# Derivation of The Wave Equation for E-field and B-field

## Table of contents

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:

\nabla ~\cdot~ \boldsymbol{B} ~&=~ 0 \\\\

\nabla ~\times~ \boldsymbol{E} ~&=~ -\frac{\partial \boldsymbol{B}}{\partial t} \\\\

\nabla ~\times~ \boldsymbol{B} ~&=~ \mu_0 \, \varepsilon_0 \frac{\partial \boldsymbol{E}}{\partial t} \end{align} $$

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:

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 \(t\):

Now we can replace the curl \( \nabla \times \boldsymbol{B} \) of the magnetic field with the help of the fourth Maxwell equation:

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

:

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:

## Wave equation for the magnetic field

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:

Now place the time derivative and the two constants on the right side in front of the Nabla operator:

Use Maxwell's third equation to substitute the curl \( \nabla \times \boldsymbol{E} \) of the electric field on the right-hand side:

The time derivative is combined and the double cross product on the left side is replaced using Eq. 2

:

The divergence \(\nabla \cdot \boldsymbol{B}\) is zero according to the second Maxwell equation. Thus you get: