Physics Illusion 8: Input and Output of Maxwell's equations

                            Motion of a magnet generates a current from changing electrical field.

Maxwell's equations can be formulated (assuming unit speed of light):

1. $\dot B + \nabla\times E = 0$
2. $\dot E  - \nabla\times B = - J$
3. $\dot\rho + \nabla\cdot J =0$,

where $E(x,t)$ is the electric field, $B(x,t)$ the magnetic field, $\rho (x,t)$ charge density and $J$ electric current density, given as $J=\rho v$ with $v$ charge velocity, or by a constitutive law such as Ohm's law

• $J =\sigma (E + v\times B)$,

where $\sigma$ is a conductivity and $v$ the velocity of the conducting medium. The dot indicates differentiation with respect to time $t$, while $x$ is a Euclidean space coordinate.

Differentiating 2 with respect to time and applying $\nabla\times$ to 1, gives the following wave equation for the electric field:

• $\ddot E + \nabla\times\nabla\times E = - \dot J$,  

where for a time constant charge density $\rho$ moving with velocity $v$, $\dot J =\rho\dot v$.

In the electrostatic case, Maxwell's equations reduce to:

• $\nabla\cdot E (x) = \rho (x)$ and $E(x)=\nabla\phi (x)$ with $\phi (x)$ potential,
• that is $\Delta\phi (x) =\rho (x)$.

There are two possible ways of viewing Maxwell's equations in terms of input and output:

• (I) Input: charge and current,   Output: electric and magnetic fields,
• (II) Input: electric and magnetic fields,  Output: charge and current.

Here (I) corresponds to solving wave equations with $\rho$ and $J$ given input producing as output $E$ and $B$ as action at distance with a time delay because of finite speed of light, or the static Poisson equation with potential $\phi (x)$ as output by action at distance from $\rho (x)$ without time delay aspect.

Further $(II)$ correspond to instantaneous local action by electric/magnetic fields to produce $\dot J(x.t)$ or $\rho (x)$. This is the physical process of electromagnetic induction. 

In electromagnetics, both cases appear, with an example of (II) given in the above picture.

Maxwell's equations are Galilean invariant modulo a second order factor $\frac{v^2}{c^2}$ with $c$ the speed of light, see Many Minds Relativity Chapter 17. The speed difference between human observers can differ by at most a few $km/s$, while $c\approx 300.000\, km/s$, and thus the factor is smaller than $10^{-10}$. Maxwell's equations can thus in practice be viewed by human observers to be Galilean invariant, which makes Lorentz invariance irrelevant.