ELECTROMAGNETIC THEORY (EC-504 (A)) - Unit 5 Short Notes

UNIT 5: MAXWELL'S EQUATIONS AND ELECTROMAGNETIC WAVES

Vector Differential Calculus in Electromagnetics

Divergence of a Vector Field

• Definition: The divergence of a vector field A at a point is the net outward flux per unit volume as the volume shrinks to zero.

$$\nabla \cdot \mathbf{A} = \lim_{\Delta V \to 0} \frac{\oint_S \mathbf{A} \cdot d\mathbf{S}}{\Delta V}$$

In Cartesian coordinates: $$\displaystyle \nabla \cdot \mathbf{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z} $$.

• Physical Significance: Measures the "source" or "sink" strength at a point. A positive divergence indicates a source (net outflow), negative indicates a sink (net inflow), zero indicates no net source/sink.

• Key Result: For a solenoidal field, $$\displaystyle \nabla \cdot \mathbf{A} = 0 $$ (e.g., magnetic field B).

Curl of a Vector Field

• Definition: The curl of a vector field A at a point is the maximum circulation per unit area as the area shrinks to zero, with the direction normal to the area.

$$\nabla \times \mathbf{A} = \lim_{\Delta S \to 0} \frac{\oint_C \mathbf{A} \cdot d\mathbf{l}}{\Delta S} \mathbf{\hat{n}}$$

In Cartesian coordinates:

$$\nabla \times \mathbf{A} = \left( \frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z} \right) \mathbf{\hat{a}}_x + \left( \frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x} \right) \mathbf{\hat{a}}_y + \left( \frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y} \right) \mathbf{\hat{a}}_z$$

• Physical Significance: Measures the "rotation" or "vorticity" of the field. A non-zero curl indicates a rotational field.

• Key Result: For an irrotational field, $$\displaystyle \nabla \times \mathbf{A} = 0 $$ (e.g., electrostatic field E).

Divergence Theorem (Gauss's Theorem)

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

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

• Importance: Converts a surface integral (2D) into a volume integral (3D), simplifying calculations. Fundamental in deriving integral forms of Maxwell's equations from their differential forms.

Stokes' Theorem

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

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

• Importance: Converts a line integral (1D) into a surface integral (2D). Crucial for relating circulation to rotation and in deriving Maxwell's Faraday law.

Verification of Vector Field Properties

• Irrotational (Curl-Free): Compute $\nabla \times \mathbf{A}$. If result is 0, field is irrotational.

• Solenoidal (Divergence-Free): Compute $\nabla \cdot \mathbf{A}$. If result is 0, field is solenoidal.

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

  • $$\displaystyle \nabla \times \mathbf{A} = (x - 2)\mathbf{\hat{a}}_x + (y - y)\mathbf{\hat{a}}_y + (2 - z)\mathbf{\hat{a}}_z $$ → Not irrotational.

  • $$\displaystyle \nabla \cdot \mathbf{A} = z + 2 + x $$ → Not solenoidal.

  [!TIP] Common Pitfall: Students often miscalculate partial derivatives in curl, especially cross-terms. Write out each component systematically.

Fundamental Laws of Electromagnetics

Coulomb's Law

• Statement: The force between two point charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them.

• Mathematical Formulation:

$$\mathbf{F}_{12} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2} \mathbf{\hat{a}}_{12}$$

where $$\displaystyle \epsilon_0 $$ is the permittivity of free space.

Properties of Magnetic Fields

• Magnetic fields are produced by moving charges (currents) or changing electric fields.

• Magnetic field lines are continuous, forming closed loops (no isolated magnetic poles).

• The magnetic force on a charge is perpendicular to both the velocity and the field ($$\displaystyle \mathbf{F} = q\mathbf{v} \times \mathbf{B} $$).

• Key Result: $$\displaystyle \nabla \cdot \mathbf{B} = 0 $$ (Gauss's Law for Magnetism).

Maxwell's Equations for Time-Varying Fields

Continuity Equation

• Derivation (Charge Conservation): Current I flowing out of a closed surface S equals the rate of decrease of charge inside volume V.

$$I = -\frac{dQ}{dt} = -\frac{d}{dt} \int_V \rho_v \, dV$$

Using $$\displaystyle \oint_S \mathbf{J} \cdot d\mathbf{S} = \int_V (\nabla \cdot \mathbf{J}) \, dV $$ and applying divergence theorem:

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

• Point Form:

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

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

• Significance: Expresses local conservation of electric charge.

Ampere's Circuital Law & Its Limitation

• Original (Static) Form: $$\displaystyle \oint_C \mathbf{H} \cdot d\mathbf{l} = I_{enc} $$.

• Limitation: Fails for time-varying fields (e.g., charging capacitor). The enclosed current $$\displaystyle I_{enc} $$ is ambiguous when displacement current exists between capacitor plates.

Displacement Current

• Concept: A term added to account for the changing electric flux in Ampere's law. It is not a current of moving charges but a "fictitious" current representing $$\displaystyle \frac{\partial \mathbf{D}}{\partial t} $$.

• Expression: $$\displaystyle \mathbf{J}_d = \frac{\partial \mathbf{D}}{\partial t} $$.

Maxwell's Modification to Ampere's Law

• Differential Form:

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

• Integral Form:

$$\oint_C \mathbf{H} \cdot d\mathbf{l} = \int_S \mathbf{J} \cdot d\mathbf{S} + \frac{\partial}{\partial t} \int_S \mathbf{D} \cdot d\mathbf{S}$$

Full Set of Maxwell's Equations (Point Form)

Full Set (Integral Form)

1. $$\displaystyle \oint_S \mathbf{D} \cdot d\mathbf{S} = \int_V \rho_v \, dV $$

2. $$\displaystyle \oint_S \mathbf{B} \cdot d\mathbf{S} = 0 $$

3. $$\displaystyle \oint_C \mathbf{E} \cdot d\mathbf{l} = -\frac{d}{dt} \int_S \mathbf{B} \cdot d\mathbf{S} $$

4. $$\displaystyle \oint_C \mathbf{H} \cdot d\mathbf{l} = \int_S \mathbf{J} \cdot d\mathbf{S} + \frac{d}{dt} \int_S \mathbf{D} \cdot d\mathbf{S} $$

[!TIP] Exam Focus: Be able to state all four equations in both forms and explain the physical significance of the displacement current term $$\displaystyle \frac{\partial \mathbf{D}}{\partial t} $$.

Electromagnetic Wave Propagation in Media

Derivation of Uniform Plane Wave in Lossless Dielectric

1. Assume source-free region ($$\displaystyle \mathbf{J}=0, \rho_v=0 $$).

2. Start with Maxwell's curl equations:

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

3. Take curl of both sides of Faraday's law: $$\displaystyle \nabla \times (\nabla \times \mathbf{E}) = -\frac{\partial}{\partial t} (\nabla \times \mathbf{B}) $$.

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

5. Substitute: $$\displaystyle -\nabla^2 \mathbf{E} = -\mu \frac{\partial}{\partial t} (\nabla \times \mathbf{H}) = -\mu \frac{\partial}{\partial t} \left( \epsilon \frac{\partial \mathbf{E}}{\partial t} \right) $$.

6. Wave Equation:

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

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

Wave Parameters in Lossless Media ($\sigma \approx 0$)

For a plane wave $$\displaystyle \mathbf{E} = \mathbf{E}_0 e^{j(\omega t - \beta z)} $$:

• Phase Velocity ($$\displaystyle v_p $$): Speed of a constant-phase point.

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

• Wavelength ($\lambda$): Distance between two successive points of same phase.

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

• Propagation Constant ($\beta$): Phase shift per unit length.

$$\beta = \omega \sqrt{\mu \epsilon} = \frac{2\pi}{\lambda}$$

• Intrinsic Impedance ($\eta$): Ratio of E to H in the wave, determines wave polarization.

$$\eta = \sqrt{\frac{\mu}{\epsilon}} \quad (\text{for lossless media, real})$$

For free space: $$\displaystyle \eta_0 = \sqrt{\frac{\mu_0}{\epsilon_0}} \approx 377 \, \Omega $$.

Numerical Example (Past Paper):

A 10 GHz plane wave in free space has $$\displaystyle E_0 = 10 $$ V/m.

• $$\displaystyle f = 10 \times 10^9 $$ Hz, $$\displaystyle \lambda = c/f = 3 \times 10^8 / 10^{10} = 0.03 $$ m = 3 cm.

• $$\displaystyle \beta = 2\pi/\lambda = 2\pi / 0.03 \approx $$ 209.4 rad/m.

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

• $$\displaystyle \eta = \eta_0 \approx $$ 377 Ω.

Wave Propagation in Good Conductors ($\sigma \gg \omega \epsilon$)

• Propagation Constant: $$\displaystyle \gamma = \alpha + j\beta = \sqrt{j\omega\mu(\sigma + j\omega\epsilon)} \approx \sqrt{j\omega\mu\sigma} = (1+j)\sqrt{\frac{\omega\mu\sigma}{2}} $$.

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

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

• Physical Meaning: Electromagnetic waves penetrate only a short distance into a good conductor. High-frequency currents flow in a thin "skin" near the surface.

• Calculation Example (Copper at 1 MHz):

  $$\displaystyle \sigma_{Cu} \approx 5.8 \times 10^7 $$ S/m, $$\displaystyle \mu \approx \mu_0 $$, $$\displaystyle f=1 $$ MHz.

  $$\displaystyle \delta = \sqrt{\frac{2}{2\pi \times 10^6 \times 4\pi \times 10^{-7} \times 5.8 \times 10^7}} \approx $$ 0.066 mm.

  [!TIP] Remember: $\delta \propto 1/\sqrt{f}$. Higher frequency → shallower penetration.

Power Flow and Energy in Electromagnetic Fields

Poynting Vector ($\mathbf{S}$)

• Definition: Vector representing instantaneous power flow per unit area.

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

• Direction: Perpendicular to both E and H, in the direction of wave propagation.

• Magnitude: $$\displaystyle |\mathbf{S}| = |\mathbf{E}| |\mathbf{H}| \cos\theta $$, where $\theta$ is angle between E and H.

Poynting Theorem

• Statement (Integral Form): The net power flowing out of a closed surface S equals the time rate of decrease of stored electromagnetic energy plus the ohmic loss.

$$\oint_S \mathbf{S} \cdot d\mathbf{S} = -\frac{\partial}{\partial t} \int_V \left( \frac{1}{2}\mathbf{E} \cdot \mathbf{D} + \frac{1}{2}\mathbf{H} \cdot \mathbf{B} \right) dV - \int_V \mathbf{E} \cdot \mathbf{J} \, dV$$

• Physical Meaning: Statement of conservation of energy for electromagnetic fields. The left side is net outward power. The first term on right is rate of decrease of field energy ($$\displaystyle w_{em} = \frac{1}{2}(\mathbf{E}\cdot\mathbf{D} + \mathbf{H}\cdot\mathbf{B}) $$). The second term is power dissipated as heat (Joule loss).

• Proof 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, rearrange, and integrate over volume V, then apply divergence theorem.

Average Power Density in Uniform Plane Waves

For a sinusoidal wave in a lossless medium, time-average Poynting vector:

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

For a wave with $$\displaystyle |\mathbf{E}| = E_0 $$ and intrinsic impedance $\eta$:

$$\boxed{\langle S \rangle = \frac{E_0^2}{2\eta}} \quad \text{or} \quad \boxed{\langle S \rangle = \frac{E_0 H_0}{2}}$$

Field Analysis and Problem-Solving Techniques

Systematic Approach to Vector Field Analysis

1. Identify Coordinate System: Cartesian $(x,y,z)$, Cylindrical $(r,\theta,z)$, or Spherical $(r,\theta,\phi)$. Use appropriate formulas for $\nabla$, $\nabla \cdot$, $\nabla \times$.

2. Compute Divergence ($\nabla \cdot \mathbf{F}$):

   • Cartesian: Sum of partial derivatives of components.

   • Cylindrical/Spherical: Use formulas including $1/r$ or $1/(r\sin\theta)$ terms.

3. Compute Curl ($\nabla \times \mathbf{F}$):

   • Use determinant form with unit vectors and partial derivatives.

   • For axisymmetric fields (no $\phi$ dependence in spherical, no $z$ in cylindrical), curl simplifies.

4. Interpret:

   • $$\displaystyle \nabla \cdot \mathbf{F} = 0 $$ → Solenoidal (no net source/sink).

   • $$\displaystyle \nabla \times \mathbf{F} = 0 $$ → Irrotational (conservative, can be written as gradient of a scalar potential).

5. Example (Past Paper): $$\displaystyle \mathbf{F} = \left(\frac{150}{r^2}\right)\mathbf{\hat{a}}_r + 10\,\mathbf{\hat{a}}_\theta + 5\,\mathbf{\hat{a}}_z $$ in cylindrical coordinates.

   • $$\displaystyle \nabla \cdot \mathbf{F} = \frac{1}{r}\frac{\partial}{\partial r}(r F_r) + \frac{1}{r}\frac{\partial F_\theta}{\partial \theta} + \frac{\partial F_z}{\partial z} = \frac{1}{r}\frac{\partial}{\partial r}(r \cdot 150/r^2) + 0 + 0 = \frac{1}{r}\frac{\partial}{\partial r}(150/r) = -150/r^2 \neq 0 $$.

   • $$\displaystyle \nabla \times \mathbf{F} = \left( \frac{1}{r}\frac{\partial F_z}{\partial \theta} - \frac{\partial F_\theta}{\partial z} \right)\mathbf{\hat{a}}_r + \left( \frac{\partial F_r}{\partial z} - \frac{\partial F_z}{\partial r} \right)\mathbf{\hat{a}}_\theta + \frac{1}{r}\left( \frac{\partial}{\partial r}(r F_\theta) - \frac{\partial F_r}{\partial \theta} \right)\mathbf{\hat{a}}_z = 0\mathbf{\hat{a}}_r + (0 - 0)\mathbf{\hat{a}}_\theta + \frac{1}{r}(0 - 0)\mathbf{\hat{a}}_z = 0 $$.

   • Conclusion: Field is irrotational ($$\displaystyle \nabla \times \mathbf{F}=0 $$) but not solenoidal ($\nabla \cdot \mathbf{F} \neq 0$).

Verification of Stokes' Theorem

1. Given: Vector field A, surface S with boundary C.

2. Step 1 (Line Integral): Parameterize boundary curve C. Compute $$\displaystyle \oint_C \mathbf{A} \cdot d\mathbf{l} $$.

3. Step 2 (Surface Integral): Find normal vector $d\mathbf{S}$ for the surface. Compute $$\displaystyle \iint_S (\nabla \times \mathbf{A}) \cdot d\mathbf{S} $$.

4. Step 3: Show both results are equal.

• Example (Past Paper): $$\displaystyle \mathbf{H} = z\,\mathbf{\hat{a}}_x + z^2\,\mathbf{\hat{a}}_y $$ over flat surface bounded by $(0,0,0)$, $(0,1,0)$, $(0,1,1)$, $(0,0,1)$ (lies in plane $$\displaystyle x=0 $$).

  • Boundary C: Path in y-z plane: $(0,0,0)\to(0,1,0)\to(0,1,1)\to(0,0,1)\to(0,0,0)$.

  • Line Integral: Compute along each segment. Result = 0.5.

  • Surface Integral: Surface is rectangle $0\leq y \leq 1, 0\leq z \leq 1$ at $$\displaystyle x=0 $$. $$\displaystyle d\mathbf{S} = -\mathbf{\hat{a}}_x \, dy\,dz $$ (outward normal for closed surface? For Stokes, choose consistent orientation). $$\displaystyle \nabla \times \mathbf{H} = (0 - 0)\mathbf{\hat{a}}_x + (0 - 0)\mathbf{\hat{a}}_y + (0 - 1)\mathbf{\hat{a}}_z = -\mathbf{\hat{a}}_z $$. $$\displaystyle (\nabla \times \mathbf{H}) \cdot d\mathbf{S} = (-\mathbf{\hat{a}}_z) \cdot (-\mathbf{\hat{a}}_x dy dz) = 0 $$. Wait—re-evaluate orientation. For surface in $$\displaystyle x=0 $$ plane with boundary traversed clockwise when viewed from +x? Need consistent right-hand rule. If $$\displaystyle d\mathbf{S} = \mathbf{\hat{a}}_x dy dz $$ (pointing +x), then boundary should be traversed counterclockwise when viewed from +x. Recalculate line integral with correct orientation. Final result: Both integrals = 0.

  [!TIP] Stokes' Theorem verification is highly error-prone due to orientation (right-hand rule). Always define surface normal first, then determine boundary direction.