Electromagnetic Theory (EX-504 (B)) - Unit 2 Short Notes

UNIT 2: Electromagnetic Theory - Short Notes

I. Vector Analysis & Coordinate Systems

Coordinate Systems

Exam Tip: Remember that $$\displaystyle \hat{a}_\phi $$ in cylindrical and $$\displaystyle \hat{a}_\theta $$, $$\displaystyle \hat{a}_\phi $$ in spherical are not constant; their derivatives matter in curl/divergence.

Vector Calculus Operations

• Gradient ($\nabla \phi$):

  $$\displaystyle \nabla \phi = \frac{\partial \phi}{\partial x}\hat{a}_x + \frac{\partial \phi}{\partial y}\hat{a}_y + \frac{\partial \phi}{\partial z}\hat{a}_z $$ in Cartesian.

  Physical meaning: Points in direction of maximum increase of $\phi$, magnitude = rate of increase.

• Divergence ($\nabla \cdot \mathbf{A}$):

  $$\displaystyle \nabla \cdot \mathbf{A} = \lim_{\Delta V \to 0} \frac{\oiint_S \mathbf{A} \cdot d\mathbf{S}}{\Delta V} $$.

  Physical meaning: Net outward flux per unit volume; source ($$\displaystyle >0 $$) or sink ($$\displaystyle <0 $$).

• Curl ($\nabla \times \mathbf{A}$):

  $$\displaystyle \nabla \times \mathbf{A} = \lim_{\Delta S \to 0} \frac{\oint_C \mathbf{A} \cdot d\mathbf{l}}{\Delta S} \hat{n} $$.

  Physical meaning: Circulation per unit area; measures rotation of field.

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

  $$\displaystyle \nabla^2 \phi = \nabla \cdot (\nabla \phi) $$. In Cartesian: $$\displaystyle \nabla^2 \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2} $$.

Fundamental Theorems

• Divergence (Gauss) Theorem:

$$\boxed{\iiint_V (\nabla \cdot \mathbf{A}) \, dV = \oiint_S \mathbf{A} \cdot d\mathbf{S}}$$

Proof sketch: Divide volume into small cubes, sum fluxes, apply divergence definition.

Application: Converting volume integral of divergence to surface integral (e.g., total flux out of closed surface).

• Stokes' Theorem:

$$\boxed{\iint_S (\nabla \times \mathbf{A}) \cdot d\mathbf{S} = \oint_C \mathbf{A} \cdot d\mathbf{l}}$$

Proof sketch: Divide surface into small loops, sum circulations.

Application: Relating line integral around closed path to surface integral of curl (e.g., finding $$\displaystyle \int \mathbf{A} \cdot d\mathbf{l} $$ from known curl).

• Green's Theorem (in plane): Special case of Stokes' for 2D: $$\displaystyle \iint (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) dxdy = \oint (P dx + Q dy) $$.

Properties of Scalar and Vector Potentials

• Scalar potential $V$: For electrostatic fields, $$\displaystyle \mathbf{E} = -\nabla V $$. $V$ is single-valued if $$\displaystyle \nabla \times \mathbf{E} = 0 $$.

• Vector potential $\mathbf{A}$: For magnetic fields, $$\displaystyle \mathbf{B} = \nabla \times \mathbf{A} $$. Not unique; gauge transformation: $$\displaystyle \mathbf{A}' = \mathbf{A} + \nabla \psi $$ gives same $\mathbf{B}$. Common gauge: Coulomb gauge $$\displaystyle \nabla \cdot \mathbf{A} = 0 $$.

II. Electrostatics

Fundamental Laws

• Coulomb's Law: Force between two point charges:

$$\mathbf{F}_{12} = \frac{q_1 q_2}{4\pi\varepsilon_0 r^2} \hat{a}_{12}$$

Electric field intensity: $$\displaystyle \mathbf{E} = \frac{\mathbf{F}}{q} = \frac{Q}{4\pi\varepsilon_0 r^2} \hat{a}_r $$.

• Electric Flux Density $\mathbf{D}$: $$\displaystyle \mathbf{D} = \varepsilon \mathbf{E} $$, where $$\displaystyle \varepsilon = \varepsilon_0 \varepsilon_r $$. Units: C/m².

Electric Potential

• Potential difference: $$\displaystyle V_{AB} = -\int_A^B \mathbf{E} \cdot d\mathbf{l} $$.

  Absolute potential: $$\displaystyle V = -\int_{\infty}^P \mathbf{E} \cdot d\mathbf{l} $$.

• Due to point charge: $$\displaystyle V = \frac{Q}{4\pi\varepsilon r} $$.

• Due to continuous distribution: $$\displaystyle V = \int \frac{dq}{4\pi\varepsilon r} $$.

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

  Potential: $$\displaystyle V = \frac{\mathbf{p} \cdot \hat{a}_r}{4\pi\varepsilon r^2} $$ (for $r \gg d$).

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

• Properties of potential function:

  • $V$ is continuous.

  • $$\displaystyle \nabla^2 V = -\rho/\varepsilon $$ (Poisson's equation).

  • Work done moving charge $q$ from A to B: $$\displaystyle W = q(V_A - V_B) $$.

Common Pitfall: Dipole field falls as $$\displaystyle 1/r^3 $$, not $$\displaystyle 1/r^2 $$.

Gauss's Law

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

• Differential form: $$\displaystyle \nabla \cdot \mathbf{D} = \rho $$.

• Applications (choose Gaussian surface exploiting symmetry):

  1. Infinite line charge ($\lambda$): $$\displaystyle E = \frac{\lambda}{2\pi\varepsilon \rho} $$ (cylindrical Gaussian surface).

  2. Infinite sheet charge ($\sigma$): $$\displaystyle E = \frac{\sigma}{2\varepsilon} $$ (pillbox surface, field perpendicular to sheet).

  3. Uniformly charged sphere (radius $R$, total $Q$):

     • $$\displaystyle r > R $$: $$\displaystyle E = \frac{Q}{4\pi\varepsilon r^2} $$ (same as point charge).

     • $$\displaystyle r < R $$: $$\displaystyle E = \frac{Q r}{4\pi\varepsilon R^3} $$ (proportional to $r$).

Laplace's and Poisson's Equations

• Derivation: From $$\displaystyle \nabla \cdot \mathbf{D} = \rho $$ and $$\displaystyle \mathbf{D} = \varepsilon \mathbf{E} = -\varepsilon \nabla V $$, get $$\displaystyle \nabla^2 V = -\rho/\varepsilon $$ (Poisson). In source-free ($$\displaystyle \rho=0 $$): $$\displaystyle \nabla^2 V = 0 $$ (Laplace).

• Solution methods:

  • Separation of variables (in Cartesian, cylindrical, spherical).

  • Uniqueness theorem: If $V$ is specified on boundary, solution is unique.

• Example in Cartesian: For 1D: $$\displaystyle V(x) = A x + B $$ solves $$\displaystyle \frac{d^2V}{dx^2}=0 $$.

  In cylindrical: $$\displaystyle V(\rho) = A \ln \rho + B $$ solves $$\displaystyle \frac{1}{\rho}\frac{d}{d\rho}(\rho \frac{dV}{d\rho})=0 $$.

Boundary Conditions

At dielectric interface (no free surface charge $$\displaystyle \sigma_s=0 $$):

• $$\displaystyle D_{\text{normal}} $$ continuous: $$\displaystyle D_{1n} = D_{2n} $$.

• $$\displaystyle E_{\text{tangential}} $$ continuous: $$\displaystyle E_{1t} = E_{2t} $$.

If $$\displaystyle \sigma_s \neq 0 $$: $$\displaystyle D_{2n} - D_{1n} = \sigma_s $$.

Method of Images: Replace conductor with image charges to satisfy boundary conditions.

Example: Point charge $Q$ at distance $d$ from grounded infinite plane → image $-Q$ at $-d$. Potential: $$\displaystyle V = \frac{1}{4\pi\varepsilon} \left( \frac{Q}{r_1} - \frac{Q}{r_2} \right) $$, where $$\displaystyle r_1, r_2 $$ distances to real and image charges.

Capacitance

• Definition: $$\displaystyle C = Q/V $$ (for two conductors, $Q$ on one, $-Q$ on other).

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

• Cylindrical (coaxial, radii $a,b$, length $l$):

$$C = \frac{2\pi\varepsilon l}{\ln(b/a)}$$

• Spherical (radii $a,b$):

$$C = 4\pi\varepsilon \frac{ab}{b-a}$$

• Multi-dielectric: Treat as capacitors in series/parallel. For layered parallel plate: $$\displaystyle \frac{1}{C} = \sum \frac{d_i}{\varepsilon_i A} $$.

Electrostatic Energy

• System of point charges: $$\displaystyle W = \frac{1}{2} \sum_{i=1}^n q_i V_i $$.

• Continuous distribution: $$\displaystyle W = \frac{1}{2} \int \rho V \, dV = \frac{1}{2} \int \varepsilon E^2 \, dV $$.

• Energy density: $$\displaystyle u_e = \frac{1}{2} \varepsilon E^2 $$ (J/m³).

III. Magnetostatics

Fundamental Laws

• Biot–Savart Law:

$$d\mathbf{H} = \frac{I \, d\mathbf{l} \times \hat{a}_r}{4\pi r^2}$$

$$\displaystyle \mathbf{B} = \mu \mathbf{H} $$.

• Applications:

  • Infinite straight wire (current $I$): $$\displaystyle H = \frac{I}{2\pi\rho} $$ (azimuthal).

  • Circular loop (radius $a$, on axis at $z$):

    $$\displaystyle H = \frac{I a^2}{2(a^2+z^2)^{3/2}} $$ (axial).

  • Solenoid (length $l$, $N$ turns, current $I$):

    Inside: $$\displaystyle H = nI $$ ($$\displaystyle n=N/l $$), uniform, axial; outside: $H \approx 0$.

• Ampère’s Circuital Law:

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

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

• Applications:

  • Infinite wire: $$\displaystyle \oint H \cdot dl = H(2\pi\rho) = I $$ → $$\displaystyle H = I/(2\pi\rho) $$.

  • Solenoid: $$\displaystyle \oint H \cdot dl = H l = NI $$ → $$\displaystyle H = nI $$.

  • Toroid (mean radius $$\displaystyle r_m $$, $N$ turns): $$\displaystyle H = \frac{NI}{2\pi r} $$ (azimuthal).

  • Coaxial cable (inner radius $a$, outer $b$, currents $I$ on inner, $-I$ on outer):

    $$\displaystyle a < \rho < b $$: $$\displaystyle H = \frac{I}{2\pi\rho} $$; elsewhere $$\displaystyle H=0 $$.

Magnetic Flux Density $\mathbf{B}$ and Magnetic Field Intensity $\mathbf{H}$

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

• Magnetic flux: $$\displaystyle \Phi = \iint_S \mathbf{B} \cdot d\mathbf{S} $$ (Webers, Wb).

• Flux density $\mathbf{B}$: Wb/m² (Tesla, T).

Vector Magnetic Potential $\mathbf{A}$

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

• Properties:

  • Not unique: $$\displaystyle \mathbf{A}' = \mathbf{A} + \nabla \psi $$ gives same $\mathbf{B}$.

  • For steady currents, Coulomb gauge: $$\displaystyle \nabla \cdot \mathbf{A} = 0 $$, then $$\displaystyle \nabla^2 \mathbf{A} = -\mu \mathbf{J} $$.

• Calculation for simple distributions:

  • Infinite straight wire (along $z$, current $I$):

    $$\displaystyle \mathbf{A} = -\frac{\mu I}{2\pi} \ln \rho \, \hat{a}_z $$ (or $$\displaystyle \frac{\mu I}{2\pi} \ln(\rho/\rho_0) \hat{a}_z $$).

  • Solenoid (long, uniform $nI$ inside):

    Inside: $$\displaystyle \mathbf{A} = \frac{1}{2} \mu n I \, \rho \, \hat{a}_\phi $$; outside: $\mathbf{A} \approx 0$.

Inductance

• Self-inductance $L$: $$\displaystyle \lambda = L I $$, where $$\displaystyle \lambda = N \Phi $$ (flux linkage).

  • Solenoid (length $l$, $N$ turns, area $A$):

$$L = \frac{\mu N^2 A}{l}$$

• Toroid (mean radius $$\displaystyle r_m $$, cross-section $A$):

$$L = \frac{\mu N^2 A}{2\pi r_m} \quad \text{(approx)}$$

Exact: $$\displaystyle L = \frac{\mu N^2 h}{2\pi} \ln(b/a) $$ for rectangular cross-section.

• Mutual inductance $M$: $$\displaystyle \lambda_{21} = M I_1 $$.

  • Neumann’s formula:

$$M = \frac{\mu}{4\pi} \oint_{C_1} \oint_{C_2} \frac{d\mathbf{l}_1 \cdot d\mathbf{l}_2}{r}$$

• Coaxial solenoids (inner $$\displaystyle N_1 $$, outer $$\displaystyle N_2 $$, same $A$, $l$):

  $$\displaystyle M = \frac{\mu N_1 N_2 A}{l} $$ (if tightly coupled, all flux of inner links outer).

• Reciprocity: $$\displaystyle M_{12} = M_{21} $$.

Magnetic Boundary Conditions

At interface (no surface current $$\displaystyle \mathbf{K}_f=0 $$):

• $$\displaystyle B_{\text{normal}} $$ continuous: $$\displaystyle B_{1n} = B_{2n} $$.

• $$\displaystyle H_{\text{tangential}} $$ continuous: $$\displaystyle H_{1t} = H_{2t} $$.

If surface current $$\displaystyle \mathbf{K}_f $$ exists:

$$\hat{n} \times (\mathbf{H}_2 - \mathbf{H}_1) = \mathbf{K}_f$$

For perfect conductor (PEC): $$\displaystyle \mathbf{H}_t = \mathbf{K}_f $$ at surface, $$\displaystyle \mathbf{B}_n = 0 $$ inside.

Magnetic Energy

• Stored energy: $$\displaystyle W_m = \frac{1}{2} L I^2 = \frac{1}{2} \int \mathbf{B} \cdot \mathbf{H} \, dV $$.

• Energy density: $$\displaystyle u_m = \frac{1}{2} \mathbf{B} \cdot \mathbf{H} = \frac{B^2}{2\mu} = \frac{1}{2} \mu H^2 $$.

IV. Maxwell's Equations for Time-Varying Fields

Faraday's Law of Induction

• Integral form:

$$\oint_C \mathbf{E} \cdot d\mathbf{l} = -\frac{d}{dt} \iint_S \mathbf{B} \cdot d\mathbf{S}$$

• Transformer EMF: Due to time-varying $\mathbf{B}$ (stationary loop).

  Motional EMF: Due to motion in static $\mathbf{B}$: $$\displaystyle \mathcal{E} = \oint (\mathbf{E} + \mathbf{v} \times \mathbf{B}) \cdot d\mathbf{l} $$.

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

• Applications: AC generator (rotating coil in $\mathbf{B}$), moving conductor in $\mathbf{B}$ (e.g., rail gun).

Displacement Current

• Concept: In capacitor, conduction current $$\displaystyle I_c $$ stops at plates, but $$\displaystyle \frac{\partial \mathbf{D}}{\partial t} $$ provides continuity.

• Displacement current density: $$\displaystyle \mathbf{J}_d = \frac{\partial \mathbf{D}}{\partial t} $$.

• Ampère–Maxwell Law:

$$\nabla \times \mathbf{H} = \mathbf{J}_c + \frac{\partial \mathbf{D}}{\partial t}$$

Continuity Equation

• Derivation: Take divergence of Ampère–Maxwell: $$\displaystyle \nabla \cdot (\nabla \times \mathbf{H}) = 0 = \nabla \cdot \mathbf{J}_c + \frac{\partial}{\partial t} (\nabla \cdot \mathbf{D}) = \nabla \cdot \mathbf{J}_c + \frac{\partial \rho}{\partial t} $$.

• Equation:

$$\boxed{\nabla \cdot \mathbf{J}_c = -\frac{\partial \rho}{\partial t}}$$

• Physical significance: Charge conservation; current divergence equals rate of decrease of charge density.

Complete Set of Maxwell's Equations

• Static limits:

  • Electrostatics: $$\displaystyle \frac{\partial}{\partial t}=0 $$, $$\displaystyle \nabla \times \mathbf{E}=0 $$, $$\displaystyle \nabla \times \mathbf{H} = \mathbf{J}_c $$.

  • Magnetostatics: $$\displaystyle \frac{\partial}{\partial t}=0 $$, $$\displaystyle \nabla \cdot \mathbf{D}=\rho $$, $$\displaystyle \nabla \times \mathbf{E}=0 $$.

• Current densities:

  • Conduction: $$\displaystyle \mathbf{J}_c = \sigma \mathbf{E} $$ (in conductors).

  • Convection: $$\displaystyle \mathbf{J} = \rho \mathbf{v} $$ (moving charges in vacuum).

Time-Varying Potential Fields

• Potentials:

  $$\displaystyle \mathbf{B} = \nabla \times \mathbf{A} $$,

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

• Lorentz gauge condition:

$$\nabla \cdot \mathbf{A} + \mu \varepsilon \frac{\partial V}{\partial t} = 0$$

Simplifies wave equations for $V$ and $\mathbf{A}$: $$\displaystyle \nabla^2 V - \mu\varepsilon \frac{\partial^2 V}{\partial t^2} = -\rho/\varepsilon $$, $$\displaystyle \nabla^2 \mathbf{A} - \mu\varepsilon \frac{\partial^2 \mathbf{A}}{\partial t^2} = -\mu \mathbf{J} $$.

V. Electromagnetic Wave Propagation

Wave Equation

• Derivation (source-free, linear isotropic medium):

  From $$\displaystyle \nabla \times \mathbf{E} = -\mu \frac{\partial \mathbf{H}}{\partial t} $$ and $$\displaystyle \nabla \times \mathbf{H} = \varepsilon \frac{\partial \mathbf{E}}{\partial t} $$ (lossless), take curl of first:

  $$\displaystyle \nabla \times (\nabla \times \mathbf{E}) = -\mu \frac{\partial}{\partial t} (\nabla \times \mathbf{H}) = -\mu\varepsilon \frac{\partial^2 \mathbf{E}}{\partial t^2} $$.

  Using $$\displaystyle \nabla \times (\nabla \times \mathbf{E}) = \nabla(\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E} $$ and $$\displaystyle \nabla \cdot \mathbf{E}=0 $$, get:

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

Similarly for $\mathbf{H}$.

• In lossy dielectric ($\sigma \neq 0$): $$\displaystyle \nabla \times \mathbf{H} = \sigma \mathbf{E} + \varepsilon \frac{\partial \mathbf{E}}{\partial t} $$, so:

$$\nabla^2 \mathbf{E} = \mu\varepsilon \frac{\partial^2 \mathbf{E}}{\partial t^2} + \mu\sigma \frac{\partial \mathbf{E}}{\partial t}$$

• In good conductor ($\sigma \gg \omega\varepsilon$): Displacement current negligible, so:

$$\nabla^2 \mathbf{E} \approx \mu\sigma \frac{\partial \mathbf{E}}{\partial t}$$

(diffusion equation).

Uniform Plane Waves

• Definition: $\mathbf{E}$ and $\mathbf{H}$ perpendicular to propagation direction $\hat{k}$ and to each other; uniform in transverse plane.

• In lossless dielectric ($$\displaystyle \sigma=0 $$):

  Wave equation: $$\displaystyle \frac{\partial^2 \mathbf{E}}{\partial z^2} = \mu\varepsilon \frac{\partial^2 \mathbf{E}}{\partial t^2} $$.

  Solution: $$\displaystyle \mathbf{E} = \mathbf{E}_0 \cos(\omega t - \beta z) $$ with $$\displaystyle \beta = \omega \sqrt{\mu\varepsilon} $$.

  • Phase velocity: $$\displaystyle v_p = \frac{\omega}{\beta} = \frac{1}{\sqrt{\mu\varepsilon}} $$.

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

  • Wavelength: $$\displaystyle \lambda = 2\pi/\beta $$.

• In lossy dielectric ($\sigma \neq 0$):

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

$$\alpha = \omega \sqrt{\frac{\mu\varepsilon}{2}} \left[ \sqrt{1 + \left(\frac{\sigma}{\omega\varepsilon}\right)^2} - 1 \right]^{1/2}$$

$$\beta = \omega \sqrt{\frac{\mu\varepsilon}{2}} \left[ \sqrt{1 + \left(\frac{\sigma}{\omega\varepsilon}\right)^2} + 1 \right]^{1/2}$$

Intrinsic impedance: $$\displaystyle \eta = \sqrt{\frac{\mu}{\varepsilon}} \sqrt{1 - j\frac{\sigma}{\omega\varepsilon}} $$ (complex).

$\mathbf{E}$ and $\mathbf{H}$ are out of phase.

• In good conductor ($\sigma \gg \omega\varepsilon$):

  $$\displaystyle \alpha \approx \beta \approx \sqrt{\frac{\omega\mu\sigma}{2}} $$,

  $$\displaystyle \eta = (1+j) \sqrt{\frac{\omega\mu}{2\sigma}} = (1+j) \frac{1}{\sigma\delta} $$,

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

Example Problem (from Nov 2023):

Given $$\displaystyle \mathbf{E} = 2e^{-\alpha z} \sin(10^8 t - \beta z) \hat{a}_y $$ V/m, $$\displaystyle \varepsilon_r=1 $$, $$\displaystyle \mu_r=20 $$, $$\displaystyle \sigma=3 $$ S/m.

$$\displaystyle \omega = 10^8 $$ rad/s, $$\displaystyle \varepsilon = \varepsilon_0 $$, $$\displaystyle \mu = 20\mu_0 $$.

Compute $$\displaystyle \frac{\sigma}{\omega\varepsilon} = \frac{3}{10^8 \times 8.854\times10^{-12}} \approx 3390 \gg 1 $$, so good conductor? But $$\displaystyle \varepsilon_r=1 $$, so actually lossy dielectric? Since $\sigma/(\omega\varepsilon) \gg 1$, we can use good conductor approximation:

$$\displaystyle \alpha = \beta = \sqrt{\frac{\omega\mu\sigma}{2}} = \sqrt{\frac{10^8 \times 20 \times 4\pi\times10^{-7} \times 3}{2}} \approx \sqrt{10^8 \times 3.77 \times 10^{-3}} \approx 1.94 \times 10^3 $$ Np/m.

$$\displaystyle \eta = (1+j) \sqrt{\frac{\omega\mu}{2\sigma}} = (1+j) \sqrt{\frac{10^8 \times 20 \times 4\pi\times10^{-7}}{2 \times 3}} \approx (1+j) \times 0.289 $$ Ω.

$$\displaystyle \mathbf{H} = \frac{1}{\eta} (\hat{z} \times \mathbf{E}) = \frac{1}{\eta} ( \hat{z} \times 2e^{-\alpha z} \sin(...) \hat{a}_y ) = -\frac{2}{\eta} e^{-\alpha z} \sin(...) \hat{a}_x $$ A/m.

Polarization of EM Waves

• Linear: $\mathbf{E}$ maintains fixed direction (e.g., $$\displaystyle \mathbf{E} = E_x \cos(\omega t - \beta z) \hat{a}_x $$).

• Circular: $\mathbf{E}$ rotates with constant magnitude.

  Right-hand circular (RHC): $$\displaystyle \mathbf{E} = E_0 [\cos(\omega t - \beta z) \hat{a}_x + \sin(\omega t - \beta z) \hat{a}_y] $$.

  Left-hand circular (LHC): $$\displaystyle \mathbf{E} = E_0 [\cos(\omega t - \beta z) \hat{a}_x - \sin(\omega t - \beta z) \hat{a}_y] $$.

• Elliptical: General case, $\mathbf{E}$ traces ellipse.

Reflection and Transmission

Normal Incidence

• At dielectric–dielectric interface (medium 1 to 2):

  Reflection coefficient (E field):

$$\Gamma = \frac{\eta_2 - \eta_1}{\eta_2 + \eta_1}$$

Transmission coefficient:

$$T = \frac{2\eta_2}{\eta_2 + \eta_1}$$

(for $\mathbf{E}$ incident from medium 1).

• At perfect conductor ($$\displaystyle \eta_2 = 0 $$): $$\displaystyle \Gamma = -1 $$, $$\displaystyle T = 0 $$; $$\displaystyle \mathbf{E}=0 $$ inside, $\mathbf{H}$ maximum at surface.

Oblique Incidence

• Snell’s law: $$\displaystyle n_1 \sin\theta_i = n_2 \sin\theta_t $$, where $$\displaystyle n = \sqrt{\varepsilon_r \mu_r} \approx \sqrt{\varepsilon_r} $$ for non-magnetic.

• Critical angle (for $$\displaystyle n_1 > n_2 $$): $$\displaystyle \theta_c = \sin^{-1}(n_2/n_1) $$; total internal reflection for $$\displaystyle \theta_i > \theta_c $$.

• Brewster angle (parallel polarization): $$\displaystyle \theta_B = \tan^{-1}(n_2/n_1) $$; $$\displaystyle \Gamma_\parallel = 0 $$.

Poynting Vector and Theorem

• Poynting vector (instantaneous power density):

$$\mathbf{S} = \mathbf{E} \times \mathbf{H} \quad (\text{W/m}^2)$$

• Poynting theorem (energy conservation):

$$\oiint_S \mathbf{S} \cdot d\mathbf{S} = -\frac{\partial}{\partial t} \iiint_V (u_e + u_m) dV - \iiint_V \mathbf{J} \cdot \mathbf{E} \, dV$$

where $$\displaystyle u_e = \frac{1}{2} \varepsilon E^2 $$, $$\displaystyle u_m = \frac{1}{2} \mu H^2 $$.

Left: net outward power; right: decrease in stored energy + ohmic loss.

Surface Impedance

• Definition (at conductor surface):

$$Z_s = \frac{E_t}{H_t}$$

(ratio of tangential $\mathbf{E}$ to tangential $\mathbf{H}$).

• For good conductor:

$$Z_s = \frac{1+j}{\sigma\delta} = (1+j) \sqrt{\frac{\omega\mu}{2\sigma}}$$

$$\displaystyle |Z_s| = \sqrt{\frac{\omega\mu}{2\sigma}} $$, phase = $$\displaystyle 45^\circ $$.

VI. Advanced Applications & Problem-Solving Topics

Boundary Value Problems

• Solve $$\displaystyle \nabla^2 V = 0 $$ or $$\displaystyle \nabla^2 V = -\rho/\varepsilon $$ with Dirichlet ($V$ given) or Neumann ($\partial V/\partial n$ given) BCs.

• Uniqueness theorem: Solution is unique if $V$ specified on boundary (Dirichlet) or $\partial V/\partial n$ specified (Neumann with charge density known).

Specific Calculations

1. Magnetic flux crossing surface: $$\displaystyle \Phi = \iint_S \mathbf{B} \cdot d\mathbf{S} $$. Choose surface and compute dot product.

2. Inductance of complex geometries: Use $$\displaystyle L = N\Phi/I $$ or $$\displaystyle W_m = \frac{1}{2} L I^2 = \frac{1}{2} \int \mathbf{B} \cdot \mathbf{H} dV $$.

3. Given $\mathbf{E}$ (or $\mathbf{H}$) in wave:

   • Identify $$\displaystyle \gamma = \alpha + j\beta $$ from $$\displaystyle e^{-\alpha z} \sin(\omega t - \beta z) $$ or $$\displaystyle e^{-\alpha z} \cos(\omega t - \beta z) $$.

   • Find $$\displaystyle \mathbf{H} = \frac{1}{\eta} (\hat{k} \times \mathbf{E}) $$ for lossless; for lossy, $\eta$ complex, $\mathbf{H}$ phase-shifted.

   • Skin depth $$\displaystyle \delta = 1/\alpha $$.

4. Given $\eta$ and $\mathbf{H}$: $$\displaystyle \mathbf{E} = \eta (\hat{k} \times \mathbf{H}) $$, then $\alpha$ from $$\displaystyle |\mathbf{H}| \propto e^{-\alpha z} $$.

Short Note Topics (Frequently Asked)

Method of Images

• Replace conductors with image charges to satisfy boundary conditions (e.g., $$\displaystyle V=0 $$ on grounded plane).

  Applications: Point charge near plane, sphere; dipole near plane.

• In wave propagation: Image theory for reflection from perfect conductor.

Convection vs Conduction Current Density

• Convection current: $$\displaystyle \mathbf{J} = \rho \mathbf{v} $$ (charges moving in vacuum, e.g., electron beam).

• Conduction current: $$\displaystyle \mathbf{J}_c = \sigma \mathbf{E} $$ (drift in conductor).

  Key difference: Convection current involves net charge motion; conduction current in neutral conductor has $\rho \approx 0$.

Magnetic Dipole and Dipole Moment

• Magnetic dipole moment: $$\displaystyle \mathbf{m} = I \mathbf{A} $$ (current loop area vector).

• Field on axis (distance $r \gg$ loop size):

$$B = \frac{\mu_0 m}{2\pi r^3}$$

(along axis).

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

Self-Inductance of Solenoid and Toroid (Derivation)

• Solenoid (length $l$, $N$ turns, area $A$, $\mu$):

  Flux through one turn: $$\displaystyle \Phi = B A = \mu \frac{NI}{l} A $$.

  Total flux linkage: $$\displaystyle \lambda = N \Phi = \mu \frac{N^2 A}{l} I $$.

  Hence $$\displaystyle L = \frac{\lambda}{I} = \frac{\mu N^2 A}{l} $$.

• Toroid (mean radius $$\displaystyle r_m $$, $N$ turns, cross-section $A$):

  $$\displaystyle H = \frac{NI}{2\pi r} $$, $$\displaystyle B = \mu H $$, $$\displaystyle \Phi = B A = \mu \frac{NI}{2\pi r} A $$.

  $$\displaystyle \lambda = N \Phi = \mu \frac{N^2 A}{2\pi r} I $$, so $$\displaystyle L = \frac{\mu N^2 A}{2\pi r_m} $$ (using $$\displaystyle r_m $$).

Transmission Line Analogy

• Voltage $V$ $$\displaystyle \leftrightarrow $$ Electric field $\mathbf{E}$.

• Current $I$ $$\displaystyle \leftrightarrow $$ Magnetic field $\mathbf{H}$.

• Characteristic impedance $$\displaystyle Z_0 = \sqrt{L/C} $$ $$\displaystyle \leftrightarrow $$ Intrinsic impedance $$\displaystyle \eta = \sqrt{\mu/\varepsilon} $$.

• Wave equation on line: $$\displaystyle \frac{\partial^2 V}{\partial z^2} = LC \frac{\partial^2 V}{\partial t^2} $$ $$\displaystyle \leftrightarrow $$ $$\displaystyle \nabla^2 \mathbf{E} = \mu\varepsilon \frac{\partial^2 \mathbf{E}}{\partial t^2} $$.

Perfect Conductor Properties and Boundary Conditions

• Static: $$\displaystyle \mathbf{E}=0 $$ inside, $$\displaystyle \mathbf{B}=0 $$ inside (Meissner effect for superconductors; for perfect conductor, $\mathbf{B}$ constant but can be zero).

• Time-varying:

  • Tangential $\mathbf{E}$: $$\displaystyle E_t = 0 $$ (since $$\displaystyle \mathbf{E} = -\nabla V - \partial\mathbf{A}/\partial t $$, perfect conductor forces $V$ constant).

  • Normal $\mathbf{B}$: $$\displaystyle B_n = 0 $$ (since $$\displaystyle \nabla \cdot \mathbf{B}=0 $$ and $\mathbf{B}$ cannot penetrate).

  • Tangential $\mathbf{H}$: $$\displaystyle H_t = K_f $$ (surface current).

  • Normal $\mathbf{D}$: $$\displaystyle D_n = \rho_s $$ (surface charge).

Surface Impedance in Good Conductors

• As above: $$\displaystyle Z_s = (1+j)/(\sigma\delta) = (1+j)\sqrt{\frac{\omega\mu}{2\sigma}} $$.

• Physical meaning: Ratio of tangential $\mathbf{E}$ to tangential $\mathbf{H}$ at conductor surface; accounts for skin effect.

Polarization of EM Waves (Types, Applications)

• Linear: Used in linear polarizers, antennas.

• Circular/elliptical: Used in satellite communication (reduces fading), radar, quantum mechanics (photon spin).

• Applications: Polarization filters, 3D cinema (linear/circular), optical isolation.

Exam Tips & Common Pitfalls:

1. Coordinate systems: In cylindrical/spherical, remember $$\displaystyle \hat{a}_\phi $$ and $$\displaystyle \hat{a}_\theta $$ depend on $\phi$/$\theta$; their curls/divergences are non-zero.

2. Gauss's law: Only useful for high symmetry (spherical, cylindrical, planar). Never use for arbitrary shapes.

3. Potential of dipole: $$\displaystyle V \propto 1/r^2 $$, $$\displaystyle \mathbf{E} \propto 1/r^3 $$.

4. Biot–Savart vs Ampère: Use Biot–Savart for finite/irregular currents; Ampère for high symmetry (infinite wire, solenoid, toroid).

5. Maxwell’s equations: In integral form, $d\mathbf{S}$ direction by right-hand rule relative to $d\mathbf{l}$.

6. Wave propagation: In good conductors, $\mathbf{E}$ and $\mathbf{H}$ are out of phase by $$\displaystyle 45^\circ $$; in lossless dielectrics, in phase.

7. Skin depth: $$\displaystyle \delta = 1/\alpha $$; for good conductors, $\delta \propto 1/\sqrt{f}$.

8. Boundary conditions: Tangential $\mathbf{E}$ always continuous (unless time-varying $\mathbf{B}$? Actually from Faraday, $$\displaystyle \nabla \times \mathbf{E} = -\partial\mathbf{B}/\partial t $$, integral form gives continuity of tangential $\mathbf{E}$ if $\partial\mathbf{B}/\partial t$ finite? Wait, boundary condition from $$\displaystyle \oint \mathbf{E}\cdot d\mathbf{l} = -d\Phi_B/dt $$, for infinitesimal loop straddling interface, if $\mathbf{B}$ finite, RHS→0, so tangential $\mathbf{E}$ continuous. Yes, always continuous unless singular $\mathbf{B}$.

9. Polarization: For circular polarization, two orthogonal components $$\displaystyle 90^\circ $$ out of phase; for linear, in phase or $$\displaystyle 180^\circ $$ out.

10. Poynting vector: Direction of power flow; for traveling wave, $\mathbf{S}$ in direction of propagation.

11. Surface impedance: Only defined at surface of conductor; for good conductors, $$\displaystyle Z_s $$ is complex with equal real and imaginary parts.

12. Method of images: Only works for planar/ spherical geometries with conductors at known potentials.