Why do people claim electrons are accelerating A lot of text books mention that one of the reasons that classical mechanics failed to explain atomic and subatomic processes is that electrons which accelerate should release energy in the form of electromagnetic radiation, which would lower the atoms overall energy level, but this does not happen. One place where I discovered this, for example, is in the description for the Bohr model. 
What I don't understand is why everyone takes for granted the fact that the electron is accelerating. I thought the electron orbits the nucleus at a, more or less, constant velocity. Are people referring to specific situations when the atom is excited? Furthermore, I was under the impression that electrons already travel at the fastest allowable speed, the speed of light.
 A: In uniform circular motion: $a=\frac{v^2}{r}$ 
A: On a uniform circular orbit, even if the speed does not change in norm, it does change in direction so that the speed vector change over time and $\frac{d\vec{v}}{dt}\neq\vec{0}$. In fact, in polar coordinates, you have
$$\vec{a} = \frac{d\vec{v}}{dt} = -\frac{v^2}{R}\,\vec{e_r}$$
Imagine a car taking a turn at constant speed: if the turn is left, you feel like you're pulled on the right because the car itself is exerting a force on your lower parts to make you turn left rather than continue in straight line. Thus you effectively experience an acceleration with respect to the ground even if your speed stay constant in norm.
A: Look at it this way. An electron is a charged particle. A moving, electrically charged particle creates a magnetic field, and the particle itself already has an electric field. If the particle is accelerating, then you're going to have a ripple effect from the electric and magnetic field, or an electromagnetic ripple, ie an electromagnetic wave. So an accelerating electron does indeed release electromagnetic waves.
A: The electrons should accelerate in the electric current also, since there is an electric force created by the potential difference, probably this is the cause of the magnetic field of the electric current.
