Unit 2 | Engineering Physics Notes | AKTU Notes

UNIT 2: Electromagnetic Field Theory

2.1 Stokes' Theorem and Divergence Theorem

These are two important mathematical theorems used in electromagnetic theory to convert between line/surface integrals and volume integrals.

Stokes' Theorem:

• Relates a line integral around a closed curve to a surface integral over the surface bounded by that curve.
• Formula: ∮ F · dl = ∬ (∇ × F) · dS
• In simple words: The circulation of a vector field around a closed loop equals the curl of that field integrated over the surface.
• Used to convert Faraday's and Ampere's laws from integral form to differential form.

Divergence Theorem (Gauss's Theorem):

• Relates the flux through a closed surface to the volume integral of divergence inside.
• Formula: ∯ F · dS = ∭ (∇ · F) dV
• In simple words: The total outward flux of a vector field through a closed surface equals the divergence of that field integrated over the volume inside.
• Used to convert Gauss's law from integral to differential form.

2.2 Basic Laws of Electricity and Magnetism

1. Gauss's Law for Electric Field:

• The total electric flux through any closed surface equals the total charge enclosed divided by ε₀.
• Formula: ∯ E · dS = Q/ε₀ (integral form) | ∇ · E = ρ/ε₀ (differential form)

2. Gauss's Law for Magnetic Field:

• Magnetic monopoles do not exist. Magnetic field lines always form closed loops.
• Formula: ∯ B · dS = 0 (integral) | ∇ · B = 0 (differential)

3. Faraday's Law of Electromagnetic Induction:

• A changing magnetic field induces an electric field (EMF).
• Formula: ∮ E · dl = −dΦB/dt (integral) | ∇ × E = −∂B/∂t (differential)

4. Ampere's Law:

• A current-carrying conductor produces a magnetic field around it.
• Formula: ∮ B · dl = μ₀I (integral form)

2.3 Continuity Equation for Current Density

The continuity equation expresses the conservation of electric charge. Charge can neither be created nor destroyed.

Derivation idea:

• If current flows out of a volume, the charge inside must decrease.
• Rate of decrease of charge inside = current flowing out.

Formula:

• ∇ · J = −∂ρ/∂t
• Where J = current density (current per unit area), ρ = charge density.
• This means: divergence of current density = rate of decrease of charge density.

In steady state: ∂ρ/∂t = 0 → ∇ · J = 0 (charge does not accumulate anywhere).

2.4 Displacement Current

This is one of Maxwell's greatest contributions. He noticed that Ampere's law was incomplete.

The Problem:

• Consider a capacitor being charged. Current flows in the wire but NOT between the capacitor plates (there is no physical current there — it's a gap).
• Ampere's law gave inconsistent results for the magnetic field around this circuit.

Maxwell's Solution — Displacement Current:

• Even though no real current flows between the capacitor plates, the electric field between the plates is changing.
• Maxwell said: a changing electric field is equivalent to a current called displacement current.
• Displacement current density: Jd = ε₀ (∂E/∂t)

Modified Ampere's Law:

• ∮ B · dl = μ₀(I + Id) = μ₀I + μ₀ε₀ (∂E/∂t)
• Differential form: ∇ × B = μ₀J + μ₀ε₀ (∂E/∂t)

Significance:

• Displacement current predicted that changing electric fields produce magnetic fields.
• This led directly to the prediction of electromagnetic waves.

2.5 Maxwell's Equations

Maxwell's equations are the four fundamental equations of electromagnetism. Together, they describe all classical electromagnetic phenomena.

Equation	Integral Form	Differential Form	Meaning
Gauss's Law (Electric)	∯ E·dS = Q/ε₀	∇·E = ρ/ε₀	Electric charges produce electric fields
Gauss's Law (Magnetic)	∯ B·dS = 0	∇·B = 0	No magnetic monopoles exist
Faraday's Law	∮ E·dl = −dΦB/dt	∇×E = −∂B/∂t	Changing magnetic field produces electric field
Ampere-Maxwell Law	∮ B·dl = μ₀(I+Id)	∇×B = μ₀J + μ₀ε₀∂E/∂t	Current and changing electric field produce magnetic field

2.6 Maxwell's Equations in Vacuum and Conducting Medium

In Vacuum (Free Space):

• No free charges: ρ = 0
• No conduction current: J = 0
• Maxwell's equations simplify to: ∇·E = 0, ∇·B = 0, ∇×E = −∂B/∂t, ∇×B = μ₀ε₀ ∂E/∂t

In Conducting Medium:

• Free charges exist: ρ ≠ 0
• Conduction current exists: J = σE (Ohm's law in vector form), where σ = conductivity.
• Modified Ampere's law: ∇×H = J + ∂D/∂t = σE + ε ∂E/∂t
• These equations explain why EM waves get attenuated (weakened) inside conductors.

2.7 Poynting Vector and Poynting Theorem

Poynting Vector (S):

• The Poynting vector represents the direction and rate of energy flow in an electromagnetic field.
• Formula: S = E × H (cross product of electric and magnetic field vectors)
• Unit: W/m² (watts per square meter)
• Direction: perpendicular to both E and H — it tells us which direction EM energy is traveling.

Poynting Theorem:

• This is the law of conservation of energy for electromagnetic fields.
• It states: The rate of energy leaving a region through electromagnetic radiation = rate of decrease of stored energy inside − power dissipated as heat.
• Mathematical form: −∂u/∂t = ∇ · S + J · E
• Where u = electromagnetic energy density, J·E = power dissipated per unit volume.

2.8 Plane Electromagnetic Waves in Vacuum and Their Transverse Nature

Electromagnetic Wave in Vacuum:

• From Maxwell's equations in free space, we can derive the wave equations for E and B.
• Wave equation: ∇²E = μ₀ε₀ ∂²E/∂t²
• This shows E (and B) travel as waves with speed: c = 1/√(μ₀ε₀) = 3 × 10⁸ m/s (speed of light).

Transverse Nature:

• In an EM wave, the electric field E and magnetic field B are perpendicular to each other.
• Both E and B are also perpendicular to the direction of wave propagation.
• This makes EM waves transverse waves (unlike sound which is longitudinal).
• If wave travels in z-direction: E is along x, B is along y, propagation along z.

2.9 Relation Between Electric and Magnetic Fields of an EM Wave

• The magnitudes of E and B are related by: E/B = c (speed of light)
• In a medium: E/B = v (speed of wave in that medium)
• E and B oscillate in phase — they reach maximum and minimum at the same time.
• The ratio E/H = √(μ/ε) is called the intrinsic impedance of the medium.
• For free space: E/H = √(μ₀/ε₀) = 377 Ω

2.10 Plane Electromagnetic Waves in Conducting Medium and Skin Depth

EM Waves in Conductors:

• When an EM wave enters a conducting medium, it gets attenuated (its amplitude decreases exponentially).
• This is because free electrons in the conductor absorb the wave energy and convert it to heat.
• The wave amplitude decays as: E = E₀ e^(−αz) where α = attenuation constant.

Skin Depth (δ):

• Skin depth is the depth at which the amplitude of the EM wave falls to 1/e (about 37%) of its surface value.
• Formula: δ = 1/α = √(2/ωμσ) = √(2ρ/ωμ)
• Where ω = angular frequency, μ = permeability, σ = conductivity.
• At higher frequencies, skin depth is smaller — the wave penetrates less into the conductor.

Skin Effect:

• At high frequencies, current flows only near the surface of a conductor (within a skin depth).
• This is called the skin effect.
• This is why high frequency cables use hollow or stranded conductors — the inside is not used anyway.