Visualizing the beauty in physics and mathematics
$\bigg ( v^2\nabla^2 - \dfrac {\partial^2}{\partial t^2} \bigg) \vec{E} = 0, \bigg ( v^2\nabla^2 - \dfrac {\partial^2}{\partial t^2} \bigg) \vec{B} = 0, v=\dfrac {1} {\sqrt {\mu \epsilon}}$
where $v$ is the speed of light (i.e. phase velocity) in a medium with permeability $\mu$, and permittivity $\epsilon$.
Electric Field vectors are orange. Magnetic Field vectors are cyan. The thick green vector representing $\partial \vec{E}/\partial t$ is associated with the spatial arrangement of the magnetic field according to the Ampere-Maxwell law (as evaluated on the green loop). The sense of circulation on the green loop (by the right-hand rule) determines the direction of change of the electric field (thumb-direction).
The thick magenta vector representing $\partial \vec{B}/\partial t$ is associated with the spatial arrangement of the electric field according to the Faraday's Law (as evaluated on the magenta loop). The sense of circulation on the magenta loop (by the right-hand rule) determines the direction of change of the magnetic field (opposite to thumb direction).
Intuitively, $\partial \vec{E}/\partial t$ tells the current value of $\vec{E}$ at that point to look like the value of $\vec{E}$ at the point to its left (in this example). In other words, the pattern of the electric field moves to the right.
Similarly, $\partial \vec{B}/\partial t$ tells the current value of $\vec{B}$ at that point to look like the value of $\vec{B}$ at the point to its left (in this example). In other words, the pattern of the magnetic field moves to the right. Thus, this electromagnetic plane wave moves to the right.
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