Maxwell's equations

Maxwell’s equations are the fundamental set of integral and differential equations in electromagnetism. In differential form, they’re given by:

$$\nabla \cdot \vec{E}=\frac{\rho }{{ϵ}_0}$$ $$\nabla \times \vec{E}=-\frac{\partial \vec{B}}{\partial t}$$ $$\nabla \cdot \vec{B}=0$$ $$\nabla \times \vec{B}=\mu (\vec{J}+{ϵ}_0\frac{\partial \vec{E}}{\partial t})$$

The divergence term is given by Gauss’ law, and the curl term is given by Ampere’s law. Under time-invariant (electrostatic, magnetostatic) conditions, the time derivative terms go to 0 and the electric field is irrotational and the magnetic field is source-free, from the Helmholtz decomposition theorem.

In the electrostatic case:

• Think about the electric field. The field lines radiate out from a positive point charge and enter a negative point charge.
• For the magnetic field, the field lines rotate around and through a given object. So this intuitively makes some sense.

In optics

Maxwell’s equations are used in wave optics. Some simplifying considerations:

• $\vec{J}=\vec{0}$. Generally materials won’t carry a current.
• $\rho =0$. Generally we deal with charge distribution-free materials.
• We have a linear medium, i.e., $\vec{D}\propto \vec{E}$ and $\vec{B}\propto \vec{H}$.
• We also have an isotropic medium, i.e., the permittivity $ϵ$ and permeability $\mu$ are scalars, not tensors.