Maxwell’s Equations in a Vacuum

The electromagnetism of vacuum space has some properties worth mentioning. It turns out that Maxwell’s equations, which describe electric and magnetic fields, take the following form in a vacuum:

\begin{array}{rlr} \vec{\nabla} \cdot \vec{E} &= 0 & [1]\\ \vec{\nabla} \cdot \vec{B} &= 0 & [2]\\ \vec{\nabla} \times \vec{E} &\displaystyle = -\frac{\partial \vec{B}}{\partial t} & [3]\\ \vec{\nabla} \times \vec{B} &\displaystyle = \mu_0\epsilon_0 \frac{\partial \vec{E}}{\partial t} & [4] \end{array}

From this, we can verify that any disturbance in the electric and magnetic fields propagates as a wave in vacuum space. How do we know this? Because analyzing these expressions yields a wave equation for both fields.

The Propagation of Electromagnetic Waves

From [4] and [5], it follows that the electric field satisfies the following relation:

\begin{array}{llr} \vec{\nabla} \times (\vec{\nabla} \times \vec{E}) &= \displaystyle \vec{\nabla} \times \left( -\frac{\partial \vec{B}}{\partial t} \right) &\\ &=\displaystyle -\frac{\partial}{\partial t}\left(\vec{\nabla} \times \vec{B}\right) = -\frac{\partial}{\partial t} \left(\mu_0\epsilon_0 \frac{\partial \vec{E}}{\partial t} \right) = -\mu_0\epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}& [6] \end{array}

Then, as any vector field satisfies the relation:

\vec{\nabla} \times \left(\vec{\nabla} \times \vec{A} \right) = \vec{\nabla}(\vec{\nabla} \cdot \vec{A}) - \nabla^2 \vec{A},\;\;\;[7]

From [2, 6] and [7], we can write:

\begin{array}{rll} &\displaystyle \vec{\nabla}(\underbrace{\vec{\nabla} \cdot \vec{E}}_{=0}) - \nabla^2 \vec{E} = \vec{\nabla} \times \left(\vec{\nabla} \times \vec{E} \right) = -\mu_0\epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} & \\ \equiv &\displaystyle \color{blue}{\nabla^2 \vec{E} = \mu_0\epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}} & [8] \end{array}

This, marked in blue, is just a wave propagation equation for the electric field.

Analogously, the same occurs for the magnetic field

\begin{array}{ll} \vec{\nabla} \times (\vec{\nabla} \times \vec{B}) &= \displaystyle \vec{\nabla} \times \left(\mu_0\epsilon_0 \frac{\partial \vec{E}}{\partial t} \right)\\ &=\displaystyle \mu_0\epsilon_0 \frac{\partial}{\partial t}\left(\vec{\nabla} \times \vec{E}\right) = \mu_0\epsilon_0 \frac{\partial}{\partial t} \left(- \frac{\partial \vec{B}}{\partial t} \right) = -\mu_0\epsilon_0 \displaystyle \frac{\partial^2 \vec{B}}{\partial t^2} \end{array}

and then

\begin{array}{rll} & \displaystyle \vec{\nabla}(\underbrace{\vec{\nabla} \cdot \vec{B}}_{=0}) - \nabla^2 \vec{B} = \vec{\nabla} \times \left(\vec{\nabla} \times \vec{B} \right) = -\mu_0\epsilon_0 \frac{\partial^2 \vec{B}}{\partial t^2} &\\ \equiv &\color{blue}{\nabla^2 \vec{B} = \displaystyle \mu_0\epsilon_0 \frac{\partial^2 \vec{B}}{\partial t^2}}& [9] \end{array}

It is from this that electromagnetic fields in vacuum are said to have many possible modes of being, and one family of these modes takes the form of an electromagnetic wave that propagates through space and time.

The Speed of Light is a Universal Constant

In other words, disturbances in the electromagnetic field always propagate with the speed c = 1/\sqrt{\epsilon_0 \mu_0}\approx 3\cdot 10^8[m/s], which is the speed of light in a vacuum. Experimentally, it is observed that this speed is the same for all inertial frames of reference, which does not align with what would be obtained if Galileo’s transformations were applied, as shown in Galileo’s Transformations and Their Limitations; because according to these, even the wave’s own structure is altered when moving from one inertial frame to another. These results are the key piece to set aside Galileo’s transformations, giving way to Lorentz’s transformations of special relativity because: a properly formulated coordinate transformation must preserve the laws of physics for all inertial observers.