Electromagnetic Theory (EX-504 (B)) - Unit 3 Short Notes

UNIT 3: ELECTROMAGNETIC THEORY - EXAM-FOCUSED SHORT NOTES

I. VECTOR CALCULUS & COORDINATE SYSTEMS

A. Vector Algebra & Calculus

• Divergence of a Vector Field $\mathbf{F}$:

  • Definition: Net outward flux per unit volume as volume shrinks to a point.

  • Physical Significance: Measures "source" or "sink" strength at a point. $$\displaystyle \nabla \cdot \mathbf{F} > 0 $$ indicates a source.

  • Cartesian: $$\displaystyle \nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} $$

  • Cylindrical: $$\displaystyle \nabla \cdot \mathbf{F} = \frac{1}{\rho}\frac{\partial (\rho F_\rho)}{\partial \rho} + \frac{1}{\rho}\frac{\partial F_\phi}{\partial \phi} + \frac{\partial F_z}{\partial z} $$

  [!TIP] Exam Focus: Calculate divergence for given $\mathbf{F}$ in Cartesian or Cylindrical coordinates. Common pitfall: forgetting scale factors in non-Cartesian systems.

• Curl of a Vector Field $\mathbf{F}$:

  • Definition: Circulation (line integral per unit area) as area shrinks to a point, indicating rotation.

  • Physical Significance: Measures "vorticity" or rotational tendency. $\nabla \times \mathbf{F} \neq 0$ implies field has a curl.

  • Cartesian: $$\displaystyle \nabla \times \mathbf{F} = \begin{vmatrix} \hat{a}_x & \hat{a}_y & \hat{a}_z \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_x & F_y & F_z \end{vmatrix} $$

  • Cylindrical: $$\displaystyle \nabla \times \mathbf{F} = \frac{1}{\rho}\begin{vmatrix} \hat{a}_\rho & \rho\hat{a}_\phi & \hat{a}_z \\ \frac{\partial}{\partial \rho} & \frac{\partial}{\partial \phi} & \frac{\partial}{\partial z} \\ F_\rho & \rho F_\phi & F_z \end{vmatrix} $$

B. Coordinate Systems

[!TIP] Exam Focus: Be able to convert a vector expression from Cartesian to Cylindrical (most common) and compute $d\mathbf{l}$, $d\mathbf{S}$, $dV$ for integration.

II. ELECTROSTATICS

A. Fundamental Laws & Equations

• Gauss's Law:

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

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

  • Application: Used to find $\mathbf{D}$ for symmetric charge distributions (point, infinite line, infinite sheet, sphere).

  • Example (Infinite Line Charge): $$\displaystyle \mathbf{D} = \frac{\rho_l}{2\pi\rho} \hat{a}_\rho $$, where $$\displaystyle \rho_l $$ is line charge density.

• Divergence Theorem (Gauss's Theorem): $$\displaystyle \oint_S \mathbf{F} \cdot d\mathbf{S} = \int_V (\nabla \cdot \mathbf{F})\, dV $$. Relates surface integral to volume integral.

B. Electric Potential & Field

• Relationship: $$\displaystyle \mathbf{E} = -\nabla V $$ (Electric field is negative gradient of potential).

• Electric Dipole (two equal/opposite charges $+Q$, $-Q$ separated by $l$):

  • Dipole Moment: $$\displaystyle \mathbf{p} = Q \mathbf{l} $$ (points from $-Q$ to $+Q$).

  • Potential at distant point ($r \gg l$): $$\displaystyle V \approx \frac{1}{4\pi\varepsilon} \frac{\mathbf{p} \cdot \hat{a}_r}{r^2} $$

  • Electric Field:

$$E_r = \frac{2p \cos\theta}{4\pi\varepsilon r^3}, \quad E_\theta = \frac{p \sin\theta}{4\pi\varepsilon r^3}$$

    > [!TIP] **Exam Focus**: Derivation of $V$ and $\mathbf{E}$ for a dipole is a **recurring 7-mark question**. Start from $$\displaystyle V = \frac{1}{4\pi\varepsilon}(\frac{Q}{r_1} - \frac{Q}{r_2}) $$ and use binomial expansion for $r \gg l$.

C. Material Media & Boundary Conditions

• Polarization $\mathbf{P}$: Dipole moment per unit volume. $$\displaystyle \mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P} $$.

• Linear Isotropic Dielectric: $$\displaystyle \mathbf{P} = \varepsilon_0 \chi_e \mathbf{E} $$, $$\displaystyle \varepsilon = \varepsilon_0 (1 + \chi_e) = \varepsilon_0 \varepsilon_r $$.

• Boundary Conditions for $\mathbf{E}$ (at interface, no free surface charge $$\displaystyle \sigma_f=0 $$):

  1. Tangential $\mathbf{E}$ continuous: $$\displaystyle E_{t1} = E_{t2} $$

  2. Normal $\mathbf{D}$ continuous: $$\displaystyle D_{n1} = D_{n2} $$

  [!TIP] Exam Focus: Apply these to find $\mathbf{E}$ or $\mathbf{D}$ in one medium given the other. For dielectric-conductor, $$\displaystyle E_t=0 $$, $$\displaystyle E_n = \sigma/\varepsilon $$ (conductor surface).

D. Energy & Capacitance

• Energy Density: $$\displaystyle W_E = \frac{1}{2} \mathbf{E} \cdot \mathbf{D} $$.

• Total Energy: $$\displaystyle W = \frac{1}{2} \int_V \mathbf{E} \cdot \mathbf{D}\, dV = \frac{1}{2} \int_S V \mathbf{D} \cdot d\mathbf{S} $$.

• Capacitance: $$\displaystyle C = Q/V $$.

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

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

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

III. MAGNETOSTATICS

A. Fundamental Laws & Equations

• Biot-Savart's Law:

$$\mathbf{H} = \frac{1}{4\pi} \int \frac{I d\mathbf{l} \times \hat{a}_R}{R^2}$$

> [!TIP] **Exam Focus**: Use to derive $\mathbf{H}$ for simple geometries like infinite straight wire ($$\displaystyle \mathbf{H} = \frac{I}{2\pi\rho} \hat{a}_\phi $$).

• Ampere's Circuital Law:

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

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

  • Applications: Infinite line current, infinite sheet current, solenoid ($$\displaystyle \mathbf{H} = nI \hat{a}_z $$ inside), toroid ($$\displaystyle \mathbf{H} = \frac{NI}{2\pi\rho} \hat{a}_\phi $$).

B. Magnetic Fields & Forces

• Relationship: $$\displaystyle \mathbf{B} = \mu_0 (\mathbf{H} + \mathbf{M}) = \mu \mathbf{H} $$ in linear media.

• Lorentz Force on charge $q$: $$\displaystyle \mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B}) $$.

• Force on current element: $$\displaystyle d\mathbf{F} = I d\mathbf{l} \times \mathbf{B} $$.

C. Magnetic Potential & Inductance

• Vector Magnetic Potential $\mathbf{A}$:

  • Defined by $$\displaystyle \mathbf{B} = \nabla \times \mathbf{A} $$.

  • For static fields: $$\displaystyle \nabla^2 \mathbf{A} = -\mu \mathbf{J} $$ (in Coulomb gauge, $$\displaystyle \nabla \cdot \mathbf{A}=0 $$).

  • Property: $\mathbf{A}$ is not unique (gauge freedom), but $\mathbf{B}$ is unique.

• Self-Inductance $L$: Flux linking circuit per unit current. $$\displaystyle L = \frac{\lambda}{I} = \frac{N\Phi}{I} $$.

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

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

• Mutual Inductance $M$: Flux in circuit 2 due to current in circuit 1. $$\displaystyle M_{12} = \frac{\lambda_{21}}{I_1} $$.

  [!TIP] Exam Focus: Short notes on self & mutual inductance are common. Know definitions and formulas for solenoid/toroid.

D. Magnetic Materials & Boundary Conditions

• Magnetization $\mathbf{M}$: Magnetic dipole moment per unit volume.

• Linear Magnetic Material: $$\displaystyle \mathbf{B} = \mu \mathbf{H} $$, $$\displaystyle \mu = \mu_0 (1 + \chi_m) = \mu_0 \mu_r $$.

• Classification:

  | Material | $$\displaystyle \chi_m $$ | $$\displaystyle \mu_r $$ | Example | | :--- | :--- | :--- | :--- | | Diamagnetic | $$\displaystyle <0 $$, small | $$\displaystyle <1 $$, ~1 | Cu, Ag, Au | | Paramagnetic | $$\displaystyle >0 $$, small | $$\displaystyle >1 $$, ~1 | Al, Pt | | Ferromagnetic | $$\displaystyle >0 $$, large | $$\displaystyle >>1 $$, nonlinear | Fe, Co, Ni | | Antiferromagnetic | $$\displaystyle <0 $$, small | $\approx 1$ | MnO | | Ferrimagnetic | $$\displaystyle >0 $$, large | $$\displaystyle >1 $$ | Ferrites |

  [!TIP] Exam Focus: Compare ferromagnetism (parallel alignment, hysteresis, large $$\displaystyle \chi_m $$) vs antiferromagnetism (alternating antiparallel, net $$\displaystyle M=0 $$). Also hard (high coercivity, permanent magnets) vs soft (low coercivity, transformer cores) magnetic materials.

• Boundary Conditions for $\mathbf{B}$ and $\mathbf{H}$ (at interface):

  1. Normal $\mathbf{B}$ continuous: $$\displaystyle B_{n1} = B_{n2} $$

  2. Tangential $\mathbf{H}$ discontinuous by surface current $\mathbf{K}$: $$\displaystyle \hat{a}_n \times (\mathbf{H}_1 - \mathbf{H}_2) = \mathbf{K} $$

  [!TIP] Exam Focus: Derive these from Maxwell's equations (integral form). Problem: "Calculate total magnetic flux crossing a surface" – use $$\displaystyle \Phi = \int \mathbf{B} \cdot d\mathbf{S} $$ with given $\mathbf{B}$.

IV. TIME-VARYING FIELDS & MAXWELL'S EQUATIONS

A. Faraday's Law of Induction

• Integral Form: $$\displaystyle \oint_C \mathbf{E} \cdot d\mathbf{l} = -\frac{d}{dt} \int_S \mathbf{B} \cdot d\mathbf{S} $$

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

• Two EMF Types:

  1. Transformer EMF: $\mathbf{E}$ induced by changing $\mathbf{B}$ (stationary loop).

  2. Motional EMF: $(\mathbf{v} \times \mathbf{B})$ force on charges in moving conductor.

  [!TIP] Exam Focus: Detailed explanation with diagrams is a recurring 7-mark question. Emphasize that Faraday's law unifies both effects.

B. Displacement Current

• Concept: Term $$\displaystyle \frac{\partial \mathbf{D}}{\partial t} $$ added to current density to satisfy continuity equation in time-varying fields, especially across capacitor gaps.

• Ampere-Maxwell Law: $$\displaystyle \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t} $$

• Application: Explains magnetic field between capacitor plates during charging.

C. Maxwell's Equations (Complete Set)

[!TIP] Exam Focus: Comprehensive explanation of all four equations in both forms and their physical significance is a major 7-mark question. Memorize the table.

D. Potential Functions for Time-Varying Fields

• Scalar Potential $V$ and Vector Potential $\mathbf{A}$:

  • $$\displaystyle \mathbf{B} = \nabla \times \mathbf{A} $$ (always true).

  • $$\displaystyle \mathbf{E} = -\nabla V - \frac{\partial \mathbf{A}}{\partial t} $$ (from $$\displaystyle \nabla \times \mathbf{E} = -\partial \mathbf{B}/\partial t $$).

  • Lorenz Gauge: $$\displaystyle \nabla \cdot \mathbf{A} + \mu \varepsilon \frac{\partial V}{\partial t} = 0 $$ simplifies wave equations.

• Wave Equations (in source-free, linear, isotropic medium):

$$\nabla^2 \mathbf{A} - \mu \varepsilon \frac{\partial^2 \mathbf{A}}{\partial t^2} = -\mu \mathbf{J}$$

$$\nabla^2 V - \mu \varepsilon \frac{\partial^2 V}{\partial t^2} = -\frac{\rho_v}{\varepsilon}$$

> [!TIP] **Exam Focus**: Discussion of potentials and derivation of wave equations from Maxwell's equations.

V. ELECTROMAGNETIC WAVE PROPAGATION

A. Wave Equation & Uniform Plane Waves

• Wave Equation Derivation (from Maxwell's in source-free medium):

$$\nabla^2 \mathbf{E} - \mu \varepsilon \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0, \quad \nabla^2 \mathbf{H} - \mu \varepsilon \frac{\partial^2 \mathbf{H}}{\partial t^2} = 0$$

• Uniform Plane Wave: $\mathbf{E}$ and $\mathbf{H}$ are perpendicular to each other and to direction of propagation ($$\displaystyle \hat{a}_k $$). Fields vary only in that direction.

  • Wave Equation Solution: $$\displaystyle \mathbf{E}(z,t) = \mathbf{E}_0 e^{-\gamma z} e^{j\omega t} $$, where $$\displaystyle \gamma = \alpha + j\beta = j\omega \sqrt{\mu \varepsilon} \sqrt{1 - j\frac{\sigma}{\omega \varepsilon}} $$.

• Propagation in Dielectrics:

  • Lossless ($$\displaystyle \sigma=0 $$): $$\displaystyle \gamma = j\beta $$, $$\displaystyle \beta = \omega \sqrt{\mu \varepsilon} $$, $$\displaystyle \eta = \sqrt{\mu/\varepsilon} $$, $$\displaystyle v_p = 1/\sqrt{\mu \varepsilon} $$.

  • Lossy ($\sigma \neq 0$): $$\displaystyle \alpha = \omega \sqrt{\frac{\mu \varepsilon}{2} \left( \sqrt{1 + (\frac{\sigma}{\omega \varepsilon})^2} - 1 \right)} $$, $$\displaystyle \beta = \omega \sqrt{\frac{\mu \varepsilon}{2} \left( \sqrt{1 + (\frac{\sigma}{\omega \varepsilon})^2} + 1 \right)} $$.

• Propagation in Conductors (Good conductor: $\sigma \gg \omega \varepsilon$):

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

  • Skin Depth $\delta$: Depth where amplitude falls to $1/e$ of surface value.

$$\boxed{\delta = \frac{1}{\alpha} = \sqrt{\frac{2}{\omega \mu \sigma}}}$$

*   **Intrinsic Impedance** $$\displaystyle \eta_c \approx (1+j)\sqrt{\frac{\omega \mu}{2\sigma}} $$ (small, highly attenuating).

> [!TIP] **Exam Focus**: Calculate $\alpha$, $\beta$, $\mathbf{H}$ from given $\mathbf{E}$ using $$\displaystyle \eta = \sqrt{\frac{j\omega \mu}{\sigma + j\omega \varepsilon}} $$ and $$\displaystyle \mathbf{H} = (1/\eta) \hat{a}_k \times \mathbf{E} $$. **This is a recurring problem.**

B. Polarization

• Definition: Orientation of the electric field vector $\mathbf{E}$ of a plane wave as a function of time at a fixed point.

• Types:

  1. Linear: $\mathbf{E}$ remains along a fixed line (phase difference $0$ or $\pi$).

  2. Circular: $\mathbf{E}$ rotates with constant magnitude (phase difference $\pm \pi/2$, equal amplitudes).

  3. Elliptical: General case (different amplitudes, any phase difference).

  [!TIP] Exam Focus: Explain polarization types with equations. For circular: $$\displaystyle E_x = E_0 \cos(\omega t - \beta z) $$, $$\displaystyle E_y = E_0 \cos(\omega t - \beta z \pm \pi/2) $$.

C. Reflection & Transmission of Plane Waves

• Normal Incidence at boundary between medium 1 ($$\displaystyle \eta_1 $$) and medium 2 ($$\displaystyle \eta_2 $$):

  • Reflection Coefficient (E-field): $$\displaystyle \Gamma = \frac{\eta_2 - \eta_1}{\eta_2 + \eta_1} $$

  • Transmission Coefficient (E-field): $$\displaystyle \tau = \frac{2\eta_2}{\eta_2 + \eta_1} $$

  • Power Coefficients: $$\displaystyle |\Gamma|^2 + |\tau|^2 \frac{\eta_1}{\eta_2} = 1 $$ (conservation).

• Oblique Incidence:

  • Brewster's Angle $$\displaystyle \theta_B $$: Angle where $$\displaystyle \Gamma_\parallel = 0 $$ (no reflection for parallel polarization). $$\displaystyle \tan \theta_B = \sqrt{\frac{\varepsilon_{r2}}{\varepsilon_{r1}}} $$ (non-magnetic).

  • Total Internal Reflection (TIR): Occurs when wave goes from denser ($$\displaystyle \eta_1 $$ smaller) to rarer ($$\displaystyle \eta_2 $$ larger) medium at $$\displaystyle \theta_i > \theta_c $$, where $$\displaystyle \sin \theta_c = \eta_1/\eta_2 $$.

  [!TIP] Exam Focus: Derive $\Gamma$ and $\tau$ using boundary conditions ($$\displaystyle E_t $$, $$\displaystyle H_t $$ continuous). Concept of Brewster's angle and TIR.

D. Poynting Vector & Power Flow

• Instantaneous Poynting Vector: $$\displaystyle \mathbf{S} = \mathbf{E} \times \mathbf{H} $$ (units: W/m²). Direction = power flow.

• Time-Average Poynting Vector (for sinusoidal fields):

$$\langle \mathbf{S} \rangle = \frac{1}{2} \text{Re} \left( \mathbf{E} \times \mathbf{H}^* \right)$$

• Poynting Theorem (Conservation of Energy):

$$\oint_S \mathbf{S} \cdot d\mathbf{S} = -\frac{\partial}{\partial t} \int_V w_{em}\, dV - \int_V \mathbf{E} \cdot \mathbf{J}\, dV$$

where $$\displaystyle w_{em} = \frac{1}{2} (\mathbf{E} \cdot \mathbf{D} + \mathbf{B} \cdot \mathbf{H}) $$ is EM energy density.

> [!TIP] **Exam Focus**: State and interpret Poynting theorem. Calculate average power flow given $\mathbf{E}$ or $\mathbf{H}$.

VI. ADDITIONAL TOPICS & SHORT NOTE SYLLABUS

A. Method of Images

• Concept: Replace conducting surfaces with imaginary "image" charges to satisfy boundary condition ($$\displaystyle V=\text{constant} $$ or $$\displaystyle E_t=0 $$) in the region of interest.

• Application: Point charge near infinite grounded conducting plane, line charge near conductor.

• Key: Image charge magnitude and location depend on geometry. Valid only for regions not containing conductors.

B. Magnetic Dipole & Magnetic Dipole Moment

• Definition: Equivalent to a small current loop. Magnetic Dipole Moment: $$\displaystyle \mathbf{m} = I \mathbf{A} $$ (area vector $\mathbf{A}$).

• Field: At large distance, $$\displaystyle \mathbf{B} \approx \frac{\mu_0}{4\pi} \frac{3(\mathbf{m}\cdot\hat{a}_r)\hat{a}_r - \mathbf{m}}{r^3} $$ (analogous to electric dipole).

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

C. Surface Impedance

• Definition: Ratio of tangential electric field to tangential magnetic field at the surface of a good conductor: $$\displaystyle Z_s = \frac{E_t}{H_t} $$.

• For Good Conductor: $$\displaystyle Z_s = \frac{E_t}{H_t} = (1+j)\frac{1}{\sigma \delta} = (1+j)\sqrt{\frac{\omega \mu}{2\sigma}} $$.

• Significance: Characterizes loss at high frequencies; relates to skin depth $\delta$.

D. Perfect Conductor

• Properties:

  1. $\sigma \to \infty$

  2. $$\displaystyle \mathbf{E} = 0 $$ inside (static & time-varying).

  3. $$\displaystyle \mathbf{B} = 0 $$ inside (static & time-varying, except possibly static field trapped).

  4. All current flows on surface: $$\displaystyle \mathbf{J}_s = \hat{a}_n \times \mathbf{H} $$.

  5. Tangential $$\displaystyle \mathbf{E}=0 $$ at surface: $$\displaystyle E_t=0 $$.

  6. Normal $$\displaystyle \mathbf{B}=0 $$ at surface: $$\displaystyle B_n=0 $$.

• Boundary Conditions: $$\displaystyle E_t=0 $$, $$\displaystyle H_t = J_s $$, $$\displaystyle D_n = \rho_s $$, $$\displaystyle B_n=0 $$.

E. Transmission Line Analogy

• Telegrapher's Equations (for voltage $V(z)$ and current $I(z)$ on a lossy line):

$$\frac{\partial V}{\partial z} = - (R + j\omega L) I, \quad \frac{\partial I}{\partial z} = - (G + j\omega C) V$$

• Wave Equation: $$\displaystyle \frac{\partial^2 V}{\partial z^2} = \gamma^2 V $$, where $$\displaystyle \gamma = \sqrt{(R+j\omega L)(G+j\omega C)} $$.

• Analogy to EM Wave:

  • $$\displaystyle V \leftrightarrow E $$, $$\displaystyle I \leftrightarrow H $$

  • $$\displaystyle R \leftrightarrow \sigma $$ (loss), $$\displaystyle L \leftrightarrow \mu $$, $$\displaystyle C \leftrightarrow \varepsilon $$, $$\displaystyle G \leftrightarrow \sigma $$ (leakage).

  • Propagation constant $\gamma$ and characteristic impedance $$\displaystyle Z_0 = \sqrt{(R+j\omega L)/(G+j\omega C)} $$ analogous to $\gamma$ and $\eta$ in EM waves.

F. Laplace's and Poisson's Equations

• Poisson's Equation: $$\displaystyle \nabla^2 V = -\frac{\rho_v}{\varepsilon} $$ (for regions with charge).

• Laplace's Equation: $$\displaystyle \nabla^2 V = 0 $$ (for charge-free regions, $$\displaystyle \rho_v=0 $$).

• Derivation: From $$\displaystyle \nabla \cdot \mathbf{D} = \rho_v $$ and $$\displaystyle \mathbf{D} = \varepsilon \mathbf{E} = -\varepsilon \nabla V $$.

• Application: Solve for $V$ in regions with specified boundary conditions (Dirichlet: $$\displaystyle V=\text{constant} $$; Neumann: $$\displaystyle \partial V/\partial n = \text{constant} $$).

G. Loss in Transmission Lines/Media

• Attenuation Constant $\alpha$: Measures amplitude decay per unit length (Np/m). $$\displaystyle \alpha = \text{Re}(\gamma) $$.

• Phase Constant $\beta$: Measures phase change per unit length (rad/m). $$\displaystyle \beta = \text{Im}(\gamma) $$.

• In Lossy Dielectric/Conductor: $$\displaystyle \alpha > 0 $$ causes exponential decay $$\displaystyle e^{-\alpha z} $$ of wave amplitude.