ELECTROMAGNETIC THEORY (EC-504 (A)) - Unit 1 Short Notes

UNIT 1: ELECTROMAGNETIC THEORY - SHORT NOTES

I. VECTOR CALCULUS FUNDAMENTALS

A. Vector Operations and Identities

Fundamental Theorems (Always True):

1. Divergence of Curl: $$\displaystyle \nabla \cdot (\nabla \times \mathbf{A}) = 0 $$

2. Curl of Gradient: $$\displaystyle \nabla \times (\nabla \phi) = 0 $$

B. Divergence Theorem (Gauss's 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 region V enclosed by S.

• Mathematical Form:

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

• Physical Significance: Relates a local property (divergence at a point, i.e., source density) to a global property (net flux through the boundary). It is the foundation for converting integral forms of Maxwell's equations to differential forms.

• Electromagnetic Application: Deriving $$\displaystyle \nabla \cdot \mathbf{D} = \rho_v $$ from $$\displaystyle \oint_S \mathbf{D} \cdot d\mathbf{S} = Q_{enc} $$.

C. Stokes' Theorem

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

• Mathematical Form:

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

• Physical Significance: Relates a global circulation (around a loop) to a local rotational property (curl) over the surface. It connects the "circulation" of a field to its "vorticity."

• Electromagnetic Importance: Used to derive the differential form of Faraday's law ($$\displaystyle \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} $$) from its integral form ($$\displaystyle \oint_C \mathbf{E} \cdot d\mathbf{l} = -\frac{d}{dt}\int_S \mathbf{B} \cdot d\mathbf{S} $$).

D. Classification of Vector Fields

Verification Method (Exam Pattern):

To prove a field $\mathbf{F}$ is irrotational and/or solenoidal:

1. Compute $\nabla \times \mathbf{F}$. If result is zero vector, field is irrotational.

2. Compute $\nabla \cdot \mathbf{F}$. If result is zero scalar, field is solenoidal.

3. Example from Dec 2024: $$\displaystyle \mathbf{A} = yz\,\hat{a}_x + 2x\,\hat{a}_y + xy\,\hat{a}_z $$.<br>

$$\displaystyle \nabla \times \mathbf{A} = (0-0)\hat{a}_x + (0-0)\hat{a}_y + (2-2)\hat{a}_z = 0 $$ → Irrotational.<br>

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

II. ELECTROSTATICS

A. Coulomb's Law

• Statement: The force F between two point charges $$\displaystyle Q_1 $$ and $$\displaystyle Q_2 $$ in free space is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

• Mathematical Form:

$$\mathbf{F}_{12} = \frac{Q_1 Q_2}{4\pi\varepsilon_0 r^2} \hat{\mathbf{a}}_{r12}$$

where $$\displaystyle \varepsilon_0 $$ is the permittivity of free space, $r$ is the distance, and $$\displaystyle \hat{\mathbf{a}}_{r12} $$ is the unit vector from $$\displaystyle Q_1 $$ to $$\displaystyle Q_2 $$.

• Electric Field Intensity (E): Force per unit charge. Due to a point charge $Q$:

$$\mathbf{E} = \frac{Q}{4\pi\varepsilon_0 r^2} \hat{\mathbf{a}}_r$$

B. Electric Flux Density (D) and Gauss's Law

• Integral Form (Gauss's Law):

$$\boxed{\oint_S \mathbf{D} \cdot d\mathbf{S} = Q_{enc}}$$

The total electric flux emerging from a closed surface S equals the **total free charge** enclosed.

• Differential Form:

$$\boxed{\nabla \cdot \mathbf{D} = \rho_v}$$

The **divergence** of **D** at a point equals the **volume charge density** $$\displaystyle \rho_v $$ at that point.

• Physical Significance of Divergence: $\nabla \cdot \mathbf{D}$ quantifies the net outflow of electric flux from an infinitesimal volume. A positive value ($$\displaystyle >0 $$) indicates a source (positive charge), a negative value ($$\displaystyle <0 $$) indicates a sink (negative charge), and zero indicates no net charge in that volume element.

III. MAGNETOSTATICS

A. Magnetic Field Intensity (H) and Magnetic Flux Density (B)

• Definitions & Relationship:

  • Magnetic Flux Density (B): Force on a moving charge. Unit: Tesla (T).

  • Magnetic Field Intensity (H): Describes how currents produce magnetic fields. Unit: A/m.

  • Constitutive Relation: $$\displaystyle \boxed{\mathbf{B} = \mu \mathbf{H}} $$, where $$\displaystyle \mu = \mu_0\mu_r $$ is the permeability of the medium.

• Properties of Magnetic Fields:

  1. Magnetic field lines are continuous (no start or end).

  2. They form closed loops.

  3. There are no isolated magnetic poles (monopoles) — $$\displaystyle \nabla \cdot \mathbf{B} = 0 $$ (Gauss's Law for Magnetism).

B. Ampere's Circuital Law

• Integral Form (Static Fields):

$$\boxed{\oint_C \mathbf{H} \cdot d\mathbf{l} = I_{enc}}$$

The line integral of **H** around a closed path C equals the **total conduction current** $$\displaystyle I_{enc} $$ passing through the surface bounded by C.

• Limitation: This form is valid only for steady (time-invariant) currents. For time-varying fields, it fails to satisfy the continuity equation ($$\displaystyle \nabla \cdot \mathbf{J} = -\frac{\partial \rho_v}{\partial t} $$). This limitation led Maxwell to introduce the displacement current term.

IV. MAXWELL'S EQUATIONS

A. Complete Set (Integral & Differential Forms)

B. Displacement Current Density

• Concept: The term $$\displaystyle \frac{\partial \mathbf{D}}{\partial t} $$ added by Maxwell to Ampere's law.

• Necessity: To make Ampere's law consistent with the law of conservation of charge (continuity equation) for time-varying fields. It "completes" the current path in regions where there is no conduction current (e.g., between capacitor plates).

• Physical Interpretation: It represents a "current" due to the time rate of change of electric flux density. It is not a flow of charges but a changing electric field that produces a magnetic field, just like a real current.

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

C. Continuity Equation

• Integral Form (Conservation of Charge):

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

The net outward conduction current through a closed surface equals the **negative rate of decrease** of total charge within the volume.

• Point Form:

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

• Significance: Expresses local conservation of electric charge. It is derived by taking the divergence of the Ampere-Maxwell law and using $$\displaystyle \nabla \cdot \mathbf{D} = \rho_v $$ and $$\displaystyle \nabla \cdot (\nabla \times \mathbf{H}) = 0 $$.

V. ELECTROMAGNETIC WAVES

A. Derivation of Uniform Plane Wave in Perfect Dielectric

1. Start with source-free Maxwell's Curl Equations ($$\displaystyle \rho_v=0, \mathbf{J}=0 $$):

$$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}, \quad \nabla \times \mathbf{H} = \frac{\partial \mathbf{D}}{\partial t}$$

2. Take curl of first equation: $$\displaystyle \nabla \times (\nabla \times \mathbf{E}) = -\frac{\partial}{\partial t}(\nabla \times \mathbf{B}) $$

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

4. Substitute $$\displaystyle \mathbf{B} = \mu \mathbf{H} $$ and $$\displaystyle \mathbf{D} = \varepsilon \mathbf{E} $$: $$\displaystyle \nabla \times \mathbf{B} = \mu \nabla \times \mathbf{H} = \mu \frac{\partial \mathbf{D}}{\partial t} = \mu \varepsilon \frac{\partial \mathbf{E}}{\partial t} $$

5. Obtain Wave Equation for E:

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

Similarly for **H**: $$\displaystyle \nabla^2 \mathbf{H} = \mu \varepsilon \frac{\partial^2 \mathbf{H}}{\partial t^2} $$

6. Solution for Uniform Plane Wave (propagating in +z direction, E polarized in x):

$$\mathbf{E}(z,t) = \mathbf{E}_0 \cos(\omega t - \beta z) \hat{\mathbf{a}}_x$$

where $\omega$ is angular frequency, $\beta$ is phase constant.

7. E-H Relationship: From Faraday's law, $$\displaystyle \mathbf{H} = \frac{1}{\eta} \hat{\mathbf{a}}_z \times \mathbf{E} $$, where $$\displaystyle \eta = \sqrt{\mu/\varepsilon} $$ is the intrinsic impedance.

B. Wave Parameters in Lossless Dielectric

• Propagation Constant: $$\displaystyle \gamma = \alpha + j\beta $$. For lossless ($$\displaystyle \sigma=0 $$), $$\displaystyle \gamma = j\beta $$.

$$\boxed{\beta = \omega \sqrt{\mu \varepsilon}}$$

• Phase Velocity: Speed of a constant-phase point.

$$\boxed{v_p = \frac{\omega}{\beta} = \frac{1}{\sqrt{\mu \varepsilon}}}$$

In **free space**: $$\displaystyle v_p = c = 3 \times 10^8 $$ m/s.

• Wavelength: Distance for $2\pi$ phase shift.

$$\boxed{\lambda = \frac{2\pi}{\beta} = \frac{v_p}{f}}$$

• Intrinsic Impedance: Ratio of E to H magnitudes in a plane wave.

$$\boxed{\eta = \sqrt{\frac{\mu}{\varepsilon}}} \quad (\text{Unit: }\Omega)$$

For **free space**: $$\displaystyle \eta_0 = \sqrt{\frac{\mu_0}{\varepsilon_0}} \approx 120\pi \approx 377\ \Omega $$.

C. Waves in Conducting Media (Good Conductors)

• General Propagation Constant: $$\displaystyle \gamma = \sqrt{j\omega\mu(\sigma + j\omega\varepsilon)} = \alpha + j\beta $$

• Skin Depth (δ): Depth at which amplitude falls to $1/e$ (~37%) of its surface value.

$$\boxed{\delta = \frac{1}{\alpha}}$$

• For Good Conductors ($\sigma \gg \omega\varepsilon$):

$$\boxed{\delta \approx \sqrt{\frac{2}{\omega \mu \sigma}}}$$

• Physical Meaning: EM waves penetrate very poorly into good conductors. High conductivity $\sigma$ leads to small $\delta$ (e.g., copper at 1 MHz: $\delta \approx 0.066$ mm).

• Wave Behavior: Fields decay exponentially as $$\displaystyle e^{-\alpha z} = e^{-z/\delta} $$. Both E and H are out of phase and have different phase constants.

D. Poynting Vector and Power Flow

• Instantaneous Poynting Vector:

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

Represents the **instantaneous power density** (power per unit area) flowing in the direction of **S**.

• Time-Average Poynting Vector (for sinusoidal steady-state):

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

where $$\displaystyle \mathbf{H}^* $$ is the complex conjugate of phasor **H**.

• Proof of Power Flow: From Lorentz force, the work done per unit time on charges in volume V is $$\displaystyle \int_V \mathbf{J} \cdot \mathbf{E} \, dV $$. Using Maxwell's equations and vector calculus, this can be shown to equal the net outward flow of electromagnetic power $$\displaystyle -\frac{\partial}{\partial t}\int_V \left( \frac{1}{2}\mathbf{E}\cdot\mathbf{D} + \frac{1}{2}\mathbf{B}\cdot\mathbf{H} \right) dV - \oint_S (\mathbf{E} \times \mathbf{H}) \cdot d\mathbf{S} $$. The surface integral term is the power leaving V, confirming $$\displaystyle \mathbf{S} = \mathbf{E} \times \mathbf{H} $$ as the power flux density.

• Poynting Theorem (Energy Conservation):

$$\boxed{\frac{\partial u_{em}}{\partial t} + \nabla \cdot \mathbf{S} = -\mathbf{J} \cdot \mathbf{E}}$$

where $$\displaystyle u_{em} = \frac{1}{2}(\mathbf{E}\cdot\mathbf{D} + \mathbf{B}\cdot\mathbf{H}) $$ is the electromagnetic energy density. The RHS is the **power per unit volume** delivered to charges (ohmic loss if $$\displaystyle \mathbf{J} = \sigma\mathbf{E} $$).

DiagramCANVAS: Sketch showing a uniform plane wave propagating in +z direction, with E_x and H_y vectors in phase, forming a right-handed set (E x H = S_z). Show exponential decay in a conductor with skin depth δ marked.

Quick Reference: Key Formulas

Exam Tips & Common Pitfalls:

1. Vector Identities: $$\displaystyle \nabla \cdot (\nabla \times \mathbf{A}) = 0 $$ is always true. $$\displaystyle \nabla \times (\nabla \phi) = 0 $$ is always true. These are used to prove field properties.

2. Maxwell's Equations: Remember the sign in Faraday's law ($-\partial\mathbf{B}/\partial t$) and the displacement current term ($+\partial\mathbf{D}/\partial t$) in Ampere's law.

3. Wave Parameters: In free space, $$\displaystyle \mu = \mu_0 $$, $$\displaystyle \varepsilon = \varepsilon_0 $$, so $$\displaystyle v_p = c $$, $$\displaystyle \eta = \eta_0 \approx 377\ \Omega $$. In a lossless dielectric, $$\displaystyle \sigma=0 $$, so $$\displaystyle \alpha=0 $$, $$\displaystyle \gamma=j\beta $$.

4. Skin Depth: $\delta \propto 1/\sqrt{f}$ and $1/\sqrt{\sigma}$. Higher frequency or better conductor → shallower penetration.

5. Poynting Vector: Direction is $\mathbf{E} \times \mathbf{H}$ (right-hand rule). For a plane wave in +z direction with $$\displaystyle \mathbf{E}=E_x\hat{a}_x $$, $$\displaystyle \mathbf{H}=H_y\hat{a}_y $$, then $$\displaystyle \mathbf{S}=S_z\hat{a}_z $$.