Unit 5: Maxwell's Equations and Electromagnetic Wave Propagation

1. Equation of Continuity of Current

The equation of continuity is a mathematical expression of the fundamental law of conservation of electric charge. It states that charge can neither be created nor destroyed; the rate at which charge leaves a volume must equal the rate at which the internal charge decreases.

Mathematical Derivation

Consider a closed surface S enclosing a volume V containing a total electric charge Q. The total electric current I flowing out of this volume through the surface S is given by the surface integral of the current density vector J:

I = ∯ J · dS

By the principle of conservation of charge, this outgoing current must equal the rate of decrease of charge within the volume V:

I = -dQ / dt

The total charge Q inside volume V is the volume integral of the charge density ρ:

Q = ∭ ρ dV

Substituting Q into the current expression gives:

∯ J · dS = -d/dt ∭ ρ dV = ∭ (-∂ρ / ∂t) dV

Applying Gauss's Divergence Theorem to convert the surface integral on the left side into a volume integral:

∯ J · dS = ∭ (∇ · J) dV

Equating both volume integrals:

∭ (∇ · J) dV = ∭ (-∂ρ / ∂t) dV

Since this equality holds for any arbitrary volume V, the integrands must be equal:

∇ · J + ∂ρ / ∂t = 0

This is the Differential Form of the Continuity Equation.

Special Cases

• Steady Currents (Steady State): For steady-state currents, the charge density ρ does not change with time (∂ρ / ∂t = 0). Thus, ∇ · J = 0. This means steady current field lines are continuous and form closed loops (no sources or sinks).
• Unsteady Currents (Time-Varying Fields): When charge density accumulates or depletes over time, ∂ρ / ∂t ≠ 0, requiring the full continuity equation.

Key Physical Insights

• ∇ · J > 0: Net positive current divergence; positive charge inside the volume is decreasing with time.
• ∇ · J < 0: Net convergence of current; charge is accumulating inside the volume.
• ∇ · J = 0: Inflow equals outflow; charge density remains constant.

2. Displacement Current

Inconsistency of Ampere's Circuital Law

Ampere's law in its original form states:

∇ × B = μ₀J

Taking the divergence of both sides gives:

∇ · (∇ × B) = μ₀ (∇ · J)

Since the divergence of any curl is identically zero (∇ · (∇ × A) = 0 for any vector A), the left side equals zero:

0 = μ₀ (∇ · J) → ∇ · J = 0

However, from the equation of continuity, ∇ · J = -∂ρ / ∂t. For time-varying fields, ∂ρ / ∂t is generally non-zero. Hence, original Ampere's law is mathematically inconsistent for time-dependent electromagnetic fields.

Maxwell's Modification

To resolve this inconsistency, James Clerk Maxwell added an additional term, called the Displacement Current Density (J_d), to Ampere's law:

∇ × B = μ₀ (J + J_d)

Taking the divergence of both sides:

∇ · (∇ × B) = μ₀ ∇ · (J + J_d) = 0

∇ · J + ∇ · J_d = 0 → ∇ · J_d = -∇ · J

Using the equation of continuity (∇ · J = -∂ρ / ∂t):

∇ · J_d = ∂ρ / ∂t

From Gauss's law for electrostatics, ∇ · E = ρ / ε₀, so ρ = ∇ · (ε₀E). Differentiating with respect to time:

∂ρ / ∂t = ∂/∂t [∇ · (ε₀E)] = ∇ · [∂(ε₀E) / ∂t]

Equating this to ∇ · J_d:

J_d = ∂D / ∂t = ε₀ (∂E / ∂t)

Where D = ε₀E is the electric displacement field. The total displacement current I_d passing through a surface S is:

I_d = ∬ J_d · dS = ε₀ (dΦ_E / dt)

where Φ_E = ∬ E · dS is the electric flux.

Example: Charging Capacitor

Consider a parallel-plate capacitor being charged by an alternating or changing current I:

• In the connecting wire, physical charge flows, generating a conduction current I_c. No displacement current exists in the wire (I_d = 0).
• In the region between the capacitor plates, no real electric charge travels across the gap (I_c = 0). However, the time-varying electric field produces a displacement current I_d.
• The value of I_d inside the gap is exactly equal to I_c in the wires, ensuring current continuity throughout the circuit.

3. Maxwell's Equations

Maxwell's equations are the foundational set of four partial differential equations that describe how electric and magnetic fields are generated and altered by each other and by charges and currents.

Differential and Integral Forms

Maxwell's Equations in Free Space (Vacuum)

In free space, where there are no free charges (ρ = 0) and no conduction currents (J = 0), and where D = ε₀E and B = μ₀H:

1. ∇ · E = 0
2. ∇ · B = 0
3. ∇ × E = -∂B / ∂t
4. ∇ × B = μ₀ε₀ (∂E / ∂t)

4. Vector and Scalar Potentials

Electromagnetic fields E and B can be formulated in terms of potential functions: the Electric Scalar Potential (V) and the Magnetic Vector Potential (A).

Magnetic Vector Potential (A)

According to Gauss's Law for Magnetism, ∇ · B = 0. Using the vector identity that the divergence of any curl is identically zero (∇ · (∇ × A) = 0), we can define B in terms of a vector potential A:

B = ∇ × A

Where A is the Magnetic Vector Potential. Units: Weber/meter (Wb/m) or Tesla-meter (T·m).

Electric Scalar Potential (V)

Substituting B = ∇ × A into Faraday's Law (∇ × E = -∂B / ∂t):

∇ × E = -∂(∇ × A) / ∂t = -∇ × (∂A / ∂t)

∇ × (E + ∂A / ∂t) = 0

Since the curl of any scalar gradient is identically zero (∇ × (∇V) = 0), the vector quantity (E + ∂A / ∂t) can be expressed as the negative gradient of a scalar potential V:

E + ∂A / ∂t = -∇V

E = -∇V - ∂A / ∂t

This equation gives the complete expression for the electric field in dynamic (time-varying) electromagnetic situations.

• In static conditions (∂A / ∂t = 0), it simplifies to the familiar electrostatic equation: E = -∇V.
• In time-varying conditions, the electric field has two sources: static charges (-∇V) and changing magnetic fields (-∂A / ∂t).

5. Gauge Transformations: Lorentz and Coulomb Gauge

Concept of Gauge Transformation

The electromagnetic fields E and B represent physical observables. However, the potentials V and A are not uniquely determined. Multiple sets of potentials (V, A) can produce the exact same physical fields E and B.

A transformation of the potentials that leaves the physical electric and magnetic fields unchanged is called a Gauge Transformation.

If we transform A and V using an arbitrary scalar function ψ(r, t):

A' = A + ∇ψ

V' = V - ∂ψ / ∂t

Checking the resulting physical fields:

B' = ∇ × A' = ∇ × (A + ∇ψ) = ∇ × A + ∇ × ∇ψ = B (since ∇ × ∇ψ = 0)

E' = -∇V' - ∂A' / ∂t = -∇(V - ∂ψ/∂t) - ∂(A + ∇ψ) / ∂t = -∇V + ∇(∂ψ/∂t) - ∂A/∂t - ∇(∂ψ/∂t) = -∇V - ∂A/∂t = E

Since E' = E and B' = B, the physical fields remain invariant under gauge transformations.

1. Coulomb Gauge

The Coulomb Gauge (also known as the Transverse Gauge) imposes the condition:

∇ · A = 0

Substituting ∇ · A = 0 into the differential equations for V yields:

∇²V = -ρ / ε₀

This is Poisson's Equation for the scalar potential. Key characteristics of Coulomb Gauge:

• The scalar potential V is determined instantly throughout space by the charge distribution ρ at the same instant (instantaneous Coulomb potential).
• It is particularly useful in electrostatics, magnetostatics, and quantum electrodynamics when radiation effects are calculated in the absence of source charges.
• It is not explicitly Lorentz covariant (relativistically invariant).

2. Lorentz Gauge

The Lorentz Gauge condition is chosen to treat spatial and temporal components symmetrically:

∇ · A + μ₀ε₀ (∂V / ∂t) = 0

or in terms of speed of light c = 1 / √(μ₀ε₀):

∇ · A + (1 / c²) (∂V / ∂t) = 0

Applying the Lorentz gauge condition uncouples the wave equations for V and A completely:

∇²V - (1 / c²) (∂²V / ∂t²) = -ρ / ε₀

∇²A - (1 / c²) (∂²A / ∂t²) = -μ₀J

Key characteristics of Lorentz Gauge:

• Decouples scalar and vector potentials into identical wave equations with source terms.
• Shows explicitly that potentials propagate at finite wave speed c (retarded potentials).
• Is explicitly Lorentz invariant, making it ideal for relativistic dynamics and electromagnetic radiation theory.

6. Wave Equations for Plane EM Waves in Vacuum and Isotropic Dielectric Medium

Derivation of Electromagnetic Wave Equations

Consider a homogeneous, isotropic, linear, non-conducting dielectric medium characterized by permittivity ε, permeability μ, charge density ρ = 0, and conductivity σ = 0 (J = 0).

Maxwell's equations in this medium are:

1. ∇ · E = 0
2. ∇ · B = 0
3. ∇ × E = -∂B / ∂t
4. ∇ × B = μ ε (∂E / ∂t)

To obtain the wave equation for the electric field E, take the curl of Faraday's Law (Eq. 3):

∇ × (∇ × E) = ∇ × (-∂B / ∂t) = -∂/∂t (∇ × B)

Using the vector identity ∇ × (∇ × E) = ∇(∇ · E) - ∇²E, and substituting ∇ · E = 0:

-∇²E = -∂/∂t [μ ε (∂E / ∂t)]

∇²E = μ ε (∂²E / ∂t²)

Similarly, taking the curl of Ampere-Maxwell law (Eq. 4) yields the wave equation for the magnetic field B:

∇²B = μ ε (∂²B / ∂t²)

Velocity of Wave Propagation

The standard 3D wave equation has the mathematical form:

∇²Ψ = (1 / v²) (∂²Ψ / ∂t²)

Comparing this with ∇²E = μ ε (∂²E / ∂t²), the phase velocity v of the electromagnetic wave in the medium is:

v = 1 / √(μ ε)

In vacuum (free space), where μ = μ₀ and ε = ε₀:

c = 1 / √(μ₀ ε₀) = 1 / √[(4π × 10^-7 H/m) × (8.854 × 10^-12 F/m)] ≈ 3 × 10⁸ m/s

This theoretical result directly established that light is an electromagnetic wave.

7. Transverse Nature of Plane EM Waves

An electromagnetic wave is transverse if both the electric field vector E and the magnetic field vector B are perpendicular to the direction of wave propagation.

Mathematical Proof

Consider a plane wave traveling along the positive z-direction. For a plane wave, spatial variations exist only along z, so ∂/∂x = 0 and ∂/∂y = 0.

The electric and magnetic fields can be written as:

E = E_x(z, t) î + E_y(z, t) ĵ + E_z(z, t) k̂

B = B_x(z, t) î + B_y(z, t) ĵ + B_z(z, t) k̂

1. Evaluating Gauss's Law for E:

∇ · E = ∂E_x/∂x + ∂E_y/∂y + ∂E_z/∂z = 0

Since ∂E_x/∂x = 0 and ∂E_y/∂y = 0, this simplifies to:

∂E_z / ∂z = 0

This implies that E_z is independent of z. A static uniform electric field along the propagation direction does not form part of the wave; hence for wave motion, E_z = 0.

2. Evaluating Gauss's Law for B:

∇ · B = ∂B_z / ∂z = 0 → B_z = 0

Since E_z = 0 and B_z = 0, neither field has a component along the direction of propagation (z-axis). Therefore, electromagnetic waves are strictly transverse waves.

3. Mutual Perpendicularity of E and B:

Using Faraday's law ∇ × E = -∂B / ∂t for a plane wave propagating in +z direction with E = E_x î:

∇ × E = î(0) - ĵ(-∂E_x/∂z) + k̂(0) = (∂E_x / ∂z) ĵ

Equating components to -∂B / ∂t = -(∂B_x/∂t î + ∂B_y/∂t ĵ):

∂B_x / ∂t = 0 → B_x = 0

∂B_y / ∂t = -∂E_x / ∂z

For a sinusoidal wave E_x = E₀ cos(kz - ωt), we get B_y = (k / ω) E₀ cos(kz - ωt) = (1 / v) E_x.

Hence, E (along x), B (along y), and wave propagation vector k (along z) form a mutually orthogonal right-handed coordinate system:

E ⊥ B ⊥ k

8. Refractive Index and Dielectric Constant

Refractive Index (n)

The refractive index n of a medium is defined as the ratio of the speed of light in vacuum (c) to the phase velocity of light in the medium (v):

n = c / v

Relation with Dielectric Constant (Relative Permittivity)

Substituting v = 1 / √(με) and c = 1 / √(μ₀ε₀):

n = [1 / √(μ₀ε₀)] / [1 / √(με)] = √(μ ε / μ₀ ε₀) = √(ε_r μ_r)

where:

• ε_r = ε / ε₀ is the relative permittivity (also called the Dielectric Constant, K).
• μ_r = μ / μ₀ is the relative magnetic permeability.

For most non-magnetic dielectric materials at optical frequencies, μ_r ≈ 1. Therefore, Maxwell's relationship between refractive index and dielectric constant simplifies to:

n = √ε_r or ε_r = n²

Important Exam Observations

• Maxwell's relation n = √ε_r holds precisely when ε_r and n are measured at the same frequency.
• At optical frequencies, rapid field oscillations cause electronic polarization to dominate, so high-frequency dielectric constant values differ from static dielectric constants (e.g., for water, static ε_r ≈ 80, but at optical frequencies n ≈ 1.33, so n² ≈ 1.77).

9. Wave Impedance

Definition

Wave Impedance (or Characteristic Impedance) Z of an electromagnetic wave is defined as the ratio of the transverse electric field amplitude to the transverse magnetic field amplitude:

Z = E / H

Since B = μH, and for a plane wave E / B = v:

Z = E / (B / μ) = μ (E / B) = μ v

Substituting v = 1 / √(με):

Z = μ / √(με) = √(μ / ε)

Intrinsic Impedance of Free Space (Z₀)

For vacuum (free space), where μ = μ₀ and ε = ε₀:

Z₀ = √(μ₀ / ε₀)

Substituting numerical values μ₀ = 4π × 10^-7 H/m and ε₀ = 8.854 × 10^-12 F/m:

Z₀ = √[(4π × 10^-7) / (8.854 × 10^-12)] ≈ 376.73 Ω ≈ 120π Ω

Wave Impedance in Dielectric Medium

In a non-magnetic isotropic dielectric (μ_r = 1):

Z = √(μ₀ / (ε₀ ε_r)) = Z₀ / √ε_r = Z₀ / n

Because n > 1 in dielectrics, the wave impedance inside a dielectric is always smaller than the wave impedance of free space.

10. Propagation Through Conducting Media, Relaxation Time, and Skin Depth

Wave Equations in Conducting Media

A conducting medium is characterized by permittivity ε, permeability μ, and finite non-zero conductivity σ (where Ohm's law J = σE applies). Assuming zero net free charge density inside the bulk conductor (ρ = 0):

1. ∇ · E = 0
2. ∇ · B = 0
3. ∇ × E = -∂B / ∂t
4. ∇ × B = μ σ E + μ ε (∂E / ∂t)

Taking the curl of Faraday's Law gives the wave equation for conducting media:

∇²E = μ σ (∂E / ∂t) + μ ε (∂²E / ∂t²)

Similarly for magnetic field:

∇²B = μ σ (∂B / ∂t) + μ ε (∂²B / ∂t²)

Relaxation Time (Charge Relaxation Time)

If a net charge density ρ₀ is initially placed inside a conductor, it dissipates rapidly towards the outer boundary due to mutual electrostatic repulsion.

Using the continuity equation and Ohm's law:

∇ · J + ∂ρ / ∂t = 0 → ∇ · (σE) + ∂ρ / ∂t = 0

σ (∇ · E) + ∂ρ / ∂t = 0

Using Gauss's law ∇ · E = ρ / ε:

(σ / ε) ρ + ∂ρ / ∂t = 0 → ∂ρ / ∂t = -(σ / ε) ρ

Integrating this differential equation with respect to time t gives:

ρ(t) = ρ₀ exp(-t / τ)

where τ is the Charge Relaxation Time:

τ = ε / σ

Physical Significance: Relaxation time τ is the time taken for the charge density inside a conductor to decay to 1/e (≈ 36.8%) of its initial value ρ₀.

• Good Conductors (e.g., Copper): σ is extremely large (~10⁷ S/m), so τ ~ 10^-19 s. Charge neutralizes almost instantaneously. Bulk charge cannot persist inside a good conductor.
• Good Insulators (e.g., Fused Quartz): σ is tiny (~10^-17 S/m), so τ can be several hours or days. Charges remain localized.

Skin Depth (Attenuated Depth)

For a sinusoidal wave E = E₀ exp[i(kz - ωt)] propagating in a conducting medium, the complex wave vector k~ is:

k~² = μ ε ω² [1 + i (σ / (ω ε))]

The complex propagation constant γ = i k~ = α + iβ, where α represents the attenuation coefficient and β represents the phase constant.

The spatial variation of the electric field magnitude is given by:

E(z) = E₀ e^{-α z}

The Skin Depth (or Depth of Penetration) δ is defined as the distance z = δ over which the wave amplitude decays to 1/e (about 37%) of its value at the surface:

δ = 1 / α

For a Good Conductor (σ >> ωε):

α ≈ √(ω μ σ / 2) = √(π f μ σ)

Therefore, the skin depth δ is:

δ = √(2 / (ω μ σ)) = √(1 / (π f μ σ))

where f is the frequency of the wave (ω = 2πf).

11. Poynting Theorem and Poynting Vector

Poynting Vector (S)

The Poynting Vector S represents the directional energy flux density (rate of energy transfer per unit area) of an electromagnetic wave.

S = E × H = (1 / μ) (E × B)

• SI Unit: Watts per square meter (W/m²).
• Direction: Perpendicular to both E and H, matching the direction of wave power propagation.

Derivation of Poynting's Theorem

Consider the work done by electromagnetic fields on charge distribution inside a volume V. The rate of work done by fields per unit volume is J · E.

From the Ampere-Maxwell law: J = ∇ × H - ∂D / ∂t. Taking dot product with E:

J · E = E · (∇ × H) - E · (∂D / ∂t)

Using the vector identity ∇ · (E × H) = H · (∇ × E) - E · (∇ × H):

E · (∇ × H) = H · (∇ × E) - ∇ · (E × H)

Substitute Faraday's law ∇ × E = -∂B / ∂t:

J · E = H · (-∂B / ∂t) - ∇ · (E × H) - E · (∂D / ∂t)

J · E = - [ E · (∂D / ∂t) + H · (∂B / ∂t) ] - ∇ · (E × H)

For linear isotropic media, D = εE and B = μH:

E · (∂D / ∂t) = ∂/∂t (½ ε E²)

H · (∂B / ∂t) = ∂/∂t (½ μ H²)

Defining electromagnetic energy density u as:

u = u_e + u_m = ½ ε E² + ½ μ H²

Substituting u and S = E × H into the equation:

J · E + ∂u / ∂t + ∇ · S = 0

Integrating over volume V and applying Gauss's Divergence Theorem to the divergence term:

- ∂/∂t ∭ u dV = ∯ S · dS + ∭ (J · E) dV

Physical Meaning of Poynting Theorem

Poynting's theorem expresses conservation of energy for electromagnetic systems:

• - ∂/∂t ∭ u dV: Rate of decrease of total stored electromagnetic energy within volume V.
• ∯ S · dS: Rate of electromagnetic power flowing out across boundary surface S.
• ∭ (J · E) dV: Rate of work done by fields on charges within volume V (converted into kinetic energy or Ohmic heat dissipation I²R).

Time-Average Poynting Vector

For sinusoidally time-varying fields represented in complex phasor notation E(r, t) = Re[E₀ e^-iωt] and H(r, t) = Re[H₀ e^-iωt], the time-averaged power flux density is:

⟨S⟩ = ½ Re(E × H*)

where H* denotes the complex conjugate of magnetic field phasor H.