\(B=-i\cfrac{\partial E}{\partial x^{'}}\)
was postulated to be the driving mechanism behind EMW. And an sustained EMW is set off by a sinusoidal voltage applied to an antenna. The expression by itself is not restricted to just continuous waves but sparks and short pulses will also be driven out into space. Conceptually, any signal can be Fourier transformed and be considered as a sum of sinusoidal waves propagating independently.
Does this expression has a more fundamental application? Is this the same \(B\) field created when a current flows down a wire?
In the case of a dipole, \(B\) is travelling along with the dipole, in this case that we are considering now, \(B\) at a fixed location in space as the charges passes by. Assuming that the time average of the change in \(B\) is given by the time averaged change in the effect of moving charges over a distance \(\Delta x\),
\(\cfrac { \Delta B }{ \Delta t } =\cfrac { 1 }{ \Delta t } \int _{ 0 }^{ \Delta t }{ -i\cfrac { \partial \, E }{ \partial x^{ ' } } } dt\)
\(\Delta t=\cfrac{\Delta x}{v}\)
where \(v\) is the velocity of the charges, we have,
\(\cfrac { \Delta B }{ \Delta t } =\cfrac { v }{ \Delta x } \int _{ 0 }^{ \Delta t }{ -i\cfrac { \partial \, E }{ \partial x^{ ' } } } dt\) --- (1)
since \(\cfrac { \partial \, E }{ \partial x^{ ' } }\) is independent of \(v\) and \(t\), more importantly, as \(\Delta t\rightarrow0\),
\(\cfrac { \Delta B }{ \Delta t } =\cfrac { v }{ \Delta x } -i\cfrac { \partial \, E }{ \partial x^{ ' } }\Delta t\)
So,
\(\cfrac { \Delta B }{ \Delta t }\cfrac{ \Delta x }{\Delta t} =-i\cfrac { \partial \, E }{ \partial x^{ ' } }v\)
\(\cfrac { \partial B }{ \partial \, t }v =-i\cfrac { \partial \, E }{ \partial x^{ ' } } v\) --- (*)
or
\(\cfrac { \partial B }{ \partial \, t }=-i\cfrac { \partial \, E }{ \partial x^{ ' } } \) --- (2)
Consider further,
\(\cfrac { \partial B }{ \partial \, t } =-i\cfrac { \partial \, E }{ \partial x^{ ' } } =-i(-\cfrac { \partial \, E }{ \partial r } )\cfrac { \partial \, r }{ \partial \, x^{ ' } } \)
where \(r\) is the radial distance from the electron to the location of \(B\). An additional negative sign appears because \(E\) decreases with increasing radial distance \(r\).
where \(-i\) rotates \(r\) to the direction of \(x^{'}\). So, (*) becomes,
\(\cfrac { \partial B }{ \partial \, t } \cfrac { \partial \, x }{ \partial \, t } =-\cfrac { \partial \, E }{ \partial r } \cfrac { \partial \, x }{ \partial \, t } \)
Integrating wrt \(r\),
\( \int { \cfrac { \partial B }{ \partial \, t } \cfrac { \partial \, x }{ \partial \, t } } \partial r=-\int { \cfrac { \partial \, x }{ \partial \, t } \partial E } \)
Integrating wrt \(t\),
\( \int { \int { \cfrac { \partial B }{ \partial \, t } \partial \, x } }\, \partial r=-\iint { \partial E }\, \partial \, x\)
\( \cfrac { \partial }{ \partial \, t } \left\{ \int { \int { B\partial \, x } } \partial r \right\} =-\int { E } \partial \, x\)
If the integration of \(x\) is along a loop and \(\partial x\) is a small arc length, then, the integral \(\int { \int { B\partial \, x } } \partial r\) generates an area through which \(B\) passes. Thus
\( \cfrac { \partial }{ \partial \, t } \oint { B } dA=-\oint { E } d \, x\)
And we have Maxwell–Faraday equation in integral form!
From (1),
\(\cfrac { \Delta B }{ \Delta t } =\cfrac { v }{ \Delta x } \int _{ 0 }^{ \Delta t }{ -i\cfrac { \partial \, E }{ \partial x^{ ' } } } dt\)
we see that,
\(\Delta B=0\)
when \(v=0\). This means that a stationary charge does not produce a change in \(B\). If we started with no background \(B\) at time \(t=0\) then \(B=0\). By this alone, we cannot conclude that only moving charges generates a \(B\) field. We can however, conclude, that moving charges can produce a change in the \(B\) field.
Lastly, equation (2),
\(\cfrac { \partial B }{ \partial \, t }=-i\cfrac { \partial \, E }{ \partial x^{ ' } } \)
is on the premise that \(v\ne0\), although \(v\) has canceled from the previous expression (*).
The situation of a moving dipole with its \(B\) field being carried along is different from the case of \(B\) at a fixed location in the presences of moving charges. The difference led to a different expression for \(B\) which was found to be consistent with Faraday's law of induction.
Good Night.
