Unit 1: Electromagnetic Theory

1. Vector Calculus and Differential Field Operators

Electromagnetic theory describes the behavior of electric charges, currents, and electromagnetic radiation through spatial vector and scalar fields. A scalar field maps a scalar value to every coordinate in space (e.g., electric potential $V(x, y, z)$ or temperature $T(x, y, z)$), whereas a vector field assigns a directed magnitude (e.g., electric field intensity ec{E}(x, y, z) or magnetic flux density ec{B}(x, y, z)).

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Vector Field Classification:
├── Irrotational / Conservative:  ∇ × A = 0  (Can be expressed as A = -∇V)
├── Solenoidal / Divergence-free: ∇ · A = 0  (Can be expressed as A = ∇ × Ψ)
└── General Helmholtz Vector:     A = -∇V + ∇ × Ψ  (Superposition of irrotational & solenoidal)

1.1 The Gradient of a Scalar Field ($

abla \phi$) The gradient operator$
abla$operating on a continuously differentiable scalar field$\phi(x,y,z)$produces a vector field pointing along the trajectory of maximum spatial rate of increase of$\phi$. Its magnitude equals that maximal directional rate of increase:

abla \cdot ec{A} = \lim_{\Delta V o 0} rac{1}{\Delta V} \oiint_S ec{A} \cdot dec{S} = rac{\partial A_x}{\partial x} + rac{\partial A_y}{\partial y} + rac{\partial A_z}{\partial z}$$

• Physical Significance:
  • abla \cdot ec{A} > 0: The point acts as a net positive source (e.g., positive electric charge creating diverging ec{E}).
  • abla \cdot ec{A} < 0: The point acts as a net negative sink (e.g., negative electric charge absorbing ec{E}).
  • abla \cdot ec{A} = 0: The vector field is solenoidal (no isolated sources or sinks; field lines form closed, continuous loops, as in ec{B}).

1.3 The Curl of a Vector Field ($

abla imes ec{A}$)
The curl evaluates the circulation density or rotational vorticity of a vector field per unit area around an infinitesimal closed loop:

abla imes ec{A} = egin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ rac{\partial}{\partial x} & rac{\partial}{\partial y} & rac{\partial}{\partial z} \ A_x & A_y & A_z \end{vmatrix} = \left(rac{\partial A_z}{\partial y} - rac{\partial A_y}{\partial z}
ight)\hat{i} + \left(rac{\partial A_x}{\partial z} - rac{\partial A_z}{\partial x}
ight)\hat{j} + \left(rac{\partial A_y}{\partial x} - rac{\partial A_x}{\partial y}
ight)\hat{k}$$

• Physical Significance:
  • abla imes ec{A} e 0: The field contains rotational circulation (e.g., magnetic field circling an electric current).
  • abla imes ec{A} = 0: The field is irrotational or conservative (e.g., static electrostatic field).

1.4 Fundamental Vector Identities and the Laplacian Operator

Two foundational null identities underpin classical electromagnetic gauge theory:

1. The Curl of a Gradient is Identically Zero:

abla \cdot (
abla imes ec{A}) = 0$$

Physical Consequence: Any solenoidal vector field ec{B} ( abla \cdot ec{B} = 0) can be represented as the curl of a magnetic vector potential: ec{B} = abla imes ec{A}.

3. The Laplacian Operator ($abla^2$):
   • For a scalar field: abla^2 \phi = abla \cdot ( abla \phi) = rac{\partial^2 \phi}{\partial x^2} + rac{\partial^2 \phi}{\partial y^2} + rac{\partial^2 \phi}{\partial z^2}
   • Vector Laplacian identity: abla^2 ec{A} = abla( abla \cdot ec{A}) - abla imes ( abla imes ec{A})

1.5 Integral Calculus: Gauss's Divergence Theorem and Stokes' Theorem

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                       [Gauss's Divergence Theorem]
              Closed Surface Integral ◄───► Volume Integral
                  ∬_S A · dS  =  ∭_V (∇ · A) dV
                                   ▲
                                   │
                       [Stokes' Curl Theorem]
                Closed Line Integral ◄───► Open Surface Integral
                   ∮_C A · dr  =  ∬_S (∇ × A) · dS

• Gauss's Divergence Theorem: Transforms a flux integral over a closed boundary surface $S$ enclosing volume $V$ into an integral of the divergence over the enclosed volume:

abla \cdot ec{E} = rac{
ho}{arepsilon_0} \quad \implies \quad
abla \cdot ec{D} =
ho_f$$

Where ec{D} = arepsilon ec{E} = arepsilon_0 ec{E} + ec{P} is the electric displacement vector, ec{P} is the electric polarization vector, and $ho_f$ is the volume density of free charges.

2.2 Electrostatic Potential and Laplace/Poisson Equations

Because static electric fields are irrotational ( abla imes ec{E} = 0), the electric field can be expressed as the negative gradient of an electrostatic scalar potential $V$:

abla \cdot (-
abla V) = rac{
ho}{arepsilon} \implies
abla^2 V = -rac{
ho}{arepsilon}$$

In charge-free regions ($ho = 0$), Poisson's equation reduces to Laplace's Equation:

abla imes ec{B} = \mu_0 ec{J} \quad \iff \quad
abla imes ec{H} = ec{J}_f$$

Because magnetic monopoles do not exist in nature, magnetic flux lines form continuous loops, establishing Gauss's Law for Magnetism:

abla^2 ec{A} = -\mu_0 ec{J}$$

3. Electrodynamics and Maxwell's Formulation

3.1 Faraday's Law of Electromagnetic Induction

Michael Faraday demonstrated that a time-varying magnetic flux induces an electromotive force (EMF) in a closed conductive circuit:

\mathcal{E} = -rac{d\Phi_B}{dt} = -rac{d}{dt} \iint_S ec{B} \cdot dec{S}

The negative sign embodies Lenz's Law (conservation of energy). Using Stokes' Theorem:

abla \cdot (
abla imes ec{H}) =
abla \cdot ec{J}$$

From vector calculus, the divergence of any curl is identically zero: abla \cdot ( abla imes ec{H}) \equiv 0. Consequently, static Ampère's law demands:

abla \cdot ec{J} + rac{\partial
ho}{\partial t} = 0 \implies
abla \cdot ec{J} = -rac{\partial
ho}{\partial t}$$

In time-varying scenarios (such as charging a capacitor), $\partial ho / \partial t e 0$, exposing a fatal mathematical contradiction in Ampère's static formulation.

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Charging Capacitor Circuit Dilemma:
             Plate +Q         Plate -Q
   ─────────────| |─────────────
   Current I_c  | | (Dielectric) Current I_c
                | |
   Conduction   | | Displacement  Conduction
   Current J_c  | | Current J_d   Current J_c

To resolve this inconsistency, James Clerk Maxwell recognized that Gauss's Law relates charge density to electric displacement:

abla \cdot ec{J} +
abla \cdot \left(rac{\partial ec{D}}{\partial t}
ight) = 0 \implies
abla \cdot \left( ec{J} + rac{\partial ec{D}}{\partial t}
ight) = 0$$

Maxwell postulated that total current density consists of conduction current density ec{J}_c and Displacement Current Density ec{J}_d:

ec{J}_d = rac{\partial ec{D}}{\partial t} = arepsilon_0 rac{\partial ec{E}}{\partial t} + rac{\partial ec{P}}{\partial t}

Displacement current represents a changing electric flux; it carries no physical charge carriers through space yet generates an identical surrounding magnetic field.

3.3 The Complete Set of Maxwell's Equations

In isotropic, linear, and homogeneous media, the constitutive relations apply:

ec{D} = arepsilon ec{E} = arepsilon_r arepsilon_0 ec{E}, \quad ec{B} = \mu ec{H} = \mu_r \mu_0 ec{H}, \quad ec{J} = \sigma ec{E}

4. Electromagnetic Wave Equations and Propagation in Media

4.1 Derivation of the Electromagnetic Wave Equation in Free Space

In source-free, non-conducting vacuum ( ho = 0, ec{J} = 0, \sigma = 0, arepsilon = arepsilon_0, \mu = \mu_0), Maxwell's equations simplify to:

1. abla \cdot ec{E} = 0
2. abla \cdot ec{B} = 0
3. abla imes ec{E} = -rac{\partial ec{B}}{\partial t}
4. abla imes ec{B} = \mu_0 arepsilon_0 rac{\partial ec{E}}{\partial t}

Taking the curl of Faraday's Law:

abla^2 ec{E} - \mu_0 arepsilon_0 rac{\partial^2 ec{E}}{\partial t^2} = 0$$

Similarly, taking the curl of the Ampère-Maxwell law yields the wave equation for the magnetic field:

abla^2 ec{E} - \mu \sigma rac{\partial ec{E}}{\partial t} - \mu arepsilon rac{\partial^2 ec{E}}{\partial t^2} = 0$$

Assuming time-harmonic solutions of the form $e^{i(kz - \omega t)}$, the complex propagation constant \gamma = lpha + ieta is:

\gamma^2 = -\omega^2 \mu arepsilon + i\omega \mu \sigma

Where:

• lpha: Attenuation constant (nepers/meter)
• eta: Phase propagation constant (radians/meter)

For a good conductor (rac{\sigma}{\omega arepsilon} \gg 1):

lpha pprox eta pprox \sqrt{rac{\omega \mu \sigma}{2}} = \sqrt{\pi f \mu \sigma}

The electric field decays exponentially as it penetrates the conductor:

ec{E}(z, t) = E_0 e^{-lpha z} \cos(eta z - \omega t) \hat{i}

• Skin Depth ($\delta$): The penetration distance over which the wave amplitude attenuates to $1/e$ (pprox 36.8\%) of its initial surface value:

\delta = rac{1}{lpha} = \sqrt{rac{2}{\omega \mu \sigma}} = rac{1}{\sqrt{\pi f \mu \sigma}}

High-Frequency Implications: At microwave and radio frequencies, electrical conduction is confined exclusively to a microscopic outer layer of the conductor, increasing effective AC resistance.

5. Poynting Vector and Energy Conservation

5.1 Poynting's Theorem

Electromagnetic fields store potential and kinetic energy in space. The instantaneous volumetric energy density $u$ is the sum of electrostatic and magnetostatic energy densities:

u = u_E + u_B = rac{1}{2}arepsilon |ec{E}|^2 + rac{1}{2\mu}|ec{B}|^2 = rac{1}{2}(ec{E} \cdot ec{D} + ec{B} \cdot ec{H})

Differentiating with respect to time:

rac{\partial u}{\partial t} = ec{E} \cdot rac{\partial ec{D}}{\partial t} + ec{H} \cdot rac{\partial ec{B}}{\partial t}

Invoking Maxwell's curl equations (rac{\partial ec{D}}{\partial t} = abla imes ec{H} - ec{J} and rac{\partial ec{B}}{\partial t} = - abla imes ec{E}):