**Maxwell’s equations describe all (classical) electromagnetic phenomena:**

∇×E = −∂B/∂t

∇×H = J + ∂D/∂t

∇·D = ρ

∇·B = 0

(Maxwell’s equations)

The first is Faraday’s law of induction, the second is Amp`ere’s law as amended by Maxwell to include the displacement current ∂D/∂t, the third and fourth are Gauss’ laws for the electric and magnetic fields.

The displacement current term ∂D/∂t in Amp`ere’s law is essential in predicting the existence of propagating electromagnetic waves.

The quantities E and H are the electric and magnetic field intensities and are measured in units of [volt/m] and [ampere/m], respectively.

The quantities D and B are the electric and magnetic flux densities and are in units of [coulomb/m2] and [weber/m2], or [tesla]. D is also called the electric displacement, and B, the magnetic induction.

The quantities ρ and J are the volume charge density and electric current density (charge flux) of any external charges (that is, not including any induced polarization charges and currents.) They are measured in units of [coulomb/m3] and [ampere/m2]. The right-hand side of the fourth equation is zero because there are no magnetic monopole charges.

The charge and current densities ρ, J may be thought of as the sources of the electromagnetic fields. For wave propagation problems, these densities are localized in space; for example, they are restricted to flow on an antenna. The generated electric and magnetic fields are radiated away from these sources and can propagate to large distances to the receiving antennas. Away from the sources, that is, in source-free regions of space,

**Maxwell’s equations take the simpler form:**

∇×E = −∂B/∂t

∇×H = ∂D/∂t

∇·D = 0

∇·B = 0

(source-free Maxwell’s equations)

## Lorentz Force

The force on a charge q moving with velocity v in the presence of an electric and magnetic field E, B is called the Lorentz force and is given by:

**F = q(E + v × B) (Lorentz force) **

Newton’s equation of motion is (for non-relativistic speeds):

**m dv/dt = F = q(E + v × B)**

where m is the mass of the charge. The force F will increase the kinetic energy of the charge at a rate that is equal to the rate of work done by the Lorentz force on the charge, that is, v · F. Indeed, the time-derivative of the kinetic energy is:

**W**_{kin} = 12mv · v

⇒ **dW**_{kin}/dt = mv · dv/dt = v · F = q v · E

**We note that only the electric force contributes to the increase of the kinetic energy—**

**the magnetic force remains perpendicular to v, that is, v · (v × B)= 0.**