Maxwell’s Equations

Maxwell’s Equations in Mathematical Symbols

 

\nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}, \nabla \cdot \mathbf {B} =0, \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}, \nabla \times \mathbf {B} =\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)

 

Maxwell’s Equations in English

 

If the strength and direction of the electric and magnetic fields are viewed as velocity vectors, like the wind velocity arrows on a weather map:

  1. The flux or flow outwards of the electric field—the divergence—is proportional to the density of electric charge.
  2. The magnetic field has no sources or sinks anywhere—no divergence—no magnetic charges.
  3. The rate of rotation and direction of the rotation axis of the electric field at any point (the curl of the field) is proportional to and in the opposite direction from the rate of change of the magnetic field with time.
  4. The rate of rotation and direction of the rotation axis of the magnetic field at any point (the curl of the field) is proportional to and in the same direction as the rate of change of the electric field with time plus the current density (current per area).

 

Definitions

 

A vector is an array of numbers expressing a magnitude and a direction—like the wind speed arrows on a weather map.
The “curl of the field” is a vector pointing along the rotation axis of the field and having the magnitude of the rate of rotation. Sound complicated? Just imagine a swirling mass of water. Drop a plastic ball in at any point—then at any instant, the direction of the axis it is spinning around and its rate of spin defines the “curl” of the velocity field of the water at the ball’s location.