Maxwell Equations Differential Form

While the integral form of Maxwell's equations describes the behavior of fields over regions and surfaces, the differential form describes the behavior at every individual point in space. The differential form, also called the point form, is derived by applying the divergence theorem and Stokes' theorem to shrink the integration volumes and paths down to a single point. This form is essential for solving field problems numerically, understanding wave equations, and analyzing fields inside inhomogeneous media.

Core Concept Explanation

The differential form of Maxwell's equations expresses field relationships at every point in space using the vector differential operators divergence (∇·) and curl (∇×). Divergence measures how much a vector field spreads outward from a point. A positive divergence of D at a point means there is a positive charge density at that point. A zero divergence of B at every point means B field lines never originate or terminate; they always form closed loops.

Curl measures the rotational tendency of a field around a point. A nonzero curl of E at a point means there is a time-varying magnetic field at that location. A nonzero curl of H at a point means there is either a conduction current density J or a time-varying D (displacement current density) at that point. Together, these four equations determine the fields everywhere given their sources.

The del operator (∇) in Cartesian coordinates is ∂/∂x ax + ∂/∂y ay + ∂/∂z az. In cylindrical and spherical coordinates, the expressions for divergence and curl are more complex, but the physical meaning remains the same. For problems involving wave propagation, the differential form is combined with the constitutive relations D = εE, B = μH, and J = σE to derive the wave equation.

Mathematical Expression

The four Maxwell equations in differential (point) form are written as follows. These are valid for general time-varying fields in linear, isotropic, homogeneous media:

∇ · D = ρv (Gauss electric)

∇ · B = 0 (Gauss magnetic)

∇ × E = −∂B/∂t (Faraday)

∇ × H = J + ∂D/∂t (Ampere-Maxwell)

Taking the curl of the Faraday equation and substituting the Ampere-Maxwell equation leads to the wave equation for E in a source-free medium:

∇²E = με · ∂²E/∂t²

The wave velocity v equals 1/√(με). In free space, this becomes 1/√(μ₀ε₀) = 3 × 10⁸ m/s, which is the speed of light. This result, derived purely from Maxwell's equations, was a monumental confirmation that light is an electromagnetic wave.

Practical Understanding

The differential form is indispensable in numerical methods like the Finite Difference Time Domain (FDTD) method, where Maxwell's curl equations are discretized in both space and time to simulate how electromagnetic waves propagate through complex geometries. Each grid point in the computational domain stores field values updated using the differential equations at every time step.

In material media, the constitutive relations D = εE, B = μH, and J = σE must be substituted into the differential equations. In lossy media (σ ≠ 0), the curl of H includes a conduction current term σE in addition to the displacement current term ε·∂E/∂t. The ratio σ/(ωε) is the loss tangent, which determines whether a material behaves more like a conductor or a dielectric at a given frequency.

Given:
In a source-free non-conducting medium with εr = 4, μr = 1,
verify that the electric field E = E₀ sin(ωt − βz) ax satisfies
Maxwell's equations and find the wave velocity.

Why this formula applies:
For a plane wave in source-free medium: ∇ × E = −∂B/∂t
and ∇ × H = ∂D/∂t. The wave velocity v = 1/√(με).

Formula:
v = 1 / √(μ₀μr × ε₀εr)
v = c / √(μr × εr)   where c = 3×10⁸ m/s

Substitution:
v = 3×10⁸ / √(1 × 4)
v = 3×10⁸ / 2

Calculation:
v = 1.5 × 10⁸ m/s

Phase constant β = ω/v = ω√(με)
For f = 1 GHz: ω = 2π×10⁹ rad/s
β = (2π×10⁹) / (1.5×10⁸) = 41.9 rad/m

Final Answer with units:
Wave velocity v = 1.5 × 10⁸ m/s (half the speed of light in this medium)
Phase constant β ≈ 41.9 rad/m at 1 GHz

Exam Tip: For source-free, lossless media (ρv = 0, J = 0), both divergence equations give zero and the two curl equations are the only active ones. The condition ∇ · E = 0 in source-free media often simplifies problems significantly. Remember that the wave equation ∇²E = με∂²E/∂t² comes directly from combining the two curl equations.

• ∇ · D = ρv: Divergence of D at a point equals local volume charge density. Positive divergence means positive charge at that point.
• ∇ · B = 0: Divergence of B is always zero at every point, confirming B field lines form closed loops with no sources or sinks.
• ∇ × E = −∂B/∂t: Curl of E at a point is driven by the local time rate of change of B. Zero for static fields.
• ∇ × H = J + ∂D/∂t: Curl of H is the vector sum of conduction and displacement current densities at that point.
• Combining the two curl equations in source-free media gives the vector wave equation ∇²E = με∂²E/∂t², with wave velocity v = 1/√(με).

Quick Revision

• Four differential forms: ∇·D = ρv ; ∇·B = 0 ; ∇×E = −∂B/∂t ; ∇×H = J + ∂D/∂t.
• Source-free condition (ρv = 0, J = 0) simplifies to two curl equations; both divergence equations give zero.
• Wave equation: ∇²E = με∂²E/∂t² ; wave velocity v = 1/√(με) = c/√(μrεr).
• Constitutive relations: D = εE, B = μH, J = σE — must be substituted for material problems.
• Loss tangent: σ/(ωε) distinguishes conductor (>> 1) from dielectric (<< 1) at frequency ω.
• Exam trap: In Faraday's differential form, the sign is negative (∇×E = −∂B/∂t). Dropping the negative sign is a common error.
• Vector identity used in wave derivation: ∇×(∇×E) = ∇(∇·E) − ∇²E. Memorize this for GATE.