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Today's Topics:
1. Re: electron velocity in an electric field. (brian whatcott)
2. Re: electron velocity in an electric field. (Don)
3. Re: electron velocity in an electric field. (brian whatcott)
----------------------------------------------------------------------
Message: 1
Date: Tue, 24 Apr 2018 12:00:06 -0500
From: brian whatcott <betwys1@sbcglobal.net>
To: phys-l@mail.phys-l.org
Subject: Re: [Phys-L] electron velocity in an electric field.
Message-ID: <ce0cbc0e-a7ce-35e2-51b4-16bcbd5618b0@sbcglobal.net>
Content-Type: text/plain; charset=utf-8; format=flowed
On 4/24/2018 4:29 AM, David Bowman wrote:
Regarding the answer to the question Brian W attempted to answer:we ignore the tiny radiative losses as the electron decelerates). In this
Q:Two parallel metal plates with a spacing of 1cm have aWe assume the energy of the electron is conserved in its travels (i.e.
potential difference of 10kv. An electron is projected from one
plate directly towards second.What is the initial velocity of
electron if it comes to rest just at the surface of second
plate?
case the PE gain equals the initial KE lost in stopping the electron by the
retarding electric field. This means the initial KE of the electron is 10
keV. To simplify the math let's keep things dimensionless. So let e =
(KE)/(m*c^2), and let b = v/c. This means
simplifies to the Newtonian result
e = 1/sqrt(1 - b^2) - 1 .
Solving for b gives
b = sqrt(2*e*(1 + e/2))/(1 + e).
If we neglect the special relativistic corrections this formula
factor of
b = sqrt(2*e).
The special relativistic kinematic corrections themselves amount to a
to 2 % since the initial kinetic energy is about 2% of the rest energy.
sqrt(1 + e/2)/(1 + e).
Now e = (10 keV)/(510.9989461(31) keV) = 0.01956951198(12) (CODATA 2014).
This means
b = 0.1949856095(6) with the relativisitic corrections, and
b = 0.197835851 without them.
Converting to SI units the initial speed is v = b*c = 58,455.215 km/s.
The relativistic correction is a factor of
sqrt(1 + e/2)/(1 + e) = 0.985592897
(i.e. about a 1.46% decrease from the Newtonian result).
It stands to reason that the Newtonian result should be off by about 1
would be somewhat more precise than the 3.1*10^(-9) relative precision
BTW, using the New SI unit definitions the answer for the initial speed
uncertainty given here because the electron mass/rest energy is more
precisely known in terms of unified mass units than in terms of electron
volts, and the new SI system (2018) numbers makes the conversions between
unified mass units and kilograms or joules a matter of definition rather
than measurement.
David,
David Bowman
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thanks to your kind contribution, I see my arithmetic in the first of
three lines given below was in error
**************************
1/2 * 9.109 383 56 (11) ? 10 ?31 kg * v^2 = 10000 / 6.241*10^18
so v^2 = 10000/6.241*10^18 * 2 / ( 9.11* 10^-31) =0.730 *10^17
and V = 2.7 * 10^8 m/s !!!
**************************
1/2 * 9.109 X 10^-31 * v^2 = 10000 / 6.241 X 10^18
actually gives v^2 = 3.518*10^15 and v = 59313 km/s or
58483 km/s with the relativistic correction you gave
which agrees with your figure at the resolution I chose.
Other errors I noticed on rechecking: The breakdown voltage for air is
around 3kV/mm (not 1 kV/mm as I offered); the breakdown voltage for
(soft) vacuum only falls below 1 kV/mm at pressures between 0.3 and 10
Torr judging by the Paschen curve for N2 given here:
https://en.wikipedia.org/wiki/Paschen%27s_law#/media/File:
Paschen_curves.svg
(As remarked by Alex F Burr ~ priv. comm)
0.3 Torr is hardly beyond the capacity of a roughing vacuum pump, and
just into the "fine" or "medium" pump range according to the table given
here:
https://cas.web.cern.ch/sites/cas.web.cern.ch/files/
lectures/platjadaro-2006/chew.pdf
...and? I would expect that a school lab could easily achieve this level.
And so this schoolhouse problem does appear to be capable of
realization. (I noted in passing the similarity to Einstein's
photoelectric experiment with clean copper sheets BTW)
With thanks
Brian W
------------------------------
Message: 2
Date: Tue, 24 Apr 2018 17:12:31 -0400
From: "Don" <dgpolvani@gmail.com>
To: <Phys-L@Phys-L.org>
Subject: Re: [Phys-L] electron velocity in an electric field.
Message-ID: <000601d3dc10$f5a90a80$e0fb1f80$@gmail.com>
Content-Type: text/plain; charset="utf-8"
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
------------------------------
Message: 3
Date: Tue, 24 Apr 2018 21:55:56 -0500
From: brian whatcott <betwys1@sbcglobal.net>
To: phys-l@mail.phys-l.org
Subject: Re: [Phys-L] electron velocity in an electric field.
Message-ID: <a9b68305-532e-af86-5439-693b690ca89a@sbcglobal.net>
Content-Type: text/plain; charset=utf-8; format=flowed
On 4/24/2018 4:12 PM, Don via Phys-l wrote:
Thanks to Brian Whatcott for the non-relativistic and engineeringdetails 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.
The Larmor formulation shows the energy dissipated by radiation from a
Don
Dr. Donald Polvani
Adjunct Faculty, Physics, Retired
Anne Arundel Community College
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Forum for Physics Educators
Phys-l@mail.phys-l.org
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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
------------------------------
Subject: Digest Footer
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------------------------------
End of Phys-l Digest, Vol 160, Issue 8
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