Electromagnetic Theory (EX-504 (B)) - Unit 4 Short Notes

UNIT 4: ELECTROMAGNETIC THEORY - SHORT NOTES

I. VECTOR ANALYSIS & COORDINATE SYSTEMS

1.1 Gradient, Divergence, and Curl

• Gradient ($\nabla f$): Operates on a scalar field $f$ to produce a vector field. Points in direction of maximum increase of $f$.

$$\nabla f = \frac{\partial f}{\partial x}\hat{a}_x + \frac{\partial f}{\partial y}\hat{a}_y + \frac{\partial f}{\partial z}\hat{a}_z$$

• Divergence ($\nabla \cdot \vec{A}$): Operates on a vector field $\vec{A}$ to produce a scalar field. Measures "outgoingness" or source strength at a point.

$$\nabla \cdot \vec{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}$$

• Curl ($\nabla \times \vec{A}$): Operates on a vector field $\vec{A}$ to produce a vector field. Measures the "rotation" or circulation density.

$$\nabla \times \vec{A} = \begin{vmatrix} \hat{a}_x & \hat{a}_y & \hat{a}_z \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ A_x & A_y & A_z \end{vmatrix}$$

1.2 Divergence Theorem (Gauss's Theorem) - 🔴 HIGH FREQUENCY

• Statement: The total outward flux of a vector field $\vec{A}$ through a closed surface $S$ is equal to the volume integral of the divergence of $\vec{A}$ over the volume $V$ enclosed by $S$.

$$\boxed{\oint_S \vec{A} \cdot d\vec{S} = \int_V (\nabla \cdot \vec{A}) \, dV}$$

• Physical Significance: Relates a surface integral (flux) to a volume integral (source density). Fundamental in converting field problems to source problems.

• Proof Sketch: Divide volume into small cubes, apply divergence definition to each, sum, and take limit as cube size → 0. Surface fluxes of interior cubes cancel, leaving only flux through outer surface $S$.

1.3 Stokes' Theorem

• Statement: The line integral of a vector field $\vec{A}$ around a closed path $C$ is equal to the surface integral of the curl of $\vec{A}$ over any surface $S$ bounded by $C$.

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

• Significance: Relates a line integral (circulation) to a surface integral (vorticity).

1.4 Coordinate Systems - 🔴 HIGH FREQUENCY

[!TIP] Exam Alert: Conversion between systems is crucial. Remember: $$\displaystyle \rho = \sqrt{x^2+y^2} $$, $$\displaystyle \phi = \tan^{-1}(y/x) $$, $$\displaystyle r = \sqrt{x^2+y^2+z^2} $$. Unit vectors are direction-dependent in $\phi$ and $\theta$.

II. ELECTROSTATICS

2.1 Coulomb's Law & Electric Field Intensity (E)

• Electric Field $\vec{E}$: Force per unit positive test charge.

$$\vec{E} = \lim_{q_0 \to 0} \frac{\vec{F}}{q_0}$$

• E due to Infinite Line Charge (λ C/m) - 🔴 HIGH FREQUENCY DERIVATION

  By symmetry, $\vec{E}$ is radial ($$\displaystyle \hat{a}_\rho $$) in cylindrical coords. Using Gauss's Law on a cylindrical surface:

$$\oint \vec{D} \cdot d\vec{S} = Q_{enc} \quad \Rightarrow \quad D_\rho (2\pi\rho L) = \lambda L$$

$$D_\rho = \frac{\lambda}{2\pi\rho} \quad \Rightarrow \quad \boxed{\vec{E} = \frac{\lambda}{2\pi\epsilon_0\epsilon_r\rho}\hat{a}_\rho}$$

*Properties:* $|\vec{E}| \propto 1/\rho$, direction radial outward/inward.

2.2 Electric Flux Density (D) & Gauss's Law - 🔴 HIGH FREQUENCY

• Gauss's Law (Integral Form): Total electric flux through a closed surface equals total enclosed free charge.

$$\boxed{\oint_S \vec{D} \cdot d\vec{S} = Q_{free,enc}}$$

• Differential Form: $$\displaystyle \nabla \cdot \vec{D} = \rho_v $$ (where $$\displaystyle \rho_v $$ is free volume charge density).

• Application: To find $\vec{D}$, choose a Gaussian surface exploiting symmetry (spherical, cylindrical, planar). $\vec{D}$ is constant on the surface and normal to it.

2.3 Electric Potential (V) & Electric Dipole - 🔴 HIGH FREQUENCY

• Potential Difference: $$\displaystyle V_{AB} = -\int_A^B \vec{E} \cdot d\vec{l} $$. $$\displaystyle \vec{E} = -\nabla V $$.

• Potential due to Point Charge: $$\displaystyle V = \frac{Q}{4\pi\epsilon r} $$ (reference at ∞).

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

  • Potential at distant point (r >> d):

$$V \approx \frac{Q d \cos\theta}{4\pi\epsilon r^2} = \frac{\vec{p} \cdot \hat{a}_r}{4\pi\epsilon r^2}$$

*   **Electric Field:** $$\displaystyle \vec{E} = \frac{1}{4\pi\epsilon r^3} [3(\vec{p}\cdot\hat{a}_r)\hat{a}_r - \vec{p}] $$

> [!TIP] Dipole field falls as $$\displaystyle 1/r^3 $$ (faster than point charge's $$\displaystyle 1/r^2 $$).

2.4 Laplace's & Poisson's Equations - 🔴 HIGH FREQUENCY

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

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

$$\boxed{\nabla^2 V = \frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} + \frac{\partial^2 V}{\partial z^2} = 0}$$

• Uniqueness Theorem: If $V$ is specified on the boundaries of a region, the solution to Laplace/Poisson equation inside that region is unique.

2.5 Boundary Conditions for E & D - 🟢 FUNDAMENTAL

At an interface between two dielectrics ($$\displaystyle \epsilon_1 $$, $$\displaystyle \epsilon_2 $$):

[!CAUTION] Common Pitfall: $$\displaystyle D_{normal} $$ is discontinuous if surface free charge exists. $$\displaystyle E_{tangential} $$ is always continuous across any material boundary.

2.6 Capacitance & Energy

• Capacitance Definition: $$\displaystyle C = Q/V $$ (for a two-conductor system).

• Energy Stored in Electrostatic Field - 🔴 IMPORTANT

$$W_E = \frac{1}{2} \int_V \rho_v V \, dV = \frac{1}{2} \int_V \vec{D} \cdot \vec{E} \, dV = \frac{1}{2} \int_S \vec{D} \cdot \vec{E} \, dV$$

> Final formula: $$\displaystyle \boxed{W_E = \frac{1}{2} C V^2 = \frac{1}{2} Q V} $$

• Capacitances:

  • Parallel Plate: $$\displaystyle C = \epsilon A/d $$

  • Coaxial: $$\displaystyle C = \frac{2\pi\epsilon}{\ln(b/a)} $$ per unit length

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

III. MAGNETOSTATICS

3.1 Biot-Savart Law - 🔴 HIGH FREQUENCY

• Statement: Magnetic field $d\vec{H}$ at point $P$ due to a current element $I d\vec{l}$ is:

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

where $$\displaystyle \hat{a}_R $$ is unit vector from current element to $P$.

• Application: Used for non-symmetric current distributions (e.g., finite wire, loop).

3.2 H due to Straight Current-Carrying Filament - 🔴 HIGH FREQUENCY DERIVATION

For infinite wire along $z$-axis, by symmetry $\vec{H}$ is azimuthal ($$\displaystyle \hat{a}_\phi $$). Apply Biot-Savart:

$$H_\phi = \int_{-\infty}^{\infty} \frac{I (dl \sin\theta)}{4\pi R^2} = \frac{I}{4\pi\rho} \int_{-\infty}^{\infty} \frac{dl \sin\theta}{R}$$

With $$\displaystyle R = \sqrt{\rho^2+z^2} $$, $$\displaystyle \sin\theta = \rho/R $$, $$\displaystyle dl = dz $$:

$$\boxed{\vec{H} = \frac{I}{2\pi\rho}\hat{a}_\phi}$$

3.3 Ampere's Circuital Law - 🔴 HIGH FREQUENCY

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

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

• Applications (choose Amperian loop exploiting symmetry):

  • Infinite Line: $$\displaystyle H(2\pi\rho) = I \Rightarrow H = I/(2\pi\rho) $$

  • Infinite Sheet (surface current $\vec{K}$): $$\displaystyle H_1 - H_2 = \hat{a}_n \times \vec{K} $$

  • Solenoid (n turns/m, ideal): $$\displaystyle H_{inside} = nI $$, $$\displaystyle H_{outside} \approx 0 $$

  • Toroid: $$\displaystyle H(2\pi r) = NI \Rightarrow H = \frac{NI}{2\pi r} $$

3.4 Relationship: B, H, and M

• Definition: $$\displaystyle \vec{B} = \mu_0(\vec{H} + \vec{M}) $$

  where $\vec{M}$ is magnetization (magnetic dipole moment per unit volume).

• In Linear, Isotropic Media: $$\displaystyle \vec{B} = \mu \vec{H} = \mu_0\mu_r \vec{H} $$

• Magnetic Flux Density $\vec{B}$: Total magnetic field, includes effects of free currents ($\vec{H}$) and bound currents ($\vec{M}$).

3.5 Magnetic Vector Potential (A) - 🟢 SHORT NOTE

• Definition: Since $$\displaystyle \nabla \cdot \vec{B} = 0 $$, we can write $$\displaystyle \vec{B} = \nabla \times \vec{A} $$.

• Properties:

  1. $\vec{A}$ is not unique (gauge freedom).

  2. For steady currents, convenient gauge: $$\displaystyle \nabla \cdot \vec{A} = 0 $$ (Coulomb gauge).

  3. $\vec{A}$ at $P$ due to volume current $\vec{J}$: $$\displaystyle \vec{A} = \frac{\mu_0}{4\pi} \int_V \frac{\vec{J}}{R} dV $$

IV. ELECTRODYNAMICS & MAXWELL'S EQUATIONS

4.1 Faraday's Law of Induction - 🔴 HIGH FREQUENCY

• Integral Form (EMF): $$\displaystyle \mathcal{E} = \oint_C \vec{E} \cdot d\vec{l} = -\frac{d}{dt} \int_S \vec{B} \cdot d\vec{S} $$

$$\boxed{\mathcal{E} = -\frac{d\Phi_B}{dt}}$$

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

• Transformer EMF vs. Motional EMF:

  • Transformer EMF: $\vec{E}$ is induced by a time-varying $\vec{B}$ field. $\vec{E}$ is non-conservative (curl $\vec{E} \neq 0$). Path integral depends on rate of change of flux.

  • Motional EMF: Conductor moves in a static $\vec{B}$ field. Charges experience Lorentz force $q(\vec{v} \times \vec{B})$. Resulting $$\displaystyle \vec{E}_{motional} = \vec{v} \times \vec{B} $$.

4.2 Displacement Current - 🔴 HIGH FREQUENCY

• Concept: Ampere's Law ($$\displaystyle \nabla \times \vec{H} = \vec{J} $$) is inconsistent for charging capacitor (continuity violated). Maxwell introduced displacement current density:

$$\vec{J}_D = \frac{\partial \vec{D}}{\partial t}$$

• Ampere-Maxwell Law (Complete):

$$\boxed{\nabla \times \vec{H} = \vec{J} + \frac{\partial \vec{D}}{\partial t}}$$

$$\oint_C \vec{H} \cdot d\vec{l} = I_{cond} + \frac{d}{dt} \int_S \vec{D} \cdot d\vec{S}$$

• Application: In capacitor gap ($$\displaystyle \vec{J}=0 $$), $$\displaystyle \frac{\partial \vec{D}}{\partial t} $$ provides the "current" needed for Ampere's Law to hold.

4.3 Maxwell's Equations (Complete Set) - 🔴 HIGHEST FREQUENCY

4.4 Continuity Equation - 🔴 HIGH FREQUENCY

• Statement: Charge is conserved. The net rate of charge flowing out of a volume equals the rate of decrease of charge within the volume.

$$\oint_S \vec{J} \cdot d\vec{S} = -\frac{d}{dt} \int_V \rho_v \, dV$$

• Derivation from Maxwell's Equations: Take divergence of Ampere-Maxwell Law:

$$\nabla \cdot (\nabla \times \vec{H}) = 0 = \nabla \cdot \vec{J} + \nabla \cdot \frac{\partial \vec{D}}{\partial t} = \nabla \cdot \vec{J} + \frac{\partial}{\partial t}(\nabla \cdot \vec{D})$$

Using Gauss's Law ($$\displaystyle \nabla \cdot \vec{D} = \rho_v $$):

$$\boxed{\nabla \cdot \vec{J} + \frac{\partial \rho_v}{\partial t} = 0}$$

This is the **point form** of the continuity equation.

V. ELECTROMAGNETIC WAVE PROPAGATION

5.1 Wave Equation in Dielectrics & Conductors - 🔴 HIGH FREQUENCY

• Derivation (in source-free, linear, isotropic medium): Take curl of Faraday's and Ampere-Maxwell laws, substitute, and use vector identity $$\displaystyle \nabla \times (\nabla \times \vec{E}) = \nabla(\nabla \cdot \vec{E}) - \nabla^2 \vec{E} $$. With $$\displaystyle \nabla \cdot \vec{D}=0 $$ (no free charge):

$$\boxed{\nabla^2 \vec{E} = \mu\epsilon \frac{\partial^2 \vec{E}}{\partial t^2}}$$

Similarly for $\vec{H}$: $$\displaystyle \nabla^2 \vec{H} = \mu\epsilon \frac{\partial^2 \vec{H}}{\partial t^2} $$

• Uniform Plane Wave Solution (in $z$-direction):

$$\vec{E}(z,t) = \vec{E}_0 e^{-\gamma z} e^{j\omega t} \quad \text{where} \quad \gamma = \alpha + j\beta$$

• Propagation Constant $\gamma$ & Intrinsic Impedance $\eta$:

$$\gamma = \alpha + j\beta = j\omega\sqrt{\mu\epsilon} \quad \text{(lossless dielectric)}$$

$$\gamma = \sqrt{j\omega\mu(\sigma + j\omega\epsilon)} \quad \text{(general)}$$

$$\eta = \sqrt{\frac{\mu}{\epsilon}} \quad \text{(lossless)} \quad ; \quad \eta = \sqrt{\frac{j\omega\mu}{\sigma + j\omega\epsilon}} \quad \text{(general)}$$

> For **good conductor** ($\sigma \gg \omega\epsilon$): $\alpha \approx \beta \approx \sqrt{\pi f \mu \sigma}$, $$\displaystyle \eta \approx (1+j)\sqrt{\frac{\omega\mu}{2\sigma}} $$

5.2 Skin Depth & Surface Impedance - 🟢 SHORT NOTE

• Skin Depth $\delta$: Depth at which amplitude decays to $1/e$ of surface value.

$$\boxed{\delta = \frac{1}{\alpha} = \sqrt{\frac{2}{\omega\mu\sigma}} \quad \text{(good conductor)}}$$

• Surface Impedance $$\displaystyle Z_s $$: Ratio of tangential $\vec{E}$ to tangential $\vec{H}$ at the surface.

$$Z_s = \frac{E_t}{H_t} = \eta \quad \text{(for good conductor, } Z_s \approx (1+j)\frac{1}{\sigma\delta})$$

5.3 Properties of EM Waves - 🔴 HIGH FREQUENCY

1. Transverse Nature: $\vec{E}$, $\vec{H}$, and direction of propagation $\hat{k}$ are mutually orthogonal.

2. $\vec{E}$ and $\vec{H}$ are in phase (lossless) or have a phase difference (lossy).

3. Intrinsic Impedance $\eta$: Ratio $E/H$ in the wave.

4. Velocity: $$\displaystyle v = 1/\sqrt{\mu\epsilon} = c/n $$ where $$\displaystyle n = \sqrt{\mu_r\epsilon_r} $$ is refractive index.

5. Poynting Vector $\vec{S}$: Direction of power flow, magnitude = power/area.

$$\vec{S} = \vec{E} \times \vec{H} \quad \text{(Instantaneous)}$$

$$\langle \vec{S} \rangle = \frac{1}{2} \text{Re}\{\vec{E} \times \vec{H}^*\} \quad \text{(Time-average)}$$

5.4 Reflection & Transmission at Normal Incidence - 🔴 HIGH FREQUENCY

For wave incident from medium 1 ($$\displaystyle \eta_1 $$) to medium 2 ($$\displaystyle \eta_2 $$):

• Reflection Coefficient (Γ):

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

• Transmission Coefficient (τ):

$$\boxed{\tau = \frac{2\eta_2}{\eta_2 + \eta_1}}$$

• Special Cases:

  • Perfect Conductor ($$\displaystyle \eta_2=0 $$): $$\displaystyle \Gamma = -1 $$ (total reflection, 180° phase shift), $$\displaystyle \tau = 0 $$.

  • Dielectric-Dielectric: Power conservation: $$\displaystyle 1 - |\Gamma|^2 = \frac{\text{Re}(\eta_1)}{\text{Re}(\eta_2)}|\tau|^2 $$.

5.5 Oblique Incidence: Brewster's & Critical Angle

• Brewster's Angle ($$\displaystyle \theta_B $$): Angle where parallel-polarized wave has no reflection ($$\displaystyle \Gamma_\parallel=0 $$).

$$\tan\theta_B = \sqrt{\frac{\epsilon_{r2}}{\epsilon_{r1}}} \quad \text{(for non-magnetic media)}$$

• Critical Angle ($$\displaystyle \theta_c $$) & Total Internal Reflection (TIR): Occurs when wave goes from denser ($$\displaystyle n_1 > n_2 $$) to rarer medium. $$\displaystyle \theta_c = \sin^{-1}(n_2/n_1) $$. For $$\displaystyle \theta_i > \theta_c $$, $$\displaystyle \Gamma=1 $$ (total reflection), wave becomes evanescent in medium 2.

5.6 Polarization - 🟢 SHORT NOTE

Describes the time-varying behavior of $\vec{E}$ vector tip.

1. Linear: $\vec{E}$ oscillates along a straight line.

2. Circular: $\vec{E}$ rotates with constant magnitude, tip describes a circle. (Requires orthogonal $$\displaystyle \vec{E}_x $$, $$\displaystyle \vec{E}_y $$ with $90°$ phase difference).

3. Elliptical: General case, tip describes an ellipse.

VI. SPECIAL TOPICS & APPLICATIONS (SHORT NOTES)

6.1 Perfect Conductor - 🟢 SHORT NOTE

• Properties:

  1. $\sigma \to \infty$, $$\displaystyle \vec{E} = 0 $$ inside (static & time-varying).

  2. $$\displaystyle \vec{D} = 0 $$ inside.

  3. $$\displaystyle \vec{B} = 0 $$ inside (for static fields; for time-varying, $\vec{B}$ penetrates slightly - skin effect).

  4. Boundary Conditions: At surface ($\hat{n}$ outward normal from conductor):

     • $$\displaystyle E_{tangential} = 0 $$

     • $$\displaystyle D_{normal} = \rho_s $$ (surface charge density)

     • $$\displaystyle B_{normal} = 0 $$

     • $$\displaystyle H_{tangential} = \vec{K} $$ (surface current density, $$\displaystyle \vec{K} = \hat{n} \times \vec{H} $$)

6.2 Summary of Boundary Conditions - 🟢 CRUCIAL

[!CAUTION] Key: $$\displaystyle E_{t} $$ & $$\displaystyle B_{n} $$ are always continuous. $$\displaystyle D_{n} $$ discontinuous with $$\displaystyle \rho_s $$. $$\displaystyle H_{t} $$ discontinuous with surface current $\vec{K}$.

6.3 Method of Images - 🟢 SHORT NOTE

• Concept: Replace conducting boundary with image charges in the region of interest to satisfy boundary conditions ($$\displaystyle V=\text{constant} $$ on conductor surface).

• Applications:

  • Point charge near infinite grounded conducting plane: Image charge $-Q$ at mirror position.

  • Point charge near grounded conducting sphere: Image charge $q'$ at specific location inside sphere.

• Use: Solves Laplace's equation for specific geometries. Only valid in region not containing the image charge.

6.4 Surface Impedance - 🟢 SHORT NOTE

• Definition: $$\displaystyle Z_s = \frac{E_t}{H_t} $$ at the surface of a material.

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

• Significance: Relates tangential electric field to surface current density ($$\displaystyle \vec{K} = \hat{n} \times \vec{H} \approx H_t $$), $$\displaystyle E_t = Z_s K $$. Used in RF and microwave engineering for loss calculations on conductors.