Locality and Globality of Conservation Laws

Locality and Globality of Conservation Laws

Conservation principles lie at the heart of any physical theory. In electromagnetism they manifest as equations that constrain how charge, energy, and momentum can change in space and time. Two complementary viewpoints—local and global—describe these constraints at different scales. Understanding the relationship between them is essential for both analytical insight and reliable numerical simulation.

A local conservation law expresses the rate of change of a density at a single point in terms of the divergence of a flux and any sources or sinks present there. In differential form it reads

[
\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{J} = 0,
]

where (\rho) is a charge density and (\mathbf{J}) the associated current density. The equation tells us that if the density at a point increases, a net inflow of the corresponding quantity must be occurring.

For electromagnetic energy, the local form is Poynting’s theorem in differential form:

[
-\frac{\partial u_{\text{em}}}{\partial t}
= \nabla \cdot \mathbf{S}

  • \mathbf{J}!\cdot!\mathbf{E},
    ]

with (u_{\text{em}}) the field energy density, (\mathbf{S}) the Poynting vector, and (\mathbf{J}!\cdot!\mathbf{E}) the power supplied to charges. This equation holds at every point in space and time, making it the foundation for analytical derivations and for the discretisation schemes used in finite‑difference time‑domain (FDTD) or finite‑element method (FEM) solvers.

Key features of local laws:

  • Differential equations: involve spatial derivatives and are valid point‑wise.
  • Flux–source balance: the change of a density equals the net flux out of an infinitesimal volume plus any local production or consumption.
  • Microscopic dynamics: reveal how fields and particles interact locally.

Global Conservation Laws

A global conservation law concerns the total amount of a quantity within a finite region. It is obtained by integrating the local equation over a volume (V) and applying the divergence theorem:

[
\frac{d}{dt}\int_V \rho, dV
= -\oint_{\partial V} \mathbf{J}!\cdot d\mathbf{a}.
]

For electromagnetic energy the integrated form of Poynting’s theorem becomes

[
-\frac{d}{dt}\int_V u_{\text{em}}, dV
= \oint_{\partial V} \mathbf{S}!\cdot d\mathbf{a}

  • \int_V \mathbf{J}!\cdot!\mathbf{E}, dV.
    ]

These expressions describe how the total charge or energy inside a closed surface changes due to fluxes across the boundary and internal sources.

Typical applications of global laws:

  • Engineering design: calculating total radiated power from an antenna or total heat loss from a device.
  • System efficiency: evaluating how much of the input power is converted into useful work versus dissipated.
  • Boundary‑condition specification: ensuring that numerical models respect overall conservation.

The Divergence Theorem: Bridging the Two Views

The mathematical bridge between local and global formulations is the divergence theorem:

[
\iiint_V (\nabla \cdot \mathbf{F}), dV
= \iint_{\partial V} \mathbf{F}!\cdot d\mathbf{a}.
]

Applying it to a local conservation equation transforms the divergence term into a surface integral, yielding the corresponding global law. Conversely, if a global integral identity holds for any volume (V), one can infer that the integrand must satisfy the differential equation point‑wise. This duality guarantees that a physically consistent theory cannot be locally correct while globally inconsistent, and vice versa.

Momentum Conservation and the Maxwell Stress Tensor

Momentum conservation in electromagnetism is more subtle because the momentum density (\mathbf{g}) and the force density (\mathbf{f}) involve tensorial quantities. The local momentum balance is

[
\frac{\partial \mathbf{g}}{\partial t}
= \nabla \cdot \mathbf{T} - \mathbf{f},
]

where (\mathbf{T}) is the Maxwell stress tensor and (\mathbf{f}) the Lorentz force density. Integrating over a volume gives the total electromagnetic force on an object:

[
\mathbf{F}{\text{total}}
= \oint{\partial V} \mathbf{T}!\cdot d\mathbf{a}

  • \frac{d}{dt}\int_V \mathbf{g}, dV.
    ]

In many practical situations (e.g., static fields or steady currents) the time‑derivative term vanishes, and the surface integral of the stress tensor alone yields the net force. This approach is widely used in calculating forces in levitation, propulsion, and optical tweezers.

Practical Implications for Numerical Simulation

When implementing a computational solver:

  1. Discretise the local equations: FDTD, FEM, or spectral methods solve the differential forms directly.
  2. Validate global conservation: After each time step, compute the total charge or energy inside the computational domain and compare it with the net flux through the boundaries plus any source terms. Deviations indicate numerical errors or incorrect boundary conditions.
  3. Use the divergence theorem to design flux‑conserving schemes: Ensuring that the discrete divergence of the flux equals the discrete source term preserves global balance automatically.

By checking both local and global consistency, one can be confident that the simulation respects the underlying physics.

Summary

  • Local laws provide a point‑wise description of how densities evolve through fluxes and sources.
  • Global laws describe the net change of a quantity within a closed region, expressed as surface fluxes plus internal contributions.
  • The divergence theorem mathematically links the two, guaranteeing that a correct local formulation implies a correct global one and vice versa.
  • For momentum, the Maxwell stress tensor encapsulates the flux of electromagnetic momentum, enabling force calculations from surface integrals.
  • In computational practice, solving the local equations while verifying global conservation ensures physically reliable results.

Recognising the duality of locality and globality enriches our understanding of conservation principles and equips us with robust tools for both theoretical analysis and practical engineering.