Question:

What are magnetostatics?
Provide a detailed explanation including relevant physical laws, assumptions, and applications.

Answer:

Introduction

Magnetostatics (also known as magnostatics) is a branch of electromagnetism that deals with magnetic fields in systems where electric currents are steady, i.e., not changing with time.

Key Concepts

• Steady Currents: The electrical currents are assumed to be constant in time. That means there is no buildup of charge, and no induced electric fields arise from changing magnetic fields. This is analogous to electrostatics, where electric charges remain stationary.
• Maxwell’s Equations in Magnetostatics:
  Under the magnetostatic assumption, Maxwell’s equations simplify to:
  ∇ ⋅ B = 0 and ∇ × B = μ₀J
  where:
  • B is the magnetic field
  • J is the current density
  • μ₀ is the permeability of free space

  The term involving the time derivative of the electric field (displacement current) is ignored because fields do not change with time.

• Biot–Savart Law:
  The magnetic field produced by a steady current can be calculated using:
  B(r) = (μ₀ / 4π) ∫ [J(r′) × (r − r′)] / |r − r′|³ d³r′
  This law expresses how each element of current contributes to the magnetic field at a point r.

Applications

Magnetostatics is widely used in:

• Analyzing magnetic fields around current-carrying wires
• Studying permanent magnets
• Designing devices such as electromagnets, transformers, and inductors in the static limit

Summary

Magnetostatics focuses on magnetic fields generated by unchanging current distributions. This simplifies the mathematics of Maxwell’s equations and is fundamental to the design and understanding of many electromagnetic systems.