ELECTROMAGNETIC THEORY (EC-504 (A)) - Unit 2 Short Notes

UNIT 2: Electromagnetic Theory - Short Notes

I. Vector Calculus Foundations

A. Divergence

• Definition: The divergence of a vector field F is a scalar quantity given by the dot product of the del operator (∇) with F.

$$\nabla \cdot \mathbf{F} = \text{div} \, \mathbf{F}$$

• Physical Significance: It measures the "outgoingness" or flux source/sink density at a point. A positive divergence indicates a source (net flux out), negative indicates a sink (net flux in), and zero indicates no net source/sink.

• Calculation in Coordinates:

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

  • Cylindrical (r,θ,z): $$\displaystyle \nabla \cdot \mathbf{F} = \frac{1}{r}\frac{\partial (r F_r)}{\partial r} + \frac{1}{r}\frac{\partial F_\theta}{\partial \theta} + \frac{\partial F_z}{\partial z} $$

  • Spherical (r,θ,φ): $$\displaystyle \nabla \cdot \mathbf{F} = \frac{1}{r^2}\frac{\partial (r^2 F_r)}{\partial r} + \frac{1}{r \sin\theta}\frac{\partial (\sin\theta \, F_\theta)}{\partial \theta} + \frac{1}{r \sin\theta}\frac{\partial F_\phi}{\partial \phi} $$

• Relation to Gauss's Law: For electric flux density D, $$\displaystyle \nabla \cdot \mathbf{D} = \rho_v $$ (point form). The divergence of D at a point equals the volume charge density there.

[!TIP] Common Pitfall: Divergence is a scalar. It is not a vector. Confusing it with curl (a vector) is a frequent error.

B. Curl

• Definition: The curl of a vector field F is a vector quantity given by the cross product of the del operator (∇) with F.

$$\nabla \times \mathbf{F} = \text{curl} \, \mathbf{F}$$

• Physical Significance: It measures the "rotationality" or circulation density (torque per unit area) of the field at a point. A non-zero curl indicates the field has a vortex or spinning component.

• Calculation in Coordinates:

  • 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 & Spherical: Use standard determinant forms with scale factors (1, r, 1) and (1, r, r sinθ) respectively.

• Relation to Faraday's Law: For electric field E, $$\displaystyle \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} $$ (point form). A time-varying magnetic field induces a circulating electric field.

C. Fundamental Theorems

1. Divergence (Gauss) Theorem:

   • Statement: The total outward flux of a vector field F through a closed surface S is equal to the volume integral of the divergence of F over the volume V enclosed by S.

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

*   **Importance:** Converts a surface integral (2D) into a volume integral (3D). Used to derive integral forms of Gauss's laws from their differential forms.

*   **Proof Outline:** Partition volume into small cubes, apply definition of divergence, sum fluxes, and take limit as cube size → 0.

2. Stokes' Theorem:

   • Statement: The line integral of a vector field F around a closed path C is equal to the surface integral of the curl of F over any surface S bounded by C.

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

*   **Importance:** Converts a line integral (1D) into a surface integral (2D). Fundamental in relating circulation to vorticity. Used to derive integral form of Faraday's law.

*   **Proof Outline:** Divide surface into small loops, apply definition of curl, sum circulation, and take limit as loop area → 0.

[!TIP] Exam Focus: Be prepared to state and explain the physical meaning of both theorems. Verifying them for a simple vector field and surface (like a cube or flat plane) is a common 6-7 mark question.

D. Classification of Vector Fields

• Irrotational (Curl-Free): $$\displaystyle \nabla \times \mathbf{F} = 0 $$. Such fields are conservative and can be expressed as the gradient of a scalar potential $\phi$: $$\displaystyle \mathbf{F} = -\nabla \phi $$. Work done is path-independent. Example: Electrostatic field E.

• Solenoidal (Divergence-Free): $$\displaystyle \nabla \cdot \mathbf{F} = 0 $$. Such fields have no net sources/sinks and can be expressed as the curl of a vector potential A: $$\displaystyle \mathbf{F} = \nabla \times \mathbf{A} $$. Example: Magnetic field B (from $$\displaystyle \nabla \cdot \mathbf{B} = 0 $$).

• Harmonic Field: Both irrotational and solenoidal: $$\displaystyle \nabla \times \mathbf{F} = 0 $$ and $$\displaystyle \nabla \cdot \mathbf{F} = 0 $$. Then $$\displaystyle \nabla^2 \mathbf{F} = 0 $$ (Laplace's equation). Example: Static electric field in a charge-free region.

• Verification Technique: For a given field F(x,y,z), compute:

  1. $\nabla \cdot \mathbf{F}$. If = 0 → Solenoidal.

  2. $\nabla \times \mathbf{F}$. If = 0 → Irrotational.

  • Example from Past Paper: $$\displaystyle \mathbf{A} = yz \, \hat{a}_x + 2x \, \hat{a}_y + xy \, \hat{a}_z $$

    • $$\displaystyle \nabla \cdot \mathbf{A} = \frac{\partial (yz)}{\partial x} + \frac{\partial (2x)}{\partial y} + \frac{\partial (xy)}{\partial z} = 0 + 0 + 0 = 0 $$ → Solenoidal.

    • $$\displaystyle \nabla \times \mathbf{A} = \begin{vmatrix} \hat{a}_x & \hat{a}_y & \hat{a}_z \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ yz & 2x & xy \end{vmatrix} = (x - 0)\hat{a}_x + (y - 0)\hat{a}_y + (2 - 0)\hat{a}_z = x\hat{a}_x + y\hat{a}_y + 2\hat{a}_z \neq \mathbf{0} $$. [Correction: This field is NOT irrotational. Past paper may have had a typo or different field. Always compute carefully.]

II. Maxwell's Equations

A. Integral and Differential Forms (In Vacuo/Free Space)

[!TIP] Memory Aid: "Gauss laws deal with divergence (sources). Faraday and Ampère deal with curl (circulation)."

B. Displacement Current

• Necessity: In static fields, Ampère's law ($$\displaystyle \nabla \times \mathbf{H} = \mathbf{J} $$) fails for regions with no conduction current but changing electric fields (e.g., capacitor gap during charging). It violates charge continuity ($$\displaystyle \nabla \cdot \mathbf{J} \neq -\partial \rho_v / \partial t $$).

• Expression: The displacement current density is $$\displaystyle \mathbf{J}_d = \frac{\partial \mathbf{D}}{\partial t} $$.

• Role: Adding $\partial \mathbf{D}/\partial t$ to Ampère's law completes it for time-varying fields, ensures consistency with charge conservation, and allows EM waves to propagate.

C. Continuity Equation

• Derivation: Take divergence of Ampère's Law (differential): $$\displaystyle \nabla \cdot (\nabla \times \mathbf{H}) = 0 = \nabla \cdot \mathbf{J} + \nabla \cdot \frac{\partial \mathbf{D}}{\partial t} $$.

  Using $$\displaystyle \nabla \cdot \mathbf{D} = \rho_v $$, we get: $$\displaystyle \nabla \cdot \mathbf{J} + \frac{\partial \rho_v}{\partial t} = 0 $$.

• Forms:

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

  • Integral: $$\displaystyle \oint_S \mathbf{J} \cdot d\mathbf{S} = -\frac{d}{dt}\int_V \rho_v \, dV = -\frac{dQ_{enc}}{dt} $$

• Physical Significance: It is the local law of conservation of electric charge. The net outward current flow from a volume equals the rate of decrease of charge within that volume.

III. Electromagnetic Wave Propagation in Lossless Dielectrics

A. Wave Equation Derivation

Assume a source-free ($$\displaystyle \rho_v=0, \mathbf{J}=0 $$), linear, isotropic, homogeneous, lossless medium with constant $\mu$, $\epsilon$.

1. Start with curl equations:

   $$\displaystyle \nabla \times \mathbf{E} = -\mu \frac{\partial \mathbf{H}}{\partial t} $$ ...(1)

   $$\displaystyle \nabla \times \mathbf{H} = \epsilon \frac{\partial \mathbf{E}}{\partial t} $$ ...(2)

2. Take curl of (1): $$\displaystyle \nabla \times (\nabla \times \mathbf{E}) = -\mu \frac{\partial}{\partial t} (\nabla \times \mathbf{H}) $$

3. Use vector identity: $$\displaystyle \nabla \times (\nabla \times \mathbf{E}) = \nabla(\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E} $$. In source-free dielectric, $$\displaystyle \nabla \cdot \mathbf{D} = \epsilon \nabla \cdot \mathbf{E} = 0 \Rightarrow \nabla \cdot \mathbf{E} = 0 $$.

4. Substitute (2) into the right side: $$\displaystyle -\mu \frac{\partial}{\partial t} (\epsilon \frac{\partial \mathbf{E}}{\partial t}) = -\mu \epsilon \frac{\partial^2 \mathbf{E}}{\partial t^2} $$

5. Result: $$\displaystyle \boxed{\nabla^2 \mathbf{E} = \mu \epsilon \frac{\partial^2 \mathbf{E}}{\partial t^2}} $$ (Wave Equation for E).

   Similarly, $$\displaystyle \boxed{\nabla^2 \mathbf{H} = \mu \epsilon \frac{\partial^2 \mathbf{H}}{\partial t^2}} $$ (Wave Equation for H).

B. Uniform Plane Waves

• Assumptions: TEM (Transverse ElectroMagnetic) wave; wavefronts are infinite planes; fields vary only in direction of propagation (say, +z); fields are transverse ($$\displaystyle E_z = H_z = 0 $$).

• Solution (Phasor Form): For propagation in +z direction:

$$\mathbf{E}(z,t) = \mathbf{E}_0 e^{-j\beta z} e^{j\omega t}, \quad \mathbf{H}(z,t) = \mathbf{H}_0 e^{-j\beta z} e^{j\omega t}$$

where $\beta$ is the **phase constant**.

• E-H Relationship: From Maxwell's curl equations:

$$\mathbf{H} = \frac{1}{\eta} (\hat{a}_z \times \mathbf{E})$$

where $$\displaystyle \eta = \sqrt{\mu/\epsilon} $$ is the **intrinsic impedance** of the medium. For a wave in +z, if $$\displaystyle \mathbf{E} = E_x \hat{a}_x $$, then $$\displaystyle \mathbf{H} = (E_x/\eta) \hat{a}_y $$.

C. Wave Parameters

[!TIP] Numerical Example (Past Paper): 10 GHz plane wave in free space, $$\displaystyle E_0 = 10 $$ V/m.

1. $$\displaystyle f = 10 \times 10^9 $$ Hz, $$\displaystyle \omega = 2\pi f = 2\pi \times 10^{10} $$ rad/s.

2. $$\displaystyle \beta_0 = \omega\sqrt{\mu_0\epsilon_0} = \frac{\omega}{c} = \frac{2\pi \times 10^{10}}{3 \times 10^8} \approx 209.44 $$ rad/m.

3. $$\displaystyle \lambda_0 = c/f = 0.03 $$ m = 3 cm.

4. $$\displaystyle v_p = c = 3 \times 10^8 $$ m/s.

5. $$\displaystyle \eta_0 \approx 377 \, \Omega $$ (standard value).

IV. Skin Effect in Conductors

A. Concept and Significance

• Skin Depth ($\delta$): The depth inside a conductor at which the amplitude of an EM wave decays to $1/e$ (~37%) of its surface value.

• Significance: At high frequencies, currents and fields are confined to a thin layer near the surface. This increases the effective AC resistance of conductors (e.g., wires, transmission lines), leading to higher power losses ($$\displaystyle I^2R $$). Crucial in RF/microwave engineering for designing low-loss conductors and shields.

B. Derivation for Good Conductors ($\sigma \gg \omega\epsilon$)

1. General wave equation in a conducting medium leads to complex propagation constant:

$$\gamma = \sqrt{j\omega\mu(\sigma + j\omega\epsilon)} = \alpha + j\beta$$

2. For good conductors ($\sigma \gg \omega\epsilon$), simplify:

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

3. Therefore:

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

4. Skin Depth:

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

Fields inside conductor: $$\displaystyle \mathbf{E}(z) = \mathbf{E}_0 e^{-\alpha z} e^{-j\beta z} \approx \mathbf{E}_0 e^{-z/\delta} e^{-jz/\delta} $$ (amplitude decays as $$\displaystyle e^{-z/\delta} $$).

C. Numerical Calculations

• Formula: $$\displaystyle \delta = \sqrt{\frac{2}{\omega\mu\sigma}} = \sqrt{\frac{1}{\pi f \mu \sigma}} $$

• Example (Past Paper): Copper at $$\displaystyle f = 1 $$ MHz.

  • $$\displaystyle \sigma_{Cu} \approx 5.8 \times 10^7 $$ S/m, $$\displaystyle \mu \approx \mu_0 = 4\pi \times 10^{-7} $$ H/m.

  • $$\displaystyle \delta = \sqrt{\frac{1}{\pi \times 10^6 \times 4\pi \times 10^{-7} \times 5.8 \times 10^7}} $$

  • $$\displaystyle \delta \approx \sqrt{\frac{1}{1.57 \times 6.28 \times 5.8 \times 10^6}} \approx \sqrt{\frac{1}{5.72 \times 10^7}} \approx 0.066 $$ mm = 66 μm.

• Variation: $\delta \propto 1/\sqrt{f}$ and $\delta \propto 1/\sqrt{\sigma}$. Higher frequency or higher conductivity → smaller skin depth.

[!TIP] Key Insight: Skin depth is very small at RF/microwave frequencies (e.g., ~1-2 μm at 1 GHz for copper). This is why hollow tubes (waveguides) are used instead of solid wires for high-power transmission.

V. Poynting Vector and Energy Flow

A. Poynting Vector

• Definition: The instantaneous electromagnetic power density vector (power flow per unit area) is given by:

$$\boxed{\mathbf{S} = \mathbf{E} \times \mathbf{H}} \quad \text{(W/m²)}$$

• Direction: Perpendicular to both E and H, in the direction of wave propagation (given by $\mathbf{E} \times \mathbf{H}$).

• Time-Average (Sinusoidal Steady-State):

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

For a uniform plane wave: $$\displaystyle \langle S \rangle = \frac{|E_0|^2}{2\eta} \hat{a}_z $$ (if propagating in +z).

B. Poynting Theorem

• Derivation (Outline): Start with $$\displaystyle \nabla \cdot (\mathbf{E} \times \mathbf{H}) = \mathbf{H} \cdot (\nabla \times \mathbf{E}) - \mathbf{E} \cdot (\nabla \times \mathbf{H}) $$. Substitute Maxwell's curl equations and use $$\displaystyle \mathbf{J} = \sigma\mathbf{E} $$ for a conducting medium.

• Differential Form:

$$\boxed{\nabla \cdot \mathbf{S} = -\mathbf{J} \cdot \mathbf{E} - \frac{\partial}{\partial t} \left( \frac{1}{2}\epsilon |\mathbf{E}|^2 + \frac{1}{2}\mu |\mathbf{H}|^2 \right)}$$

where $$\displaystyle u_{em} = \frac{1}{2}\epsilon E^2 + \frac{1}{2}\mu H^2 $$ is the EM energy density.

• Integral Form:

$$\int_V \nabla \cdot \mathbf{S} \, dV = -\int_V \mathbf{J} \cdot \mathbf{E} \, dV - \frac{\partial}{\partial t} \int_V u_{em} \, dV$$

or $$\displaystyle \oint_S \mathbf{S} \cdot d\mathbf{S} = -P_d - \frac{\partial U_{em}}{\partial t} $$

• Physical Interpretation: Conservation of Energy. The net EM power flowing out of a closed surface S (LHS) equals the total power delivered to charges within V (ohmic loss, $$\displaystyle P_d = \int \mathbf{J}\cdot\mathbf{E} dV $$) plus the rate of decrease of stored EM energy within V.

C. Applications

1. Antenna Radiation: Total radiated power $$\displaystyle P_{rad} = \oint_{S \to \infty} \langle \mathbf{S} \rangle \cdot d\mathbf{S} $$.

2. Transmission Lines/Waveguides: Calculate power flow along the guide using $$\displaystyle \mathbf{S} = \mathbf{E} \times \mathbf{H} $$.

3. Wave Impedance & Loss: For a plane wave in a conductor, $\langle S \rangle$ decreases exponentially with depth $z$ as $$\displaystyle e^{-2\alpha z} = e^{-2z/\delta} $$.

VI. Magnetic Fields: Basic Properties

A. Definitions

• Magnetic Field Intensity (H): Force-producing field, measured in A/m. Related to free currents.

• Magnetic Flux Density (B): Total magnetic field, measured in Tesla (T). Related to force on moving charges.

• Relationship in Linear Media: $$\displaystyle \boxed{\mathbf{B} = \mu \mathbf{H}} $$, where $$\displaystyle \mu = \mu_0 \mu_r $$ is the permeability.

B. Fundamental Properties

1. Gauss's Law for Magnetism: $$\displaystyle \oint_S \mathbf{B} \cdot d\mathbf{S} = 0 \quad \text{or} \quad \nabla \cdot \mathbf{B} = 0 $$.

   • Meaning: No isolated magnetic charges (monopoles) exist. Magnetic field lines are always closed loops.

2. Ampère's Circuital Law (Maxwell's): $$\displaystyle \oint_C \mathbf{H} \cdot d\mathbf{l} = I_{enc} + \frac{d}{dt}\int_S \mathbf{D} \cdot d\mathbf{S} $$ or $$\displaystyle \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t} $$.

   • Meaning: H field is produced by conduction currents J and displacement currents $\partial \mathbf{D}/\partial t$. H is not necessarily conservative (its curl is non-zero if currents or changing D exist).

3. Solenoidal Nature of B: From $$\displaystyle \nabla \cdot \mathbf{B} = 0 $$, B is always solenoidal (divergence-free). This implies the existence of a magnetic vector potential A such that $$\displaystyle \mathbf{B} = \nabla \times \mathbf{A} $$.

C. Role in Electromagnetic Waves

• In a uniform plane wave, E, H, and the propagation direction $\hat{k}$ are mutually orthogonal (TEM wave).

• E and H are in phase (in lossless dielectrics) and their magnitudes are related by the intrinsic impedance: $$\displaystyle |\mathbf{H}| = |\mathbf{E}|/\eta $$.

• Both E and H contribute to the Poynting vector $$\displaystyle \mathbf{S} = \mathbf{E} \times \mathbf{H} $$, which gives the direction and magnitude of power flow.

• The wave equations are symmetric for E and H.

[!TIP] Contrast with E-Field: Unlike E (which has sources $$\displaystyle \rho_v $$ and is irrotational in statics), B has no sources (monopoles) and is always solenoidal. This fundamental asymmetry is key to EM theory.