Electromagnetic Theory (EX-504 (B)) - Unit 5 Short Notes

UNIT 5: ELECTROMAGNETIC THEORY

I. Vector Calculus Fundamentals

Scalar & Vector Fields

• Scalar Field: A function assigning a single scalar value to every point in space (e.g., temperature T(x,y,z)).

• Vector Field: A function assigning a vector to every point in space (e.g., electric field E(x,y,z)).

Vector Operators

Fundamental Theorems

1. Divergence Theorem (Gauss's Theorem):

   The total outward flux of a vector field A through a closed surface S equals the volume integral of the divergence of A over the region V enclosed by S.

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

**Proof Sketch:** Divide `V` into small volumes, apply `∇·A ≈ (1/ΔV) ∮ A·dS` to each, sum, and take limit.

2. Stokes' Theorem:

   The line integral of a vector field A around a closed contour C equals the surface integral of the curl of A over any surface S bounded by C.

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

**Relation to Curl:** It relates the circulation (line integral) to the average curl over the surface.

Coordinate Systems: Conversion & Elements

• Cartesian: (x, y, z). Unit vectors (âx, ây, âz) constant. dV = dx dy dz.

• Cylindrical: (ρ, φ, z) where x=ρ cosφ, y=ρ sinφ, z=z. Unit vectors (âρ, âφ, âz) vary with φ. dV = ρ dρ dφ dz.

• Spherical: (r, θ, φ) where x=r sinθ cosφ, y=r sinθ sinφ, z=r cosθ. Unit vectors (âr, âθ, âφ) vary with θ, φ. dV = r² sinθ dr dθ dφ.

[!TIP] Exam Focus: Be fluent in converting ∇·A and ∇×A between coordinate systems. Remember the ρ and r² sinθ factors in divergence/curl expressions and volume elements.

II. Electrostatics

Coulomb's Law & Electric Field Intensity (E)

• Force on charge q2 due to q1: F = (q1 q2)/(4πε₀ r²) âr.

• Electric Field: E = F/q = (Q)/(4πε₀ r²) âr for a point charge.

• Field from Continuous Distribution:

  • Line charge ρL: dE = (ρL dl)/(4πε₀ r²) âr

  • Surface charge ρS: dE = (ρS dS)/(4πε₀ r²) âr

  • Volume charge ρv: dE = (ρv dV)/(4πε₀ r²) âr

Gauss's Law

• Integral Form: ∮S D·dS = Q_enclosed (where D = εE in linear media).

• Differential Form: ∇·D = ρv.

• Applications: Highly effective for symmetric charge distributions (infinite line, sphere, cylinder).

  • Infinite Line Charge: |E| = ρL/(2πε₀ ρ) (radial, cylindrical symmetry).

  • Uniformly Charged Sphere: E = (ρv r)/(3ε₀) inside (r<R), E = (Q)/(4πε₀ r²) outside (r>R).

Electric Potential (V) & Relationship

• Potential Difference: V_B - V_A = -∫_A^B E·dl.

• Point-wise Relation: E = -∇V.

• Properties: V is continuous; ∇²V = -ρv/ε (Poisson's equation); in charge-free region (ρv=0), ∇²V = 0 (Laplace's equation).

• Potential of Point Charge: V = Q/(4πε₀ r).

• Potential of Dipole (moment p = Qd): At distant point (r>>d), V ≈ (p·âr)/(4πε₀ r²).

Electric Dipole

• Dipole Moment: p = Q d (vector from -Q to +Q).

• Field (E): For r>>d, E ≈ (1/(4πε₀ r³)) [3(p·âr)âr - p].

• Potential (V): V ≈ (p·âr)/(4πε₀ r²).

Laplace's & Poisson's Equations

• Poisson's Equation: ∇²V = -ρv/ε.

• Laplace's Equation: ∇²V = 0 (in source-free regions).

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

• Solution Methods: Separation of variables in specific coordinates (Cartesian, cylindrical, spherical).

Boundary Conditions

• At an interface between two dielectrics (ε₁, ε₂):

  • Normal D: D₁n = D₂n (if no surface charge ρs). If ρs exists, D₂n - D₁n = ρs.

  • Tangential E: E₁t = E₂t (always).

• At a conductor-dielectric interface (perfect conductor):

  • Inside conductor: E=0, V=constant.

  • Just outside: E is normal to surface, |E| = ρs/ε₀, V = surface potential.

Capacitance (C)

• Definition: C = Q/V (charge stored per unit potential difference).

• Calculation: Q = ∫∫ D·dS, V = -∫ E·dl (path independent). C = Q/V.

  • Parallel Plate: C = ε A/d.

  • Cylindrical (coaxial): C = (2πε L)/ln(b/a).

  • Spherical: C = (4πε ab)/(b-a).

Energy in Electrostatic Field

• Point Charge System: W = (1/2) Σ q_i V_i.

• Energy Density: w_e = (1/2) D·E = (1/2) ε E² (J/m³).

• Total Energy: W = ∫_all space (1/2) ε E² dV.

Method of Images

• Concept: Replace conducting surfaces with fictitious "image charges" to satisfy boundary conditions in the region of interest.

• Applications:

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

  • Point charge near conducting sphere: Image charge Q' = -Q a/d at distance a²/d from center.

  • Charge between two parallel grounded planes: Infinite series of images.

[!TIP] Common Pitfall: In method of images, the image charge is not real; it's a mathematical tool. Only the region not containing conductors is solved with images.

III. Magnetostatics

Biot-Savart Law

• Statement: Magnetic field dH at point P due to current element I dl is:

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

where `R` is vector from element to `P`, `âR` is unit vector.

• Applications:

  • Infinite Straight Wire: |H| = I/(2πρ) (azimuthal, âφ direction).

  • Circular Loop (on axis): H_z = (I a²)/(2(a²+z²)^(3/2)).

  • Solenoid (ideal, infinite): Inside H = nI âz (uniform), outside H≈0.

Ampere's Circuital Law

• Integral Form: ∮C H·dl = I_enclosed.

• Differential Form: ∇×H = J.

• Applications (high symmetry):

  • Infinite straight wire: Same result as Biot-Savart.

  • Solenoid: ∮ H·dl ≈ H L = (nL) I → H = nI inside.

  • Toroid: H (2πr) = N I → H = (N I)/(2πr) âφ (inside core).

Magnetic Flux Density B & Magnetic Field Intensity H

• Relationship: B = μ H in linear, isotropic media, where μ = μ₀ μ_r.

• Magnetization M: Magnetic dipole moment per unit volume. B = μ₀ (H + M).

• Fundamental Law: ∇·B = 0 (Magnetic flux continuity; no magnetic monopoles).

Magnetic Dipole

• Dipole Moment: m = I A ân (for a current loop), or for a bar magnet, from pole strength qm and length l: m = qm l.

• Field (B): At distant point (r>>size), B ≈ (μ₀/(4π r³)) [3(m·âr)âr - m].

Magnetic Boundary Conditions

• At interface between two media (μ₁, μ₂):

  • Normal B: B₁n = B₂n (always, from ∇·B=0).

  • Tangential H: H₁t - H₂t = K (surface current density). If K=0, H₁t = H₂t.

Self & Mutual Inductance

• Self Inductance (L): λ = L I, where λ is flux linkage (NΦ for a coil). L = NΦ / I.

  • Solenoid: L = (μ N² A)/l.

  • Toroid: L = (μ N² h)/(2π) ln(b/a).

• Mutual Inductance (M): λ₂ = M I₁. M = N₂ Φ₂₁ / I₁. M₁₂ = M₂₁ = M.

• Calculation: Use Φ = ∫ B·dS from Biot-Savart or Ampere's law.

Energy in Magnetic Field

• Energy Density: w_m = (1/2) B·H = (1/2) μ H² (J/m³).

• Total Energy (for inductor): W = (1/2) L I² = ∫_all space (1/2) μ H² dV.

Vector Magnetic Potential (A)

• Definition: B = ∇×A.

• Properties: Not unique (gauge freedom). For steady currents, common choice is Coulomb gauge: ∇·A = 0.

• Expression for A: A(r) = (μ₀/(4π)) ∫ (J(r')/|r-r'|) dV' (analogous to V from ρv).

[!TIP] Exam Focus: Distinguish clearly between B (flux density, fundamental) and H (field intensity, auxiliary). Remember ∇·B=0 always, ∇×H=J for magnetostatics. Inductance formulas for solenoid/toroid are must-know.

IV. Time-Varying Fields & Maxwell's Equations

Faraday's Law of Induction

• Statement (Integral): ∮C E·dl = -d/dt ∫S B·dS.

  • LHS: Induced electromotive force (emf) around closed loop C.

  • RHS: Negative rate of change of magnetic flux through surface S bounded by C.

• Differential Form: ∇×E = -∂B/∂t.

• Types of emf:

  • Transformer emf: E induced by changing B in a stationary loop (∇×E ≠ 0).

  • Motional emf: E induced by motion of a conductor in a static B ((v×B) force on charges).

• Lenz's Law: The induced emf/current opposes the change in flux that produced it (sign in Faraday's law).

Displacement Current Density (J_D)

• Concept: Term added to Ampere's law to correct inconsistency for charging capacitors.

• Definition: J_D = ∂D/∂t.

• Necessity: In a capacitor, ∇×H = J fails between plates where J=0 but D changes. With J_D, ∇×H = J + ∂D/∂t holds universally.

Maxwell's Equations (Differential Form in Vacuum)

$$\boxed{ \begin{aligned} \nabla \cdot \mathbf{E} &= \frac{\rho_v}{\varepsilon_0} &\text{(Gauss)} \\ \nabla \cdot \mathbf{B} &= 0 &\text{(Flux Continuity)} \\ \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} &\text{(Faraday)} \\ \nabla \times \mathbf{H} &= \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t} &\text{(Ampere-Maxwell)} \end{aligned} }$$

Physical Interpretation:

1. Gauss (E): Electric charges are sources/sinks of E.

2. Gauss (B): No isolated magnetic poles; B lines are continuous.

3. Faraday: A changing B field induces a circulating E field.

4. Ampere-Maxwell: Electric currents (J) and changing D fields produce a circulating H field.

Continuity Equation

• Derivation: Take divergence of Ampere-Maxwell law: ∇·(∇×H) = 0 = ∇·J + ∇·(∂D/∂t) = ∇·J + ∂(∇·D)/∂t. Using Gauss's law (∇·D = ρv), we get:

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

• Meaning: Conservation of electric charge. Local decrease in charge density must be accompanied by a divergent current outflow.

Convection vs. Conduction Current

• Conduction Current (J_c): Due to drift of charges in a conductor. J_c = σ E (Ohm's law, σ is conductivity).

• Convection Current (J_v): Due to motion of charges in free space (e.g., electron beam, ionized gas). J_v = ρ_v v (where v is velocity).

• Key Difference: Conduction requires a medium with σ; convection can occur in vacuum.

[!TIP] Critical Link: The displacement current ∂D/∂t is not a "real" current of moving charges; it's a term with units of current density that ensures current continuity and allows electromagnetic waves.

V. Electromagnetic Wave Propagation

Wave Equation Derivation (in Linear, Homogeneous, Isotropic Media)

1. Start with Maxwell's curl equations:

   ∇×E = -∂B/∂t, ∇×H = J + ∂D/∂t.

2. For a source-free region (J=0, ρv=0), D=εE, B=μH.

3. Take curl of Faraday: ∇×(∇×E) = ∇×( -μ ∂H/∂t ).

4. Use vector identity: ∇×(∇×E) = ∇(∇·E) - ∇²E. Since ∇·E=0 (source-free), -∇²E = -μ ∂/∂t (∇×H).

5. Substitute Ampere: ∇×H = ε ∂E/∂t.

6. Result: ∇²E = μ ε ∂²E/∂t². Similarly for H.

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

This is the **wave equation**. Wave speed `v = 1/√(με)`.

Uniform Plane Waves (UPW)

• Characteristics: E and H are perpendicular to each other and to the direction of propagation (k). TEM (Transverse Electromagnetic) waves.

• In Lossless Dielectric (σ=0):

  • Propagation constant γ = jβ, β = ω√(με).

  • Intrinsic impedance η = √(μ/ε) (real, e.g., free space η₀ ≈ 377 Ω).

  • Phase velocity u_p = 1/√(με).

  • E and H in phase; |E|/|H| = η.

• In Good Conductors (σ >> ωε):

  • γ = α + jβ ≈ (1+j)√(ωμσ/2).

  • α = β ≈ √(ωμσ/2) (high attenuation).

  • Intrinsic impedance η_c = (1+j) √(ωμ/(2σ)) (complex, small magnitude).

  • E and H out of phase by 45°.

  • Skin Effect: Wave penetrates only a small depth δ.

Propagation Constants (γ = α + jβ)

• Lossy Dielectric (σ finite, σ < ωε):

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

Calculate `α` and `β` from `α = ω√(με/2) [√(1+(σ/(ωε))²) - 1]^(1/2)`, `β = ω√(με/2) [√(1+(σ/(ωε))²) + 1]^(1/2)`.

• Loss Tangent: tan δ = σ/(ωε). Determines if medium is low-loss (tanδ << 1) or good conductor (tanδ >> 1).

Polarization

• Describes the time-varying behavior of the electric field vector at a fixed point.

• Linear: E vector oscillates along a straight line.

• Circular: E vector rotates with constant magnitude; Ex and Ey equal amplitude, 90° out of phase.

• Elliptical: General case; E vector traces an ellipse.

Reflection & Transmission at Normal Incidence

• Consider UPW incident from medium 1 (η₁) to medium 2 (η₂).

• Reflection Coefficient (Γ): Γ = (η₂ - η₁)/(η₂ + η₁).

• Transmission Coefficient (τ): τ = 1 + Γ = (2η₂)/(η₁ + η₂).

• For perfect conductor (η₂=0): Γ = -1 (total reflection, 180° phase shift), τ=0.

• For lossless dielectrics (η real): |Γ|<1, |τ|>0.

Skin Depth (δ) & Surface Impedance (Z_s)

• Skin Depth: Depth at which amplitude decays to 1/e of surface value. δ = 1/α.

  • For good conductor: δ = √(2/(ωμσ)).

• Surface Impedance: Ratio of tangential E to tangential H at the surface of a good conductor.

$$Z_s = \frac{E_t}{H_t} = (1+j) \frac{\delta \eta_c}{2} \approx (1+j) \frac{1}{\sigma \delta}$$

(since `η_c` is small). `Z_s` relates to power flow into the conductor.

Poynting Vector & Theorem

• Instantaneous Poynting Vector: S = E × H (W/m²). Direction: power flow.

• Complex Poynting Vector (Time-Harmonic): S_avg = (1/2) Re[E × H*].

• Poynting Theorem (Power Conservation):

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

where `w_em = (1/2)(εE² + μH²)` is stored EM energy density.

*   LHS: Net outward power flow.

*   RHS: Rate of decrease of stored energy + power dissipated as heat (`J·E`).

Properties of EM Waves

• Transverse nature (E, H, k mutually orthogonal).

• Speed in vacuum: c = 1/√(μ₀ε₀) ≈ 3×10⁸ m/s.

• In medium: v = c/√(μ_r ε_r).

• Wavelength λ = v/f = 2π/β.

• Transport energy and momentum.

[!TIP] Key Calculation: Given E field in a lossy medium, find H using η = E/H (intrinsic impedance). First determine medium type (lossless, lossy, conductor) from σ, ε, μ, ω, compute γ and η, then H = E/η (consider phase).

VI. Special Topics & Applications

Perfect Conductor

• Properties:

  • Interior: E=0, B=0, ρv=0 (charges reside on surface).

  • Surface: Tangential E=0; normal B=0; surface current K = H_t (from H₁t - H₂t = K, inside H=0).

• Wave Interaction: Total reflection (Γ=-1), no field penetration, E and H at surface are related by η₀.

Surface Impedance (Z_s)

• Definition: Z_s = E_t / H_t at the surface of a conductor.

• Significance: For good conductors at high frequency, it's a useful lumped parameter describing the ratio of tangential electric field to surface current density (J_s = H_t). Z_s ≈ (1+j)/(σδ).

• Relation to Skin Depth: Re(Z_s) = 1/(σδ) (surface resistance).

Transmission Line Analogy

• Similarity: The telegrapher's equations for voltage V(z) and current I(z) on a lossy transmission line are identical in form to the wave equations for E and H components of a uniform plane wave.

  • ∂²V/∂z² = γ² V, ∂²I/∂z² = γ² I.

• Interpretation:

  • Propagation constant γ = √((R+jωL)(G+jωC)) ↔ γ = √(jωμ(σ+jωε)).

  • Characteristic impedance Z₀ = √((R+jωL)/(G+jωC)) ↔ Intrinsic impedance η = √(jωμ/(σ+jωε)).

• Analogy: V ↔ E (transverse electric), I ↔ H (transverse magnetic). R ↔ σ (loss), L ↔ μ, C ↔ ε, G ↔ σ (leakage).

[!TIP] Short Note Focus: Past papers frequently ask for concise notes on Method of Images (solve for conductors), Convection vs Conduction Current (J = ρv vs J = σE), Magnetic Dipole (moment m, field), Self-inductance of Solenoid/Toroid (formulas), and Transmission Line Analogy (compare equations). Be ready to write 3-5 line definitions with key formulas.

High-Yield Problem Types from Past Papers

1. Calculate ∇·A and ∇×A for a given vector field in Cartesian, cylindrical, or spherical coordinates.

2. Derive E for infinite line charge using Gauss's law; state properties of V.

3. Derive E and V for an electric dipole.

4. Derive H for a straight wire using Biot-Savart law.

5. Calculate total magnetic flux crossing a given surface in cylindrical coordinates.

6. Derive Continuity Equation from Maxwell's equations.

7. Given E field in a lossy medium (with ε_r, μ_r, σ), find α, β, and H field.

8. Apply boundary conditions to find E and H in a wave normally incident on an interface (calculate Γ, τ).

9. Find α, β from given medium parameters or intrinsic impedance.

10. Short Notes: Method of images, magnetic dipole, self-inductance formulas, surface impedance, perfect conductor properties.