## Maxwell's Equations, Revisited
Maxwell's equations are differential equations in both space and time. It models how electric and magnetic field vectors (**E** and **B**) change in both space and time. That means that variables $x,y,z$ and $t$ figure prominently in the four equations. Let's use our fake machinery as a metaphor for what it would mean to transform Maxwell's equations between co-moving inertial rest frames.
Let's go back to the airport. Traveler Guy is back to standing on the sidewalk and he's so bored with himself (because he can't motivate himself to walk on the sidewalk like God intended) that he's just admiring a magnet that he's holding.
We can apply Maxwell's equations to this situation to see what the Couch People would observe.
Look at that again: a pure magnetic field in the Away frame appears to the Home frame to be a mixture of both an electric and a magnetic field.
>Electric and magnetic fields, are completely different between two frames: E & M fields are not invariant with respect to relative motion.
The two electromagnetic paradoxes of the last lesson are completely understood as manifestations of the mixing of electric and magnetic fields into one another as viewed from co-moving inertial rest frames. Neither has an independent, permanent existence. Only the combined object, the Electromagnetic Field, is constant.
Think of it this way. Suppose we hold a rectangular block up to a light so its shadow appears on the wall. In some orientation, the shadow will appear elongated or distorted. In some orientation, the shadow will appear rectilinear. The block has not changed, but the shape of the projection of the block on the wall depends on how the block itself is oriented. The block is like the permanent electromagnetic field and the shadows are like the projection of the block, sometimes elongated (say, an electric field)...sometimes, recliner (say, a magnetic field).
In the end, the 26 year old, the less-than-unknown Albert Einstein showed that:
* Newton's laws of motion need to be modified
* Maxwell's equations are by themselves correct.
```{admonition} Please answer question 1 📺
:class: danger
Fields in motion?
```