Gauss's Law:
$$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$$
Where:
- ∇ is the divergence operator
- E is the electric field
- ρ is the charge density
- ε0 is the permittivity of free space
Gauss's Law for Magnetism:
$$\nabla \cdot \mathbf{B} = 0 $$
Where:
- ∇ is the divergence operator
- B is the magnetic field
Faraday's Law (in steady-state conditions, it becomes zero):
$$\nabla \times \mathbf{E} = 0$$
Where:
- ∇ × is the curl operator
- E is the electric field
Ampere's Law with Maxwell's Addition (steady-state form):
$$\nabla \times \mathbf{B} = \mu_0 \mathbf{J}$$
Where:
- ∇ × is the curl operator
- B is the magnetic field
- μ0 is the permeability of free space
- J is the electric current density
In summary, for steady-state conditions, Maxwell's equations reduce to the simpler forms of Gauss's law, Gauss's law for magnetism, zero Faraday's law, and modified Ampere's law.
