Electromagnetic Theory (EX-504 (B)) - Unit 1 Short Notes

UNIT 1: FOUNDATIONS OF ELECTROMAGNETIC THEORY

I. VECTOR ANALYSIS

Coordinate Systems

• Cartesian (x, y, z): Unit vectors $$\displaystyle \hat{a}_x, \hat{a}_y, \hat{a}_z $$. Differential length $$\displaystyle d\mathbf{l} = dx\,\hat{a}_x + dy\,\hat{a}_y + dz\,\hat{a}_z $$, area $$\displaystyle d\mathbf{S} = dy\,dz\,\hat{a}_x + dx\,dz\,\hat{a}_y + dx\,dy\,\hat{a}_z $$, volume $$\displaystyle d\tau = dx\,dy\,dz $$.

• Cylindrical $(\rho, \phi, z)$: Unit vectors $$\displaystyle \hat{a}_\rho, \hat{a}_\phi, \hat{a}_z $$ (position-dependent). Scale factors: $$\displaystyle h_\rho=1, h_\phi=\rho, h_z=1 $$. Differential length $$\displaystyle d\mathbf{l} = d\rho\,\hat{a}_\rho + \rho\,d\phi\,\hat{a}_\phi + dz\,\hat{a}_z $$.

• Spherical $(r, \theta, \phi)$: Unit vectors $$\displaystyle \hat{a}_r, \hat{a}_\theta, \hat{a}_\phi $$. Scale factors: $$\displaystyle h_r=1, h_\theta=r, h_\phi=r\sin\theta $$. Differential length $$\displaystyle d\mathbf{l} = dr\,\hat{a}_r + r\,d\theta\,\hat{a}_\theta + r\sin\theta\,d\phi\,\hat{a}_\phi $$.

Vector Operations

• Gradient ($\nabla V$): Points in direction of max increase. In Cartesian: $$\displaystyle \nabla V = \frac{\partial V}{\partial x}\hat{a}_x + \frac{\partial V}{\partial y}\hat{a}_y + \frac{\partial V}{\partial z}\hat{a}_z $$.

• Divergence ($\nabla \cdot \mathbf{A}$): Net flux per unit volume (source density). In Cartesian: $$\displaystyle \nabla \cdot \mathbf{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z} $$.

• Curl ($\nabla \times \mathbf{A}$): Circulation per unit area (rotationality). In Cartesian: $$\displaystyle \nabla \times \mathbf{A} = \begin{vmatrix} \hat{a}_x & \hat{a}_y & \hat{a}_z \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ A_x & A_y & A_z \end{vmatrix} $$.

• Laplacian ($$\displaystyle \nabla^2 $$):

  • Scalar: $$\displaystyle \nabla^2 V = \nabla \cdot (\nabla V) $$.

  • Vector: $$\displaystyle \nabla^2 \mathbf{A} = \nabla(\nabla \cdot \mathbf{A}) - \nabla \times (\nabla \times \mathbf{A}) $$.

Fundamental Theorems

• Divergence (Gauss) Theorem: $$\displaystyle \boxed{\oint_S \mathbf{A} \cdot d\mathbf{S} = \int_V (\nabla \cdot \mathbf{A})\,d\tau} $$ Relates surface integral to volume integral.

• Stokes’ Theorem: $$\displaystyle \boxed{\oint_C \mathbf{A} \cdot d\mathbf{l} = \int_S (\nabla \times \mathbf{A}) \cdot d\mathbf{S}} $$ Relates line integral to surface integral.

Vector Identities

• $$\displaystyle \nabla \cdot (\nabla \times \mathbf{A}) = 0 $$

• $$\displaystyle \nabla \times (\nabla V) = 0 $$

• $$\displaystyle \nabla \cdot (\mathbf{A} \times \mathbf{B}) = \mathbf{B} \cdot (\nabla \times \mathbf{A}) - \mathbf{A} \cdot (\nabla \times \mathbf{B}) $$

• $$\displaystyle \nabla \times (\mathbf{A} \times \mathbf{B}) = (\mathbf{B} \cdot \nabla)\mathbf{A} - (\mathbf{A} \cdot \nabla)\mathbf{B} + \mathbf{A}(\nabla \cdot \mathbf{B}) - \mathbf{B}(\nabla \cdot \mathbf{A}) $$

• Product rules: $$\displaystyle \nabla \cdot (V\mathbf{A}) = V(\nabla \cdot \mathbf{A}) + \mathbf{A} \cdot (\nabla V) $$, $$\displaystyle \nabla \times (V\mathbf{A}) = V(\nabla \times \mathbf{A}) + (\nabla V) \times \mathbf{A} $$.

[!TIP] Exam Focus: Be fluent in converting vector operations between Cartesian and cylindrical coordinates. Past papers often ask for explicit calculations in cylindrical (e.g., NOV 2023 Q1ii).

II. ELECTROSTATICS

Coulomb’s Law & Electric Field Intensity ($\mathbf{E}$)

• Point charge: $$\displaystyle \mathbf{E} = \frac{Q}{4\pi\varepsilon_0 r^2}\,\hat{a}_r $$.

• Line charge ($\lambda$): $$\displaystyle \mathbf{E} = \frac{\lambda}{2\pi\varepsilon_0 \rho}\,\hat{a}_\rho $$ (infinite straight).

• Surface charge ($$\displaystyle \rho_s $$): $$\displaystyle \mathbf{E} = \frac{\rho_s}{2\varepsilon_0}\,\hat{a}_n $$ (infinite plane).

• Volume charge ($$\displaystyle \rho_v $$): $$\displaystyle \mathbf{E} = \frac{1}{4\pi\varepsilon_0} \int_V \frac{\rho_v\,(\mathbf{r} - \mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|^3}\,d\tau' $$.

Electric Potential ($V$)

• Definition: $$\displaystyle \boxed{V(\mathbf{r}) = -\int_{\infty}^{\mathbf{r}} \mathbf{E} \cdot d\mathbf{l}} $$ (reference at infinity).

• Relation to $\mathbf{E}$: $$\displaystyle \mathbf{E} = -\nabla V $$.

• Potential due to point charge: $$\displaystyle V = \frac{Q}{4\pi\varepsilon_0 r} $$.

• Electric Dipole (moment $$\displaystyle \mathbf{p} = Q\mathbf{d} $$):

  • Potential: $$\displaystyle \boxed{V = \frac{\mathbf{p} \cdot \hat{a}_r}{4\pi\varepsilon_0 r^2}} $$ (far field).

  • Field: $$\displaystyle \mathbf{E} = \frac{1}{4\pi\varepsilon_0 r^3} \left[ 3(\mathbf{p}\cdot\hat{a}_r)\hat{a}_r - \mathbf{p} \right] $$.

• Properties: Equipotential surfaces are perpendicular to $\mathbf{E}$; $V$ satisfies Laplace/Poisson.

Gauss’s Law

• Integral: $$\displaystyle \boxed{\oint_S \mathbf{D} \cdot d\mathbf{S} = Q_{\text{enc}}} $$.

• Differential: $$\displaystyle \boxed{\nabla \cdot \mathbf{D} = \rho_v} $$.

• Applications: Use symmetry (spherical, cylindrical, planar) to find $\mathbf{D}$ easily.

Polarization

• Polarization vector $\mathbf{P}$: dipole moment per unit volume.

• $$\displaystyle \mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P} $$.

• Linear dielectrics: $$\displaystyle \mathbf{P} = \varepsilon_0 \chi_e \mathbf{E} $$, $$\displaystyle \mathbf{D} = \varepsilon \mathbf{E} = \varepsilon_0 \varepsilon_r \mathbf{E} $$, where $$\displaystyle \varepsilon_r = 1 + \chi_e $$.

• Types: Electronic, ionic, orientation, space charge.

Continuity & Currents

• Conduction current density: $$\displaystyle \mathbf{J} = \sigma \mathbf{E} $$.

• Convection current density: $$\displaystyle \mathbf{J}_c = \rho_v \mathbf{v} $$ (moving charges in vacuum/fluid).

Laplace’s & Poisson’s Equations

• Poisson: $$\displaystyle \boxed{\nabla^2 V = -\frac{\rho_v}{\varepsilon_0}} $$.

• Laplace: $$\displaystyle \boxed{\nabla^2 V = 0} $$ (source-free regions).

• Uniqueness Theorem: Solution to Laplace/Poisson with given boundary conditions is unique.

Boundary Conditions for $\mathbf{E}$

• Normal component: $$\displaystyle \boxed{D_{1n} - D_{2n} = \rho_s} $$.

• Tangential component: $$\displaystyle \boxed{E_{1t} - E_{2t} = 0} $$.

[!TIP] At conductor surface: $$\displaystyle E_t=0 $$, $$\displaystyle D_n = \rho_s $$.

Capacitance

• Definition: $$\displaystyle \boxed{C = \frac{Q}{V}} $$.

• Parallel plate: $$\displaystyle C = \varepsilon \frac{A}{d} $$.

• Coaxial: $$\displaystyle C = \frac{2\pi\varepsilon l}{\ln(b/a)} $$.

• Spherical: $$\displaystyle C = 4\pi\varepsilon \frac{ab}{b-a} $$.

• Energy: $$\displaystyle \boxed{W = \frac{1}{2} C V^2 = \frac{1}{2} \int \rho V\,d\tau = \frac{1}{2} \int \varepsilon E^2\,d\tau} $$.

Method of Images

• Replace conductors with image charges to satisfy boundary conditions.

• For a point charge near grounded conducting plane: image charge $-Q$ at mirror position.

• For a charge inside a grounded spherical shell: image charge $$\displaystyle Q' = -\frac{a}{r}Q $$ at distance $$\displaystyle a^2/r $$ from center.

III. CURRENTS, CONTINUITY, AND CONVECTION

Current Density

• Conduction: $$\displaystyle \mathbf{J} = \sigma \mathbf{E} $$ (Ohm’s law, microscopic).

• Convection: $$\displaystyle \mathbf{J}_c = \rho_v \mathbf{v} $$ (e.g., electron beam).

Continuity Equation

• From charge conservation: $$\displaystyle \oint_S \mathbf{J} \cdot d\mathbf{S} = -\frac{\partial}{\partial t} \int_V \rho_v\,d\tau $$.

• Differential form: $$\displaystyle \boxed{\nabla \cdot \mathbf{J} = -\frac{\partial \rho_v}{\partial t}} $$.

[!TIP] For steady currents ($$\displaystyle \partial \rho_v / \partial t = 0 $$), $$\displaystyle \nabla \cdot \mathbf{J} = 0 $$.

Displacement Current Density

• $$\displaystyle \mathbf{J}_d = \frac{\partial \mathbf{D}}{\partial t} $$.

• Added to Ampere’s law for consistency with continuity: $$\displaystyle \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t} $$.

• In capacitors: $$\displaystyle I_d = \varepsilon \frac{d\Phi_E}{dt} $$ completes circuit.

IV. MAGNETOSTATICS

Biot–Savart’s Law

• $$\displaystyle \boxed{d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I\,d\mathbf{l} \times \hat{a}_r}{r^2}} $$.

• For infinite straight wire: $$\displaystyle B = \frac{\mu_0 I}{2\pi \rho}\,\hat{a}_\phi $$.

• For circular loop (axis): $$\displaystyle B = \frac{\mu_0 I R^2}{2(R^2 + z^2)^{3/2}}\,\hat{a}_z $$.

Ampere’s Circuital Law

• Integral: $$\displaystyle \boxed{\oint_C \mathbf{H} \cdot d\mathbf{l} = I_{\text{enc}}} $$.

• Differential: $$\displaystyle \boxed{\nabla \times \mathbf{H} = \mathbf{J}} $$.

• Applications:

  • Infinite wire: $$\displaystyle H = \frac{I}{2\pi \rho} $$.

  • Solenoid (ideal): $$\displaystyle H = nI\,\hat{a}_z $$ inside, ~0 outside.

  • Toroid: $$\displaystyle H = \frac{NI}{2\pi \rho}\,\hat{a}_\phi $$.

$\mathbf{B}$ & $\mathbf{H}$

• $$\displaystyle \mathbf{B} = \mu \mathbf{H} = \mu_0 \mu_r \mathbf{H} $$.

• $\mathbf{B}$: magnetic flux density (webers/m²). $\mathbf{H}$: magnetic field intensity (A/m).

Magnetic Vector Potential ($\mathbf{A}$)

• Definition: $$\displaystyle \boxed{\mathbf{B} = \nabla \times \mathbf{A}} $$.

• For steady currents: $$\displaystyle \mathbf{A}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int_V \frac{\mathbf{J}(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|}\,d\tau' $$.

• Gauge: Coulomb gauge ($$\displaystyle \nabla \cdot \mathbf{A} = 0 $$) often used.

Magnetic Dipole

• Moment: $$\displaystyle \mathbf{m} = I\mathbf{A} $$ (area vector).

• Field (far): $$\displaystyle \mathbf{B} = \frac{\mu_0}{4\pi r^3} \left[ 3(\mathbf{m}\cdot\hat{a}_r)\hat{a}_r - \mathbf{m} \right] $$.

• Torque: $$\displaystyle \mathbf{N} = \mathbf{m} \times \mathbf{B} $$.

Inductance

• Self-inductance: $$\displaystyle \boxed{L = \frac{N\Phi}{I}} $$ (flux linkage per unit current).

• Mutual inductance: $$\displaystyle \boxed{M = \frac{N_2 \Phi_{21}}{I_1}} $$.

• Formulas:

  • Solenoid: $$\displaystyle L = \mu \frac{N^2 A}{l} $$.

  • Toroid: $$\displaystyle L = \mu \frac{N^2 h}{2\pi} \ln\frac{b}{a} $$.

• Energy: $$\displaystyle \boxed{W = \frac{1}{2} L I^2 = \frac{1}{2} \int \mathbf{B} \cdot \mathbf{H}\,d\tau} $$.

Boundary Conditions for $\mathbf{B}$ & $\mathbf{H}$

• Normal component: $$\displaystyle \boxed{B_{1n} = B_{2n}} $$ ($$\displaystyle \nabla \cdot \mathbf{B}=0 $$).

• Tangential component: $$\displaystyle \boxed{\mathbf{H}_{1t} - \mathbf{H}_{2t} = \mathbf{K} \times \hat{a}_n} $$, where $\mathbf{K}$ is surface current density (A/m).

V. ELECTRODYNAMICS (TIME-VARYING FIELDS)

Faraday’s Law of Induction

• Integral: $$\displaystyle \boxed{\oint_C \mathbf{E} \cdot d\mathbf{l} = -\frac{d\Phi_B}{dt}} $$.

• Differential: $$\displaystyle \boxed{\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}} $$.

• Motional EMF: $$\displaystyle \mathbf{E} = \mathbf{v} \times \mathbf{B} $$ (for moving conductor in $\mathbf{B}$).

• Transformer EMF: Induced $\mathbf{E}$ from time-varying $\mathbf{B}$ (no motion).

Maxwell’s Equations (Complete Set)

Poynting Vector and Theorem

• Poynting vector: $$\displaystyle \boxed{\mathbf{S} = \mathbf{E} \times \mathbf{H}} $$ (power flow density, W/m²).

• Poynting theorem: $$\displaystyle \boxed{\frac{\partial}{\partial t}(u_e + u_m) + \nabla \cdot \mathbf{S} = -\mathbf{J} \cdot \mathbf{E}} $$.

• Energy densities: $$\displaystyle u_e = \frac{1}{2}\mathbf{D}\cdot\mathbf{E} $$, $$\displaystyle u_m = \frac{1}{2}\mathbf{B}\cdot\mathbf{H} $$.

[!TIP] $\mathbf{J}\cdot\mathbf{E}$ is power delivered to charges (ohmic loss).

VI. ELECTROMAGNETIC WAVES

Wave Equation

• From Maxwell’s in source-free, homogeneous, isotropic medium ($$\displaystyle \rho_v=0 $$, $$\displaystyle \mathbf{J}=0 $$):

$$\boxed{\nabla^2 \mathbf{E} = \mu\varepsilon \frac{\partial^2 \mathbf{E}}{\partial t^2}},\quad \boxed{\nabla^2 \mathbf{H} = \mu\varepsilon \frac{\partial^2 \mathbf{H}}{\partial t^2}}$$

• For uniform plane wave: $$\displaystyle \frac{\partial^2 E_z}{\partial x^2} = \mu\varepsilon \frac{\partial^2 E_z}{\partial t^2} $$.

Uniform Plane Waves

• $\mathbf{E}$, $\mathbf{H}$, propagation direction $\hat{k}$ mutually orthogonal.

• Intrinsic impedance: $$\displaystyle \eta = \sqrt{\mu/\varepsilon} $$ (lossless).

• $$\displaystyle E/H = \eta $$, direction given by $$\displaystyle \mathbf{E} \times \mathbf{H} = \hat{k} \cdot S $$.

Propagation in Different Media

• Lossless Dielectric ($$\displaystyle \sigma=0 $$):

  • $$\displaystyle \gamma = j\beta $$, $$\displaystyle \eta = \sqrt{\mu/\varepsilon} $$, $$\displaystyle v_p = 1/\sqrt{\mu\varepsilon} $$.

• Good Conductor ($\sigma \gg \omega\varepsilon$):

  • $$\displaystyle \gamma \approx (1+j)\sqrt{\frac{\omega\mu\sigma}{2}} $$, $$\displaystyle \eta \approx (1+j)\sqrt{\frac{\omega\mu}{2\sigma}} $$.

  • Skin depth: $$\displaystyle \boxed{\delta = \frac{1}{\alpha} = \sqrt{\frac{2}{\omega\mu\sigma}}} $$.

• Lossy Dielectric (general $\sigma, \varepsilon, \mu$):

  • $$\displaystyle \gamma = \alpha + j\beta = j\omega\sqrt{\mu\varepsilon}\sqrt{1 - j\frac{\sigma}{\omega\varepsilon}} $$.

  • $$\displaystyle \eta = |\eta| \angle \theta/2 $$, where $$\displaystyle \theta = \tan^{-1}\left(\frac{\sigma}{\omega\varepsilon}\right) $$.

Wave Parameters

• Propagation constant: $$\displaystyle \gamma = \alpha + j\beta $$.

• Phase constant: $$\displaystyle \beta = \omega\sqrt{\frac{\mu\varepsilon}{2}\left(\sqrt{1 + (\frac{\sigma}{\omega\varepsilon})^2} + 1\right)} $$.

• Attenuation constant: $$\displaystyle \alpha = \omega\sqrt{\frac{\mu\varepsilon}{2}\left(\sqrt{1 + (\frac{\sigma}{\omega\varepsilon})^2} - 1\right)} $$.

• Phase velocity: $$\displaystyle v_p = \omega/\beta $$.

• Group velocity: $$\displaystyle v_g = d\omega/d\beta $$.

Polarization

• Linear: $\mathbf{E}$ direction constant.

• Circular: $$\displaystyle E_x = E_0\cos(\omega t - \beta z) $$, $$\displaystyle E_y = E_0\sin(\omega t - \beta z) $$ (or phase difference $$\displaystyle 90^\circ $$).

• Elliptical: General case with unequal amplitudes and arbitrary phase difference.

Reflection & Refraction at Plane Boundaries

• Normal Incidence (medium 1→2):

  • Reflection coefficient: $$\displaystyle \boxed{\Gamma = \frac{\eta_2 - \eta_1}{\eta_2 + \eta_1}} $$.

  • Transmission coefficient: $$\displaystyle \boxed{\tau = \frac{2\eta_2}{\eta_2 + \eta_1}} $$ (for $E$).

  • Standing Wave Ratio: $$\displaystyle \boxed{\text{SWR} = \frac{1+|\Gamma|}{1-|\Gamma|}} $$.

• Oblique Incidence:

  • TE (perpendicular): $$\displaystyle \Gamma_\perp = \frac{\eta_2 \cos\theta_i - \eta_1 \cos\theta_t}{\eta_2 \cos\theta_i + \eta_1 \cos\theta_t} $$.

  • TM (parallel): $$\displaystyle \Gamma_\parallel = \frac{\eta_2 \cos\theta_t - \eta_1 \cos\theta_i}{\eta_2 \cos\theta_t + \eta_1 \cos\theta_i} $$.

  • Brewster’s angle ($$\displaystyle \theta_B $$): $$\displaystyle \Gamma_\parallel=0 $$ → $$\displaystyle \tan\theta_B = \sqrt{\frac{\varepsilon_{r2}}{\varepsilon_{r1}}} $$ (lossless).

  • Total internal reflection: $$\displaystyle \theta_i > \theta_c = \sin^{-1}(\sqrt{\varepsilon_{r1}/\varepsilon_{r2}}) $$ (for $$\displaystyle \varepsilon_{r2}<\varepsilon_{r1} $$).

Surface Impedance

• For good conductors: $$\displaystyle Z_s = \frac{E_t}{H_t} $$ at surface.

• $$\displaystyle Z_s = \eta \coth(\gamma d) \approx (1+j)\sqrt{\frac{\omega\mu}{2\sigma}} = (1+j)\frac{1}{\sigma\delta} $$ for thick conductor.

Transmission Line Analogy

VII. ADVANCED TOPICS & SPECIAL CASES

Perfect Conductor

• $$\displaystyle \mathbf{E}_t = 0 $$, $$\displaystyle \mathbf{B}_n = 0 $$.

• $$\displaystyle \mathbf{H}_t = \mathbf{K} \times \hat{a}_n $$, $$\displaystyle \mathbf{D}_n = \rho_s $$.

• Wave reflection at boundary: $$\displaystyle \Gamma = -1 $$ (for $E$ normal incidence), no field inside.

Convection Current Density in Moving Media

• $$\displaystyle \mathbf{J}_c = \rho_v \mathbf{v} $$ (distinct from $$\displaystyle \mathbf{J} = \sigma \mathbf{E} $$).

• Total current in moving dielectric: $$\displaystyle \mathbf{J}_{\text{total}} = \sigma \mathbf{E} + \rho_v \mathbf{v} $$.

Magnetic Dipole Moment

• For planar loop: $$\displaystyle \mathbf{m} = I\mathbf{A} $$.

• Torque: $$\displaystyle \mathbf{N} = \mathbf{m} \times \mathbf{B} $$.

• Potential energy: $$\displaystyle U = -\mathbf{m}\cdot\mathbf{B} $$.

Self-Inductance of Solenoid & Toroid

• Solenoid (air core): $$\displaystyle \boxed{L = \mu_0 \frac{N^2 A}{l}} $$.

• Toroid (core): $$\displaystyle \boxed{L = \mu \frac{N^2 h}{2\pi} \ln\frac{b}{a}} $$ (mean radius method).

Method of Images (Recap)

• Electrostatics: Image charges to satisfy $$\displaystyle V=\text{const} $$ on conductor.

• Magnetostatics: Image currents for perfect conductors (e.g., current near conducting plane → image current opposite direction).

Energy in Magnetic Field

• $$\displaystyle \boxed{W = \frac{1}{2} \int_V \mathbf{B} \cdot \mathbf{H}\,d\tau} $$.

• For inductors: $$\displaystyle W = \frac{1}{2} L I^2 $$.

Scalar & Vector Potentials

• Scalar potential $V$: $$\displaystyle \mathbf{E} = -\nabla V - \frac{\partial \mathbf{A}}{\partial t} $$.

• Vector potential $\mathbf{A}$: $$\displaystyle \mathbf{B} = \nabla \times \mathbf{A} $$.

• Gauge: Coulomb gauge ($$\displaystyle \nabla \cdot \mathbf{A} = 0 $$) simplifies wave equations.

• Wave equations: $$\displaystyle \nabla^2 V - \mu\varepsilon \frac{\partial^2 V}{\partial t^2} = -\frac{\rho_v}{\varepsilon} $$, $$\displaystyle \nabla^2 \mathbf{A} - \mu\varepsilon \frac{\partial^2 \mathbf{A}}{\partial t^2} = -\mu\mathbf{J} $$.

[!TIP] Common Pitfalls:

1. Confusing $\mathbf{E}$ and $\mathbf{H}$ directions in plane waves (use $$\displaystyle \mathbf{E} \times \mathbf{H} = \hat{k} S $$).

2. Forgetting displacement current in Ampere’s law for capacitors.

3. Misapplying boundary conditions: normal $D$ discontinuous with $$\displaystyle \rho_s $$, tangential $E$ continuous.

4. Using point charge formula for non-symmetric distributions.

5. In wave propagation, confusing intrinsic impedance $\eta$ with resistance.