switching in radar), attenuators/modulators (variable R), phase shifters (switched lines for phased arrays), receiver protectors/limiters; plus photodiode duty at optical λ (Unit-VIII — wide i-region absorbs).
UGC NET focus
Gunn = bulk RWH effect, no junction, f = v/L; IMPATT = avalanche 90° + transit 90°, powerful but noisiest; crystal detector square-law ∝ power; PIN = bias-controlled RF resistor → switch/attenuator/phase shifter; match device ↔ mechanism ↔ application tables.
§ 7.17Radar: Block Diagram, Frequencies & Range Equation
Principle & block diagram (pulsed radar)
- Transmit a pulse, time the echo: R = c·t/2 (the 2 = round trip; 1 µs ↔ 150 m).
- Blocks: trigger/synchronizer → modulator → transmitter (magnetron/klystron/TWT/SSPA) → duplexer (TR/ATR tubes or circulator — shares one antenna, protects the receiver) → antenna (rotating/phased) → low-noise RF amp → mixer + LO (superheterodyne) → IF amplifier → detector → video → display (classic PPI) ; AFC/STC/AGC support.
- Timing vocabulary: pulse width τ (range resolution ΔR = cτ/2), PRF f_p, maximum unambiguous range \( R_{unamb} = c/2f_p \); duty cycle = τf_p; P_avg = P_peak·τ·f_p.
- Doppler/CW & MTI: moving target shifts frequency \( f_d = 2v_r/\lambda \) — CW radar (speed guns), MTI/pulse-Doppler filters clutter; blind speeds when f_d = nf_p.
Frequencies & power
| Band | Frequency | Typical use |
|---|---|---|
| L | 1–2 GHz | long-range air surveillance, ATC |
| S | 2–4 GHz | terminal ATC, weather (2.7–3 GHz) |
| C | 4–8 GHz | weather, tracking |
| X | 8–12 GHz | airborne, marine, police, missile seekers |
| Ku/K/Ka | 12–18 / 18–27 / 27–40 GHz | mapping, automotive (24/77 GHz), short range |
Powers: peak kW–MW pulsed (magnetron/klystron) with low duty (~0.1%) → average W–kW; solid-state arrays distribute power across modules.
Radar range equation
Derivation chain → the fourth-root law
$$ S_{r} = \frac{P_tG}{4\pi R^{2}}\cdot\sigma\cdot\frac{1}{4\pi R^{2}}\cdot A_e \ \Longrightarrow\ \boxed{R_{max} = \left[\frac{P_t\,G^{2}\lambda^{2}\,\sigma}{(4\pi)^{3}\,S_{min}}\right]^{1/4}} \quad (A_e = G\lambda^2/4\pi,\ \text{monostatic}) $$
- R ∝ P_t^{1/4}: doubling range costs 16× power (or +12 dB) — the headline implication; echo power ∝ 1/R⁴.
- σ = radar cross-section (m²; stealth minimizes it); S_min set by receiver noise kT₀BF and required SNR; practical losses L and integration of n pulses refine the equation.
- Secondary (transponder) radar and bistatic forms scale as 1/R² per leg.
UGC NET focus
R = ct/2 and R_unamb = c/2f_p numericals; the (1/4-power) range equation with the 16×-for-2× consequence; duplexer function; Doppler 2v/λ; band-letter matching (X = 8–12 GHz); range resolution cτ/2.
§ 7.18Unit-7 Formula Sheet
| Topic | Formula | Notes |
|---|---|---|
| Key identities | ∇×∇V = 0; ∇·(∇×A) = 0 | conservative / solenoidal tests |
| Gauss | ∮D·dS = Q; ∇·D = ρ | sphere-inside E ∝ r |
| Sheet / line fields | σ/2ε₀ ; λ/2πε₀ρ | plate gap: σ/ε₀ |
| Poisson / Laplace | ∇²V = −ρ/ε ; ∇²V = 0 | uniqueness theorem |
| Capacitances | εA/d; 2πε/ln(b/a); 4πε ab/(b−a) | plate / coax / spheres |
| Biot–Savart set | I/2πρ; I/2a; nI; NI/2πρ | wire / loop centre / solenoid / toroid |
| Faraday | e = −N dφ/dt; e = Blv; ωNAB sinωt | Lenz minus sign |
| Inductance | L = Nφ/I = µN²A/l (solenoid) | M = k√(L₁L₂); series ±2M |
| Wave speed | v = 1/√(µε); c = 3×10⁸ | η₀ = 377 Ω; η = η₀/√ε_r |
| Loss tangent | tan δ = σ/ωε | ≫1 conductor, ≪1 dielectric |
| Skin depth | δ = 1/√(πfµσ) | α = β; η at 45°; ∝ 1/√f |
| Poynting | S_avg = E₀²/2η | S = E×H W/m² |
| Normal incidence | Γ = (η₂−η₁)/(η₂+η₁); τ = 1+Γ | conductor: Γ = −1 |
| Brewster / critical | tan θ_B = n₂/n₁; sin θ_c = n₂/n₁ | TIR → evanescent |
| Optics set | I = I₀cos²θ; β = λD/d; a sinθ = mλ (min) | grating d sinθ = mλ (max); 1.22λ/D |
| Coherence length | l_c = λ²/Δλ | visibility (I_max−I_min)/(I_max+I_min) |
| Line constants | Z₀ = √(L/C); γ = √(ZY) | distortionless R/L = G/C |
| Γ / VSWR | Γ = (Z_L−Z₀)/(Z_L+Z₀); S = (1+|Γ|)/(1−|Γ|) | P_refl = |Γ|²P_inc |
| λ/4 transformer | Z_in = Z₀²/Z_L; Z₀′ = √(Z₀Z_L) | λ/2 repeats Z_L |
| Stub reactances | short: jZ₀tanβl; open: −jZ₀cotβl | maxima λ/2 apart |
| Waveguide cutoff | f_c = (c/2)√((m/a)²+(n/b)²); TE₁₀: c/2a | λ_c = 2a; no TEM/TM₁₀ |
| Guide dispersion | λ_g = λ/√(1−(f_c/f)²); v_pv_g = c² | Z_TE > η > Z_TM |
| Hertzian dipole | R_rad = 80π²(dl/λ)²; D = 1.5 | far field ∝ sinθ/r |
| λ/2 dipole | R ≈ 73 Ω; D = 1.64 (2.15 dBi) | monopole: 36.5 Ω |
| Gain–aperture | G = 4πA_e/λ² | D ≈ 41253/(θ_E θ_H) |
| Friis | P_r/P_t = G_tG_r(λ/4πR)² | FSPL dB = 32.44+20logf_MHz+20logR_km |
| Reflex klystron | T = (n + ¾)T_RF | repeller-voltage tuning |
| Gunn / IMPATT | f = v_d/L ; f = v_sat/2L | RWH bulk NDR / avalanche+transit |
| Radar basics | R = ct/2; R_unamb = c/2f_p; f_d = 2v/λ | ΔR = cτ/2 |
| Range equation | R_max = [P_tG²λ²σ/((4π)³S_min)]^{1/4} | 2× range = 16× power |
§ 7.19Quick Revision Notes — Unit 7 in 25 Points
Rapid-fire recap (last-day revision)
- curl(grad) = 0, div(curl) = 0; divergence theorem ↔ volume sources, Stokes ↔ circulation; ∇×E = 0 ⇒ E = −∇V (statics).
- Gauss results: sphere E ∝ r inside / 1/r² outside; line λ/2πε₀ρ; sheet σ/2ε₀ (distance-free); conductor interior E = 0, surface field σ/ε₀.
- Boundaries: tangential E continuous, normal D jumps by ρ_s; normal B continuous, tangential H jumps by K.
- ∇²V = −ρ/ε (Poisson) → 0 (Laplace); uniqueness justifies images: charge above ground plane ≡ mirror −Q; no interior extrema for harmonic V.
- Field bank: wire I/2πρ, loop centre I/2a, solenoid nI, toroid NI/2πρ; inside a wire H ∝ ρ; parallel currents attract (F/L = µ₀I₁I₂/2πd → defines the ampere).
- Faraday e = −Ndφ/dt; motional Blv; generator ωNABsinωt; Lenz = opposition = energy conservation; L_solenoid = µN²A/l; series L₁+L₂±2M.
- Maxwell's four — and the displacement current ∂D/∂t that fixes continuity and predicts waves; v = 1/√(µε).
- Plane wave: E ⊥ H ⊥ k, E/H = η, η₀ = 377 Ω; in dielectric λ and η shrink by √ε_r.
- tan δ = σ/ωε sorts media; good conductor: α = β = √(πfµσ), δ = 1/√(πfµσ) ∝ 1/√f, η∠45° (H lags E).
- S = E×H; S_avg = E₀²/2η (peak!); theorem = inflow → stored (½εE² + ½µH²) + dissipated σE².
- Γ = (η₂−η₁)/(η₂+η₁), τ = 1+Γ; conductor Γ = −1 → standing wave, nulls each λ/2 from the surface.
- tan θ_B = n₂/n₁ (reflected fully s-polarized); sin θ_c = n₂/n₁; beyond θ_c the transmitted wave is evanescent (no average power).
- Polarization: in-phase = linear; equal + 90° = circular; Malus I₀cos²θ, unpolarized halves; QWP linear↔circular.
- Fringe width λD/d; visibility from I_max/min; coherence length λ²/Δλ; single-slit minima a sinθ = mλ vs grating maxima d sinθ = mλ; Rayleigh 1.22λ/D.
- Z₀ = √(L/C) lossless; distortionless R/L = G/C; Γ_L = (Z_L−Z₀)/(Z_L+Z₀); VSWR S = (1+|Γ|)/(1−|Γ|); reflected power |Γ|².
- Z_in(λ/4) = Z₀²/Z_L (match via √(Z₀Z_L)); λ/2 repeats; shorted stub jZ₀tanβl (L below λ/4); Smith chart: full turn = λ/2, opposite point = admittance.
- Waveguide = high-pass; TE₁₀ dominant, f_c = c/2a, λ_c = 2a; no TEM, no TM₁₀; below cutoff evanescent.
- Dispersion block: λ_g = λ/√(1−(f_c/f)²); v_p > c, v_g < c, v_pv_g = c²; Z_TE > η > Z_TM; probe at E-max λ_g/4 from short.
- Retarded potentials (t − R/c) explain radiation; Hertzian dipole: far field ∝ sinθ/r, R_rad = 80π²(dl/λ)², D = 1.5.
- λ/2 dipole card: 73 Ω, D = 1.64 = 2.15 dBi (0 dBd), HPBW 78°; λ/4 monopole halves to 36.5 Ω, doubles D; folded dipole ≈ 292 Ω.
- D = 4πU_max/P_rad ≈ 41253/(θ_Eθ_H); G = ηD; G = 4πA_e/λ²; Friis P_r/P_t = G_tG_r(λ/4πR)²; far field beyond 2D²/λ; path loss +6 dB per doubling of R or f.
- Reflex klystron: one cavity + repeller, modes n + ¾, repeller-voltage electronic tuning, mW LO (historic).
- Magnetron: crossed fields, cavity anode, π-mode + strapping, electron spokes — kW–MW oscillator (ovens 2.45 GHz, pulse radar). TWT: helix slow-wave, continuous interaction → widest bandwidth amplifier (satellites).
- Gunn = bulk GaAs valley transfer (RWH), f ≈ v/L, low-noise LO; IMPATT = avalanche (90°) + transit (90°) → −R, most powerful but noisiest; crystal detector = square-law Schottky (V ∝ P, mixers); PIN = bias-controlled RF resistor → switch/attenuator/phase-shifter/limiter.
- Radar: R = ct/2; R_unamb = c/2f_p; ΔR = cτ/2; f_d = 2v/λ; duplexer shares the antenna; bands L/S/C/X(8–12)/Ku…; R_max ∝ (P_tG²λ²σ/S_min)^{1/4} — double range = 16× power.