Unit 1: Electromagnetic Theory

1. Vector Calculus Foundations in Electromagnetism

Electromagnetic theory relies on vector differential operators to describe spatial variations in scalar potentials and vector fields across three-dimensional space:

Gradient of a Scalar Field ($\nabla V$)

For an electrostatic potential $V(x, y, z)$:
$\vec{E} = -\nabla V = -\left( \hat{i} \frac{\partial V}{\partial x} + \hat{j} \frac{\partial V}{\partial y} + \hat{k} \frac{\partial V}{\partial z} \right)$
The gradient vector points in the direction of the maximal rate of spatial increase of $V$, with its magnitude representing the steepest directional derivative.

Divergence of a Vector Field ($\nabla \cdot \vec{A}$)

$\nabla \cdot \vec{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}$

• Physically measures net outward flux per unit volume (source density).
• If $\nabla \cdot \vec{A} > 0$, the point acts as a source; if $\nabla \cdot \vec{A} < 0$, it is a sink.
• If $\nabla \cdot \vec{A} = 0$, the field is solenoidal (e.g., magnetic induction $\vec{B}$).

Curl of a Vector Field ($\nabla \times \vec{A}$)

$\nabla \times \vec{A} = \det \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ A_x & A_y & A_z \end{bmatrix}$

• Measures circulation density or local rotational vortex strength per unit area.
• If $\nabla \times \vec{A} = 0$, the field is irrotational (conservative) (e.g., electrostatic field $\vec{E}$).

Fundamental Integral Theorems:

1. Gauss's Divergence Theorem: $\iiint_V (\nabla \cdot \vec{A}) \, dV = \oiint_S \vec{A} \cdot d\vec{S}$
2. Stokes' Theorem: $\iint_S (\nabla \times \vec{A}) \cdot d\vec{S} = \oint_C \vec{A} \cdot d\vec{r}$
3. Null Identities: $\nabla \times (\nabla V) = 0$ (curl of gradient is identically zero); $\nabla \cdot (\nabla \times \vec{A}) = 0$ (divergence of curl is identically zero).

2. Charge Conservation and the Continuity Equation

The principle of conservation of electric charge asserts that the net outward electric current through a closed boundary surface equals the rate of decrease of enclosed charge:
$I_{\text{out}} = \oiint_S \vec{J} \cdot d\vec{S} = -\frac{d Q_{\text{enc}}}{dt} = -\frac{d}{dt} \iiint_V \rho \, dV$
Applying Gauss's divergence theorem transforms the surface integral into a volume integral:
$\iiint_V \left( \nabla \cdot \vec{J} + \frac{\partial \rho}{\partial t} \right) dV = 0$
Since this holds for an arbitrary volume $V$, the differential Equation of Continuity is:
$\nabla \cdot \vec{J} = -\frac{\partial \rho}{\partial t}$
For stationary direct currents (steady state, $\frac{\partial \rho}{\partial t} = 0$), the equation simplifies to $\nabla \cdot \vec{J} = 0$.

3. Inconsistency of Ampere's Law and Maxwell's Displacement Current

Ampere's Circuital Law for magnetostatics states:
$\nabla \times \vec{B} = \mu_0 \vec{J}$
Taking the divergence of both sides yields:
$\nabla \cdot (\nabla \times \vec{B}) = \mu_0 (\nabla \cdot \vec{J})$
Because the divergence of any curl is identically zero, the left side vanishes: $0 = \mu_0 (\nabla \cdot \vec{J}) \implies \nabla \cdot \vec{J} = 0$.
However, by the continuity equation, $\nabla \cdot \vec{J} = -\frac{\partial \rho}{\partial t} \neq 0$ for time-dependent electromagnetic phenomena (such as charging a capacitor). Thus, Ampere's law was mathematically inconsistent.

Maxwell's Formulation of Displacement Current:

Using Gauss's law $\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0} \implies \rho = \epsilon_0 (\nabla \cdot \vec{E})$, substituting into the continuity equation gives:
$\nabla \cdot \vec{J} + \frac{\partial}{\partial t} [\epsilon_0 (\nabla \cdot \vec{E})] = 0 \implies \nabla \cdot \left( \vec{J} + \epsilon_0 \frac{\partial \vec{E}}{\partial t} \right) = 0$
Maxwell recognized that the total current density consists of two components:
$\vec{J}_{\text{total}} = \vec{J} + \vec{J}_d, \quad \text{where } \vec{J}_d = \frac{\partial \vec{D}}{\partial t} = \epsilon_0 \frac{\partial \vec{E}}{\partial t}$
Here, $\vec{J}_d$ is the displacement current density. Between capacitor plates where no conduction electrons flow ($\vec{J} = 0$), the time-varying electric field generates displacement current, ensuring current continuity throughout the circuit.

4. Maxwell's Equations (Differential and Integral Forms)

In linear, isotropic media with permittivity $\epsilon$ and permeability $\mu$:

5. Electromagnetic Wave Propagation in Free Space

In non-conducting free space (vacuum: $\rho = 0, \vec{J} = 0, \epsilon = \epsilon_0, \mu = \mu_0$), Maxwell's equations reduce to:

1. $\nabla \cdot \vec{E} = 0$
2. $\nabla \cdot \vec{B} = 0$
3. $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$
4. $\nabla \times \vec{B} = \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}$

Derivation of the Electromagnetic Wave Equation:

Taking the curl of Faraday's law:
$\nabla \times (\nabla \times \vec{E}) = -\frac{\partial}{\partial t} (\nabla \times \vec{B})$
Using the vector identity $\nabla \times (\nabla \times \vec{E}) = \nabla(\nabla \cdot \vec{E}) - \nabla^2 \vec{E}$, and substituting $\nabla \cdot \vec{E} = 0$ and $\nabla \times \vec{B} = \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}$:
$-\nabla^2 \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} \implies \nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}$
Similarly, for the magnetic induction vector:
$\nabla^2 \vec{B} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{B}}{\partial t^2}$

Comparing with the standard three-dimensional wave equation $\nabla^2 \psi = \frac{1}{v^2} \frac{\partial^2 \psi}{\partial t^2}$ establishes that electromagnetic disturbances propagate as transverse waves with speed:
$c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} = \frac{1}{\sqrt{(4\pi \times 10^{-7})(8.854 \times 10^{-12})}} \approx 2.998 \times 10^8 \text{ m/s}$
This historic derivation confirmed that light is fundamentally an electromagnetic wave.

6. Energy Density and the Poynting Vector

Electromagnetic fields carry both mechanical momentum and radiant energy. The total instantaneous electromagnetic energy density $u$ stored in free space is:
$u = u_E + u_B = \frac{1}{2} \epsilon_0 E^2 + \frac{1}{2\mu_0} B^2$
In a propagating plane wave, electric and magnetic energy densities are identically equal: $u_E = u_B$.

Poynting Vector ($\vec{S}$):

The spatial flow rate of electromagnetic energy per unit area per second (power flux density in $\text{W/m}^2$) is quantified by the Poynting Vector:
$\vec{S} = \vec{E} \times \vec{H} = \frac{1}{\mu_0} (\vec{E} \times \vec{B})$

• Points strictly in the direction of wave propagation $(\hat{k} = \hat{E} \times \hat{B})$.
• Poynting's Theorem:
  $-\nabla \cdot \vec{S} = \frac{\partial u}{\partial t} + \vec{J} \cdot \vec{E}$
  The net power entering a volume equals the rate of increase of stored electromagnetic energy plus the Joule dissipation rate $(\vec{J} \cdot \vec{E})$.