This is obviously a joke.
From the previously post "Magnetic Field In General, HuYaa",
\(\cfrac { \partial B }{ \partial \, t } =-i\cfrac { \partial \, E }{ \partial x^{ ' } } \)
\(\cfrac { \partial B }{ \partial \, x } \cfrac { \partial x }{ \partial t } =-i\cfrac { \partial \, E }{ \partial x^{ ' } } \)
Since, \(-i.i=1\),
\(\cfrac { \partial B }{ \partial \, x } \cfrac { \partial x }{ \partial t } =-i\cfrac{1}{(-i.i)}\cfrac { \partial \, E }{ \partial t } \cfrac { \partial t }{ \partial x^{ ' } } \)
\(\cfrac { \partial B }{ \partial \, x } \cfrac { \partial x }{ \partial t } =\cfrac { \partial \, E }{ \partial t } \cfrac { 1 }{ i } \cfrac { \partial t }{ \partial x^{ ' } } \)
and \(i\) rotates \(x^{'}\) to \(r\), the radial distance,
\( \cfrac { \partial B }{ \partial \, x } \cfrac { \partial x }{ \partial t } =\cfrac { \partial \, E }{ \partial t } \cfrac { \partial t }{ \partial r } =\cfrac { \partial \, E }{ \partial t } \cfrac { \partial x }{ \partial r } \cfrac { \partial t }{ \partial x } \)
\( \cfrac { \partial B }{ \partial \, x } (\cfrac { \partial x }{ \partial t } )^{ 2 }=\cfrac { \partial \, E }{ \partial t } \cfrac { \partial x }{ \partial r } \)
Integrating by \(x\) and by \(r\),
\( \iint { \cfrac { \partial B }{ \partial \, x } (\cfrac { \partial x }{ \partial t } )^{ 2 }d x\,d r=\iint { \cfrac { \partial \, E }{ \partial t } \cfrac { \partial x }{ \partial r }d x\,d r } } \)
Since, for an electromagnetic wave,
\( (\cfrac { \partial x }{ \partial t } )^{ 2 }=c^ 2=(\cfrac { 1 }{ \sqrt { \varepsilon _{ o }\mu _{ o } } } )^{ 2 }=\cfrac { 1 }{ \varepsilon _{ o }\mu _{ o } } \)
\( \iint { d B } \,d r=\varepsilon _{ o }\mu _{ o }\iint { \cfrac { \partial \, E }{ \partial t } d x\,d x } \)
\( \int { B } \,dr=\varepsilon _{ o }\mu _{ o }\cfrac { \partial }{ \partial t } \left\{\iint {E }\,(dx)^2 \right\} \)
If \(\int{B}\,dr\) is around a square loop with length \(dx\) and height also \(dx\), bear in mind that \(r\) is perpendicular to \(x\), then,
\( \oint { B } \,dr=\varepsilon _{ o }\mu _{ o }\cfrac { \partial \, }{ \partial t } \left\{ \oint { E } \,dA \right\} \)
Which is Maxwell's displacement current. This is the last of Maxwell's Equations not considering current density, \(j_e\).
The maths is not rigorous but it seems to indicate that dynamically all that is needed is,
\(\cfrac { \partial B }{ \partial \, t } =-i\cfrac { \partial \, E }{ \partial x^{ ' } } \)
to define the \(B\) field established by moving charges. I have to think about this one.
