UNIT 4: ELECTROMAGNETIC THEORY – EXAM-FOCUSED SHORT NOTES

I. VECTOR CALCULUS FUNDAMENTALS FOR EM

1. Gradient, Divergence, and Curl Operators

*Curl in Cartesian:

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

Cylindrical & Spherical formulas are lengthy; focus on physical significance.

Physical Significance:

• Divergence \( \nabla \cdot \mathbf{A} \): Measures net flux outflow per unit volume.

  • \( \nabla \cdot \mathbf{A} > 0 \): Source (field diverges).

  • \( \nabla \cdot \mathbf{A} < 0 \): Sink (field converges).

  • \( \nabla \cdot \mathbf{A} = 0 \): Solenoidal field (no net source/sink).

• Curl \( \nabla \times \mathbf{A} \): Measures circulation or rotation per unit area.

  • Non-zero curl ⇒ Rotational field (vortices).

  • Zero curl ⇒ Irrotational (conservative) field.

2. Fundamental Theorems

• Divergence (Gauss's) Theorem:

  \[ \boxed{\oint_S \mathbf{A} \cdot d\mathbf{S} = \int_V (\nabla \cdot \mathbf{A}) \, dV} \]

  Interpretation: Total outward flux through closed surface \( S \) equals volume integral of divergence over \( V \).

  Importance in EM: Used to convert surface integrals (e.g., Gauss’s law) to volume integrals.

• Stokes's Theorem:

  \[ \boxed{\oint_C \mathbf{A} \cdot d\mathbf{l} = \int_S (\nabla \times \mathbf{A}) \cdot d\mathbf{S}} \]

  Interpretation: Line integral around closed curve \( C \) equals surface integral of curl over any surface \( S \) bounded by \( C \).

  Importance in EM: Foundation for deriving Maxwell’s equations (e.g., Faraday’s law, Ampere’s law).

[!TIP] Orientation Rule: For Stokes’s theorem, the direction of \( C \) and the normal to \( S \) must follow the right-hand rule: if fingers curl along \( C \), thumb points in the direction of \( d\mathbf{S} \).

3. Classification of Vector Fields

Verification Procedure:

1. Compute \( \nabla \times \mathbf{A} \) and \( \nabla \cdot \mathbf{A} \).

2. If both zero ⇒ Both irrotational and solenoidal (e.g., \( \mathbf{A} = yz\,\hat{a}_x + xz\,\hat{a}_y + xy\,\hat{a}_z \)).

3. If only \( \nabla \cdot \mathbf{A} = 0 \) ⇒ Solenoidal only.

4. If only \( \nabla \times \mathbf{A} = 0 \) ⇒ Irrotational only.

Example from DEC 2024:

\[ \mathbf{F} = \left(\frac{150}{r^2}\right)\hat{a}_r + 10\,\hat{a}_\theta + 5\,\hat{a}_z \quad \text{(cylindrical coordinates)} \]

• Divergence:

  \[ \nabla \cdot \mathbf{F} = \frac{1}{r}\frac{\partial (r F_r)}{\partial r} + \frac{1}{r}\frac{\partial F_\theta}{\partial \theta} + \frac{\partial F_z}{\partial z} = \frac{1}{r}\frac{\partial (150/r)}{\partial r} + 0 + 0 = -\frac{150}{r^3} \neq 0 \]

• Curl:

  \[ \nabla \times \mathbf{F} = \frac{1}{r}\left[ \frac{\partial (r F_\theta)}{\partial r} - \frac{\partial F_r}{\partial \theta} \right] \hat{a}_\phi = \frac{1}{r}\frac{\partial (10r)}{\partial r} = \frac{10}{r} \neq 0 \]

  Conclusion: Neither irrotational nor solenoidal.

II. ELECTROSTATICS

• Coulomb’s Law:

  \[ \mathbf{F} = \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r^2} \hat{a}_r \]

  Force between two point charges; superposition principle applies for multiple charges.

• Electric Field Intensity \( \mathbf{E} \):

  \[ \mathbf{E} = \frac{\mathbf{F}}{q_0} = \frac{1}{4\pi\varepsilon_0} \sum_i \frac{q_i}{r_i^2} \hat{a}_{r_i} \quad \text{(point charges)} \]

  For continuous distribution:

  \[ \mathbf{E} = \int \frac{\rho_l \, dl}{4\pi\varepsilon_0 r^2} \hat{a}_r \quad \text{(line)}, \quad \mathbf{E} = \int \frac{\rho_s \, dS}{4\pi\varepsilon_0 r^2} \hat{a}_r \quad \text{(surface)}, \quad \mathbf{E} = \int \frac{\rho_v \, dV}{4\pi\varepsilon_0 r^2} \hat{a}_r \quad \text{(volume)}. \]

• Electric Flux Density \( \mathbf{D} \):

  \[ \boxed{\mathbf{D} = \varepsilon \mathbf{E}} \quad \text{(linear isotropic media)} \]

  In free space: \( \mathbf{D} = \varepsilon_0 \mathbf{E} \). Accounts for material polarization.

III. MAGNETOSTATICS

• Magnetic Field Intensity \( \mathbf{H} \) & Magnetic Flux Density \( \mathbf{B} \):

  \[ \mathbf{B} = \mu \mathbf{H}, \quad \text{where } \mu = \mu_0 \mu_r. \]

• Biot–Savart Law:

  \[ d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I \, d\mathbf{l} \times \hat{a}_r}{r^2} \]

  Magnetic field due to current element.

• Ampère’s Circuital Law:

  \[ \oint_C \mathbf{H} \cdot d\mathbf{l} = I_{\text{enc}} \]

  For steady currents; relates \( \mathbf{H} \) to free current.

• Properties of Magnetic Fields:

  • Solenoidal: \( \nabla \cdot \mathbf{B} = 0 \) (no magnetic monopoles).

  • Field lines are closed loops.

IV. MAXWELL’S EQUATIONS (DYNAMIC FIELDS)

Displacement Current:

• Term \( \frac{\partial \mathbf{D}}{\partial t} \) added to Ampère’s law for time-varying fields.

• Necessity: Ensures continuity equation \( \nabla \cdot \mathbf{J} = -\frac{\partial \rho_v}{\partial t} \) is satisfied.

• Physical meaning: Represents effective current due to changing electric field (e.g., between capacitor plates).

Continuity Equation (Charge Conservation):

\[ \boxed{\nabla \cdot \mathbf{J} = -\frac{\partial \rho_v}{\partial t}} \quad \text{(point form)} \]

\[ \oint_S \mathbf{J} \cdot d\mathbf{S} = -\frac{dQ_{\text{enc}}}{dt} \quad \text{(integral form)} \]

V. ELECTROMAGNETIC WAVE PROPAGATION

1. Derivation of Uniform Plane Wave in Lossless Dielectric

Assumptions:

• Linear, isotropic, homogeneous, lossless dielectric (\( \sigma = 0 \)).

• Uniform plane wave: Fields vary only in propagation direction (\( z \)), transverse to it:

  \[ \mathbf{E} = E_x(z,t)\,\hat{a}_x, \quad \mathbf{H} = H_y(z,t)\,\hat{a}_y. \]

• No sources (\( \rho_v = 0, \mathbf{J} = 0 \)).

Steps:

1. From Maxwell’s curl equations:

   \[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \quad \Rightarrow \quad \frac{\partial E_x}{\partial z} = -\mu \frac{\partial H_y}{\partial t} \]

   \[ \nabla \times \mathbf{H} = \frac{\partial \mathbf{D}}{\partial t} \quad \Rightarrow \quad \frac{\partial H_y}{\partial z} = \varepsilon \frac{\partial E_x}{\partial t} \]

2. Differentiate first w.r.t. \( z \), substitute second:

   \[ \frac{\partial^2 E_x}{\partial z^2} = -\mu \frac{\partial}{\partial t}\left(\frac{\partial H_y}{\partial z}\right) = -\mu \varepsilon \frac{\partial^2 E_x}{\partial t^2} \]

3. Wave equation:

   \[ \boxed{\frac{\partial^2 E_x}{\partial z^2} = \mu \varepsilon \frac{\partial^2 E_x}{\partial t^2}} \]

   Similarly for \( H_y \): \( \frac{\partial^2 H_y}{\partial z^2} = \mu \varepsilon \frac{\partial^2 H_y}{\partial t^2} \).

2. Wave Parameters & Characteristics

Numerical (DEC 2024):
10 GHz plane wave in free space, \( E_0 = 10 \, \text{V/m} \).

• \( v_p = c = 3 \times 10^8 \, \text{m/s} \)

• \( \lambda = \frac{c}{f} = \frac{3 \times 10^8}{10 \times 10^9} = 0.03 \, \text{m} = 3 \, \text{cm} \)

• \( \beta = \frac{2\pi}{\lambda} = \frac{2\pi}{0.03} \approx 209.44 \, \text{rad/m} \)

• \( \eta = \eta_0 \approx 377 \, \Omega \)

3. Skin Effect & Depth of Penetration

• Skin depth \( \delta \): Distance at which field amplitude falls to \( 1/e \) of surface value in a good conductor (\( \sigma \gg \omega\varepsilon \)).

\[ \boxed{\delta = \sqrt{\frac{2}{\omega \mu \sigma}}} \]

• Significance: At high frequencies, current flows only in a thin layer near surface; effective resistance increases.

Numerical (DEC 2024): Copper at 1 MHz.

Given: \( \sigma = 5.8 \times 10^7 \, \text{S/m}, \, \mu \approx \mu_0 = 4\pi \times 10^{-7} \, \text{H/m}, \, f = 1 \, \text{MHz} \).

\[ \omega = 2\pi f = 2\pi \times 10^6 \, \text{rad/s} \]

\[ \omega\mu = 2\pi \times 10^6 \times 4\pi \times 10^{-7} = 8\pi^2 \times 10^{-1} \approx 7.895 \, \text{(SI units)} \]

\[ \omega\mu\sigma = 7.895 \times 5.8 \times 10^7 \approx 4.58 \times 10^8 \]

\[ \delta = \sqrt{\frac{2}{4.58 \times 10^8}} \approx \sqrt{4.37 \times 10^{-9}} \approx 6.61 \times 10^{-5} \, \text{m} = 66.1 \, \mu\text{m} \]

[!TIP] Common Pitfall: Skin depth formula applies only to good conductors (\( \sigma \gg \omega\varepsilon \)). For lossy dielectrics, use general \( \gamma = \alpha + j\beta \) with \( \alpha = \beta = \omega\sqrt{\frac{\mu\varepsilon}{2}\left(\sqrt{1+\left(\frac{\sigma}{\omega\varepsilon}\right)^2}-1\right)} \).

VI. POWER FLOW AND POYNTING VECTOR

• Poynting Vector \( \mathbf{S} \):

  \[ \boxed{\mathbf{S} = \mathbf{E} \times \mathbf{H}} \quad \text{(units: W/m²)} \]

  Direction: Power flow direction (perpendicular to both \( \mathbf{E} \) and \( \mathbf{H} \)).

Proof from Maxwell’s Equations (Poynting’s Theorem):

1. Start with vector identity:

   \[ \nabla \cdot (\mathbf{E} \times \mathbf{H}) = \mathbf{H} \cdot (\nabla \times \mathbf{E}) - \mathbf{E} \cdot (\nabla \times \mathbf{H}) \]

2. Substitute Maxwell’s:

   \[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}, \quad \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t} \]

   \[ \Rightarrow \nabla \cdot (\mathbf{E} \times \mathbf{H}) = -\mathbf{H} \cdot \frac{\partial \mathbf{B}}{\partial t} - \mathbf{E} \cdot \mathbf{J} - \mathbf{E} \cdot \frac{\partial \mathbf{D}}{\partial t} \]

3. For linear media (\( \mathbf{B} = \mu\mathbf{H}, \mathbf{D} = \varepsilon\mathbf{E} \)):

   \[ \mathbf{E} \cdot \frac{\partial \mathbf{D}}{\partial t} = \frac{\partial}{\partial t}\left(\frac{1}{2}\varepsilon E^2\right), \quad \mathbf{H} \cdot \frac{\partial \mathbf{B}}{\partial t} = \frac{\partial}{\partial t}\left(\frac{1}{2}\mu H^2\right) \]

4. Thus:

   \[ \nabla \cdot \mathbf{S} = -\mathbf{E} \cdot \mathbf{J} - \frac{\partial}{\partial t}\left(\frac{1}{2}\varepsilon E^2 + \frac{1}{2}\mu H^2\right) \]

   where \( u = \frac{1}{2}(\varepsilon E^2 + \mu H^2) \) is EM energy density.

Poynting Theorem (Integral Form):

\[ \boxed{\oint_S \mathbf{S} \cdot d\mathbf{S} = -\int_V \mathbf{E} \cdot \mathbf{J} \, dV - \frac{\partial}{\partial t}\int_V u \, dV} \] Interpretation: Net power flowing out of volume \( V \) equals power dissipated (\( \mathbf{E}\cdot\mathbf{J} \)) plus rate of decrease of stored energy.

VII. SPECIAL TOPICS & APPLICATIONS

1. Wave Propagation in Conducting Media

• Good conductor (\( \sigma \gg \omega\varepsilon \)):

  \[ \alpha \approx \beta \approx \sqrt{\frac{\pi f \mu \sigma}{2}}, \quad \delta = \frac{1}{\alpha} = \sqrt{\frac{2}{\omega \mu \sigma}} \]

  Fields attenuate rapidly; wave is non-propagating (evanescent) for thick conductors.

• Lossy dielectric (\( \sigma \sim \omega\varepsilon \)): Use general formulas:

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

2. Standard Problem-Solving Steps

• Given \( f \) and medium properties (\( \varepsilon, \mu, \sigma \)):

  1. Compute \( \omega = 2\pi f \).

  2. For lossless: \( v_p = 1/\sqrt{\mu\varepsilon} \), \( \lambda = v_p/f \), \( \beta = \omega\sqrt{\mu\varepsilon} \), \( \eta = \sqrt{\mu/\varepsilon} \).

  3. For good conductor: \( \delta = \sqrt{2/(\omega\mu\sigma)} \), \( \alpha = \beta = 1/\delta \).

• Verifying Theorems:

  • Stokes’s: Compute \( \nabla \times \mathbf{A} \), surface integral \( \int_S (\nabla \times \mathbf{A})\cdot d\mathbf{S} \), and line integral \( \oint_C \mathbf{A}\cdot d\mathbf{l} \) with consistent orientation.

  • Divergence: Compute \( \nabla \cdot \mathbf{A} \) and volume integral \( \int_V (\nabla \cdot \mathbf{A}) dV \), compare with \( \oint_S \mathbf{A}\cdot d\mathbf{S} \).

• Field Classification: Compute \( \nabla \times \mathbf{A} \) and \( \nabla \cdot \mathbf{A} \) in appropriate coordinates.

\boxed{\text{End of Unit 4 Notes}}