Video - Electromagnetic Wave Equation Simply Explained

The goal is to derive the wave equation for the electric field \(\boldsymbol{E}\) and the magnetic field \(\boldsymbol{B}\) using Maxwell's equations in free space.

We use the four Maxwell equations of electrodynamics in charge-free (\(\rho = 0\)) and current-free (\(\boldsymbol{j} = 0\)) space:

Another ingredient necessary for the derivation of the two wave equations is the following relation for the curl of the curl of the vector field \(\boldsymbol{F}\) (double cross product):

The first summand here is the gradient of the divergence of \(\boldsymbol{F}\) and the second summand is the divergence of the gradient of \(\boldsymbol{F}\). This is just a mathematical relation that can be derived.

Wave equation for the electric field

Maxwell's equations 1 in vacuum are coupled differential equations. To get the wave equation for the E-field, we have to decouple the third Maxwell equation. Apply the curl "\(\nabla \times \)" on both sides of the third Maxwell equation:

Double cross product applied to third Maxwell equation

The time derivative together with the minus sign may be placed before the nabla operator, because the nabla operator ( spatial derivative) does not depend on the time \(t\):

Third Maxwell equation with time derivative shifted to front

The time derivative outside the parenthesis can be placed after the magnetic and electric field constants. Two time derivatives can be compactly combined to the second time derivative:

One side of the wave equation is derived, namely the second time derivative of the electric field. Now all that remains is to rewrite the left-hand side into the correct form, as in a wave equation. Use the double cross product 2:

Decoupled third Maxwell equation with eliminated double cross product

On the left hand side of Eq. 7 contains the divergence \(\nabla \cdot \boldsymbol{E}\) of the electric field. According to the first Maxwell equation, the divergence of the electric field in charge-free space is always zero. Thus Eq. 7 simplifies to:

To derive the wave equation for the magnetic field \(\boldsymbol{B}\), we need to decouple Maxwell's fourth equation in Eq. 1. This is done analogously to the E-field. Apply the curl "\(\nabla \times\)" on both sides of the fourth Maxwell equation:

Double cross product applied to fourth Maxwell equation

+ Perfect for high school and undergraduate physics students + Contains over 500 illustrated formulas on just 140 pages + Contains tables with examples and measured constants + Easy for everyone because without vectors and integrals