Saturday, 13 January 2018

electromagnetism - Falling Electron


Suppose there are two objects in the universe. Earth, with a gravitational acceleration of g = 9.8m/s/s, and a typical electron.


The electron is dropped from a certain height, say 1000m above the Earth's surface.


The initial energy of the electron is only the potential energy, mgh=meg×1000, where me is the mass of the electron.


As the electron falls towards the earth, it will be accelerated and thus will radiate energy. Will this cause the electron to slow down, and thus will the electron take a longer time to hit the ground than that expected by the equation s=0.5at2, due to energy loss through radiation.


If so, what acceleration will the electron actually fall at? How long will it take to hit the ground?



Answer



Assuming non-relativistic velocities, the power radiated by a charge accelerating at constant acceleration a is given by the Larmor formula:


P=e2a26πϵ0c3



To do the calculation properly is surprisingly complicated, but it's easy show that the effect of the radiation on the electrons fall is negligible. If the electron falls a distance h then the time it takes is given by:


h=12gt2


so:


t=2hg


If we assume the electron is accelerating at a constant rate of g, the total energy radiated is just power times time or:


Erad=e2g26πϵ0c32hg


In your question h is 1000m, so:


Erad=7.83×1051J


The potential energy change is, as you say, just mgh:


Epot=megh=8.94×1027J



So the ratio of the radiated energy to the potential energy is about 1024, and therefore the effect of the radiation on the electron's fall is entirely negligible.


Response to comment:


The power radiated from the electron produces a force that opposes the acceleration due to gravity. Assume we can ignore the deviations from accelerating at a constant rate g, then in a small time dt the energy radiated is Pdt. The energy is force times distance (dx) so to get the force we divide by the distance:


F=Pdtdx=Pv=P2gh


using v2=2as. The acceleration produced by this force is just F/me, so the net acceleration on the electron is:


anet=gPme2gh


So the electron does accelerate slightly more slowly than g, but the difference between the acceleration and g is inversely proportional to distance fallen so it gets increasingly negligible the further the electron falls.


You've probably spotted that the above equation says the force should be infinite at the moment you release the particle. That's because as you approach the moment of release it's no longer safe to make the approximation that you can ignore the change in the acceleration due to radiation.


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