In classical physics, when electric charges accelerate, they are expected to radiate. Electromagnetic waves, are by their nature the result of changing electric and magnetic fields. But is it possible to have acceleration of charges without having changing electric and magnetic fields? One thought that comes to mind (but with changing charge densities) is when charges on a conductor are transferred to a rubber conveyor belt, like in a Van de Graff Generator: In this case we have charges being transported via a conveyor belt where they travel much faster than the drift velocity in the conducting wire. Therefore, charges must accelerate when the go onto the conveyor belt and "decelerate" when they go off the conveyor belt. Charge density in this case is not constant but accumulated in one region at the expense of another. But what if we had a c-shaped conductive conveyor belt that pulled a constant current off one end of a conductive wire, only to transfer those charges back onto the other end of the conductive wire? ⊂⊃ We would have two small regions where the charges would be accelerated, but the charge densities and current densities in this case are constant. This is due to convective acceleration (as distinct from local/Eulerian acceleration). From http://web.mit.edu/fluids-modules/www/potential_flows/LecturesHTML/lec02/lecture2.html: http://web.mit.edu/fluids-modules/www/potential_flows/LecturesHTML/lec02/img37.gif In this case, would: 1) the electric field in these regions look like that of a constant current (i.e. no curl of the electric field), or 2) would a curl of the electric field show up? Note that the acceleration of charges in each region is unidirectional, so we would not expect the emitted electric field to look like that of an oscillating charge. However, doesn't the acceleration of charge in these regions mean that there should be a transverse electric field distortion? In the image below (from http://www.tapir.caltech.edu/~teviet/Waves/empulse.html) we have a charge subject to an impulse of a finite time duration. The result is a component of the electric field that is transverse. If we have charges in a current jumping from one conductor to another conductor in relative motion (such as in a slip ring), then we should expect that these charges will be accelerating. But in our case this acceleration occurs in a "pillbox" that is moving inertially (if moving at all). So if we imagine a steady flow of electrons between these conductors, we should expect a curl of the electric field be produced due to the individual acceleration of each electron. We would in effect have an electric field that could induce current in a separate wire loop. But if that is true, then the only time we might expect the transverse distortion of the electric field to reverse is if we were to reverse the current. So this induction in question would not be alternating, as if being produced by an ever-increasing magnetic field (which clearly ISN'T happening here). A steady direct current could easily be supplied by a battery, and such current crossing the gaps between two relatively moving conductors would consist of accelerating charges, and so CLEARLY we have a problem. The Maxwell-Faraday equation (one of Maxwell's equations - see above) says that the curl of an electric field occurs simultaneously with a changing magnetic field, but here the current densities are constant and so our magnetic fields should also be constant (despite acceleration of charge across the gaps between the two relatively moving conductors). If the curl of the electric field is non-vanishing in these conditions due to the convective acceleration of charges, then it would violate the Maxwell-Faraday equation. But then, if we were to enforce the Maxwell-Faraday equation, then that is to say that despite the acceleration of charges, the induced electric field would be curl free. Why wouldn't the accelerating charges in the gaps induce an emf in a external conductive loop? Or would they actually do so without changing magnetic fields (which would violate the Maxwell-Faraday equation)? Also another thing to note is: Taking this as the definition of the electric field, will the last term be non-zero due to the convective acceleration of charges in the gaps, or will it be zero due to constant nature of the current densities? Any clarification is much welcome. - Kevin M.