magnetoquasistatic

Magnetoquasistatic (MQS) is an electromagnetic approximation to Maxwell’s equations used when magnetic effects from time-varying currents dominate and the displacement current can be neglected. In this regime the fields evolve slowly enough that wave propagation and capacitive effects are not essential to the solution, and the primary interest is in how changing currents produce magnetic fields and induced electric fields.

In the MQS formulation, Ampere’s law is simplified by dropping the displacement current term, giving curl H ≈ J, with B = μH and J = σE. Faraday’s law, curl E = −∂B/∂t, remains intact. Often the electric field is expressed as E = −∂A/∂t − ∇φ, with B = ∇×A. The continuity of charge still holds, ∇·J = −∂ρ/∂t, so charge dynamics are not entirely neglected.

The approximation is valid when the system’s characteristic dimensions and time scales satisfy conditions that suppress wave effects and displacement currents. A common criterion is that the conduction current dominates over the displacement current, σ ≫ ωε, where ω is the characteristic frequency and ε is the permittivity. In practice, MQS applies to low- to moderate-frequency operation of inductors, transformers, and other devices where inductive coupling and eddy currents are the primary concerns and capacitive effects are negligible.

Applications include the design and analysis of transformers and inductors, magnetic shielding, eddy-current suppression, and time-domain simulations of electromagnetic devices. MQS provides a simpler, computationally efficient model compared with full Maxwell equations, while still capturing essential magnetic coupling and induced electric fields. Limitations arise at higher frequencies, large structures, or when significant capacitive or wave-propagation effects cannot be ignored.