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Re: [Phys-L] electron velocity in an electric field.

I made an estimate of the effect of EM radiation on initial electron speed using the paper Brian Whatcott recommended (Daniel Schroeder's AAPT 1999 paper "Purcell Simplified" which referenced and recommended Edward Purcell's treatment of EM radiation in his book "Electricity and Magnetism"). Purcell develops a formula for the total electromagnetic power radiated by a charge undergoing constant acceleration (termed the "Larmor formula"). The formula is:

Power radiated = P_R = q^2*a^2/(6*pi*e_0*c^3). Where:

q = electronic charge = 1.602 x 10^-19 C
a = constant acceleration
e_0 = permittivity of free space = 8.854 x 10^-12 C^2/N m^2
c = speed of light = 2.998 x 10^8 m/s

For the acceleration I got (using Newtonian mechanics):

a = q*V/(d*m) where:

V = voltage across the two plates = 10 kv = 10^4 v
d = spacing between the two plates = 1 cm = 0.01 m
m = electronic mass = 9.109 x 10^-31 kg

This yields a = 1.759 x 10^17 m/s^2 and, with the other parameters specified above, results in:

P_R = 1.766 x 10^-19 w

To get the energy radiated (E_R), we need the time (t) the electron takes going between the two plates. Since I am only making an estimate, I again used the Newtonian result:

t = (v_f - v_i)/a where:

v_f = final speed = 0
v_i = initial speed (assuming radiation is a very small effect this is given approximately by David Bowman's special relativity result 5.846 x 10^7 m/s)

So, approximately, t = 3.323 x 10^-10 s and the total energy radiated (E_R) is:

E_R = P_R*t = 5.868 x 10^-29 J = 3.663 x 10^-10 eV since 1 eV = 1.602 x 10^-19 J

Therefore, E_R/(q*V) = 3.663 x 10^-14, so we expect the change in initial speed due to radiation to, indeed, be very small. Using David Bowman's special relativity formulation for the initial speed:

b = sqrt(2*e*(1 + e/2))/(1 + e) where:

b = v_i/c and e = q*V/(m*c^2 (assuming no radiation)

I get, using a number of approximations and small parameter expansions, the change in initial speed (delta v_i) is given by:

delta v_i/v_i = (1/2)*(E_R/(q*V)) = 1.83 X 10^-14 or about 2 x 10^-14. So the change in initial speed (delta v_i) is indeed very small due to radiation.

Can anyone make a similar approximate estimate of general relativity effects?

Don

Dr. Donald Polvani
Adjunct Faculty, Physics, Retired
Anne Arundel Community College
Arnold, MD 21012

-----Original Message-----
From: Phys-l [mailto:phys-l-bounces@mail.phys-l.org] On Behalf Of brian
whatcott
Sent: Tuesday, April 24, 2018 10:56 PM
To: phys-l@mail.phys-l.org
Subject: Re: [Phys-L] electron velocity in an electric field.

On 4/24/2018 4:12 PM, Don via Phys-l wrote:
Thanks to Brian Whatcott for the non-relativistic and engineering details of
this problem and David Bowman for the special relativity theoretical
correction. I'm curious if anyone can estimate the relative importance of the
radiation and general relativity corrections. Including radiation must result in
a higher initial electron velocity to supply the extra required initial KE. But it's
been too many years for me to remember how to calculate the magnitude of
the radiation, and I have no idea what general relativity does.

Don

Dr. Donald Polvani
Adjunct Faculty, Physics, Retired
Anne Arundel Community College

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The Larmor formulation shows the energy dissipated by radiation from a
decelerating charge. An E&M text by Purcell was used by Dan Schroeder
(Weber State) as a reference source for a talk he gave at a meeting of the
AAPT (1999). Here is the text he spoke to.

<http://physics.weber.edu/schroeder/mrr/MRRtalk.html>

The final paragraph is titled Quantitative Treatment of Radiation,

where he diagrams the Larmor formula for this case.


Brian W

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