What is the electromagnetic wave equation in 3D

The electromagnetic wave equation in 3D describes how electric and magnetic fields propagate through space over time in the presence of no charges or currents. It is derived from Maxwell’s equations and represents the behavior of electromagnetic waves such as light, radio waves, and microwaves in a vacuum or homogeneous medium.

The general form of the wave equation for both the electric field E and the magnetic field B in free space (vacuum) is:

Electric Field Wave Equation:

Magnetic Field Wave Equation:

Where:

• E is the electric field vector.

• B is the magnetic field vector.

• c is the speed of light in a vacuum ($c = 3 times 10^8 , text{m/s}$).

• $nabla^2$ is the Laplacian operator, which in Cartesian coordinates is expressed as:

  $$nabla^2 = frac{partial^2}{partial x^2} + frac{partial^2}{partial y^2} + frac{partial^2}{partial z^2}$$

• $frac{partial^2}{partial t^2}$ is the second derivative with respect to time.

Explanation:

• The first term ($nabla^2$) describes spatial variation (how the fields change across space).

• The second term ($frac{partial^2}{partial t^2}$) describes the time variation (how the fields change over time).

• The constant $frac{1}{c^2}$ links the spatial and time changes and ensures that the wave propagates at the speed of light.

These wave equations describe how both electric and magnetic fields propagate as waves in a vacuum. The electric and magnetic fields are perpendicular to each other and to the direction of wave propagation, forming a transverse electromagnetic wave.