Module 2: Electromagnetic Field Theory

Complete Notes with Formulas and Clear Explanation

1. Continuity Equation for Current Density

Law of conservation of electric charge in local (differential) form:

∇ · J + ∂ρ/∂t = 0

ρ → charge density, J → current density
Meaning: The rate at which charge decreases inside a volume equals the current flowing out.

2. Displacement Current

Original Ampère’s law (steady currents): ∇ × B = μ₀ J → fails when charges are accumulating (e.g., charging capacitor).

Maxwell introduced displacement current density:

J_D = ε₀ ∂E/∂t

Total current through any surface = conduction current + displacement current.

3. Modified Ampère’s Law (Ampère-Maxwell Law)

∇ × B = μ₀ (J + ε₀ ∂E/∂t)

In vacuum or non-conducting medium (J = 0): ∇ × B = μ₀ ε₀ ∂E/∂t

4. Maxwell’s Equations in Vacuum (Non-conducting Medium)

Name	Differential Form	Physical Meaning
Gauss’s law (E)	∇ · E = 0	No free charges in vacuum
Gauss’s law (B)	∇ · B = 0	No magnetic monopoles
Faraday’s law	∇ × E = −∂B/∂t	Changing B → induced E
Ampère-Maxwell law	∇ × B = μ₀ ε₀ ∂E/∂t	Changing E → induced B

5. Energy Stored in Electromagnetic Field

Electric field energy density: u_E = ½ ε₀ E²
Magnetic field energy density: u_B = B²/(2μ₀)

Total energy density u = ½ ε₀ E² + B²/(2μ₀)

Total energy in volume V: U = ∫ u dV

6. Poynting Vector

Energy flux (power per unit area):

S = (1/μ₀) (E × B)

Direction of S = direction of energy flow. Units: W/m²

7. Poynting Theorem (Conservation of Energy)

− dU/dt = ∮ S · dA + ∫ E · J dV

Decrease in field energy = energy flowing out + energy dissipated as heat/work.

8. Plane Electromagnetic Waves in Vacuum

Wave equations derived from Maxwell’s equations:

∇²E = μ₀ ε₀ ∂²E/∂t² ∇²B = μ₀ ε₀ ∂²B/∂t²

Speed of wave:

c = 1 / √(μ₀ ε₀) = 3 × 10⁸ m/s

9. Transverse Nature of Plane EM Waves

E, B and propagation direction (k) are mutually perpendicular
E × B gives direction of propagation
Magnitude relation: E = c B or E₀ = c B₀
E and B are in phase

Instantaneous Poynting vector: S = (E² / (c μ₀)) k̂

10. Average Energy and Momentum Carried by EM Waves

Average intensity (time-averaged power per unit area):

I = ⟨S⟩ = (1/2) c ε₀ E₀² = (c B₀²)/(2 μ₀) = (E₀ B₀)/(2 μ₀)

Momentum density of EM field = energy density / c
Momentum delivered per unit area per second = I / c

11. Radiation Pressure

Surface	Radiation Pressure
Perfect absorber (black body)	P = I / c
Perfect reflector	P = 2I / c

12. Skin Depth in Conductors

EM wave inside good conductor decays exponentially:

E(z,t) = E₀ e^−z/δ cos(ωt − kz) (approximately)

Skin depth (distance where amplitude drops to 1/e):

δ = √(2 / (ω μ σ))

ω = angular frequency, σ = conductivity, μ ≈ μ₀
At high frequencies or good conductors → δ is very small → current flows only near surface (skin effect).

Summary of Key Formulas

Speed of light	c = 1/√(μ₀ ε₀)
Displacement current	J_D = ε₀ ∂E/∂t
Ampère-Maxwell law	∇ × B = μ₀ J + μ₀ ε₀ ∂E/∂t
Poynting vector	S = (E × B)/μ₀
Average intensity	I = ½ c ε₀ E₀²
E–B relation	E₀ = c B₀
Radiation pressure (absorber)	P = I/c
Radiation pressure (reflector)	P = 2I/c
Skin depth	δ = √(2/ωμσ)

These equations form the complete foundation of classical electromagnetism and explain all electromagnetic waves including radio waves, microwaves, light, X-rays, etc.