Ramo's theorem [Archive] - Physics Forums

Peter

May29-04, 11:57 AM

William R. Frensley

May31-04, 04:22 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Peter wrote:\n> I\'m a mathematics student, currently doing my thesis in new battery\n> technologies. We are modelling an existing battery. Someone told me\n> that Ramo\'s theorem could be useful for me. However, I have\n> difficulties finding useful information about this theorem. Is there\n> someone who has a proper explanation of this theorem and who perhaps\n> also knows the derivation of Ohm\'s law from this theorem?\n>\nThe same theorem was published by Simon Ramo (still known to engineers\nas the co-author of the standard engineering text on electromagentic\ntheory, and to a few as the "R" in "TRW") and by William Shockley\n(of the transistor) within a year of each other. The references are:\n\nShockley, W., 1938, Currents to Conductors Induced by a Moving\nPoint Charge, Journal of Applied Physics, vol. 9, pp. 635-6.\n\nRamo, Simon, 1939, Currents Induced by Electron Motion, Proceedings\nof the IRE, vol. 27, pp. 584-5.\n\nThe subject of the theorem is the terminal currents produced by a charge\nmoving through a space in which there are two or more metallic electrodes.\n(A topic that should have been of sufficient interest in the era of vacuum\ntube electronics that it is surprising that the publication dates were as\nlate as 1938-9.) The content of the theorem is this: In a simple case\nwith a charge Q moving between the plates of a large parallel-plate capacitor\nwith plate separation d, I =D Qv/d. Here v is the velocity of the charge\nnormal to the plates, and I is the current that would be measured by an\nammeter connecting the plates.\n\nThe general form of the theorem is that the current induced in electrode\nj is given by Q v (dot) (grad)(partial phi)/(partial Vj), where phi is\nthe electrostatic potential of the whole system, and Vj is the voltage\non electrode j. We can view the (partial phi)/(partial Vj) as the\nelectrostatic potential that occurs when we put a unit voltage on\nelectrode j and keep all other electrodes at 0 potential. (And, by the\nway, one can derive this theorem in a much simpler way than Ramo or\nShockley did by invoking the principle of virtual work.)\n\nOne really can\'t derive Ohm\'s law from the Shockley-Ramo theorem,\nbecause Ohm\'s law is a statement about charge carrier motion, and the\ntheorem is a statement about the electrical consequences of such motion.\nIf you start with Ohm\'s law in its most microscopic form: v =D\n(mobility)(electric field), then you can use the Shockley-Ramo theorem\nto sum over all the charges in a macroscopic resistor to get I=DV/R.\nBut, to get the drift velocity proporitonal to the electric field, you\nneed to consider a microscopic model of charge carrier motion which is\ndominated by inelastic collisions. This is done in terms of what\nsolid-state physicists know as the Boltzmann equation (a broader class\nof transport equations than what the chemical disciplines consider to be\nthe Boltzmann equation).\n\n- Bill Frensley\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Peter wrote:
> I'm a mathematics student, currently doing my thesis in new battery
> technologies. We are modelling an existing battery. Someone told me
> that Ramo's theorem could be useful for me. However, I have
> difficulties finding useful information about this theorem. Is there
> someone who has a proper explanation of this theorem and who perhaps
> also knows the derivation of Ohm's law from this theorem?
>
The same theorem was published by Simon Ramo (still known to engineers
as the co-author of the standard engineering text on electromagentic
theory, and to a few as the "R" in "TRW") and by William Shockley
(of the transistor) within a year of each other. The references are:

Shockley, W., 1938, Currents to Conductors Induced by a Moving
Point Charge, Journal of Applied Physics, vol. 9, pp. 635-6.

Ramo, Simon, 1939, Currents Induced by Electron Motion, Proceedings
of the IRE, vol. 27, pp. 584-5.

The subject of the theorem is the terminal currents produced by a charge
moving through a space in which there are two or more metallic electrodes.
(A topic that should have been of sufficient interest in the era of vacuum
tube electronics that it is surprising that the publication dates were as
late as 1938-9.) The content of the theorem is this: In a simple case
with a charge Q moving between the plates of a large parallel-plate capacitor
with plate separation d, I =D Qv/d. Here v is the velocity of the charge
normal to the plates, and I is the current that would be measured by an
ammeter connecting the plates.

The general form of the theorem is that the current induced in electrode
j is given by Q v (dot) (grad)(partial \phi)/(partial Vj), where \phi is
the electrostatic potential of the whole system, and Vj is the voltage
on electrode j. We can view the (partial \phi)/(partial Vj) as the
electrostatic potential that occurs when we put a unit voltage on
electrode j and keep all other electrodes at potential. (And, by the
way, one can derive this theorem in a much simpler way than Ramo or
Shockley did by invoking the principle of virtual work.)

One really can't derive Ohm's law from the Shockley-Ramo theorem,
because Ohm's law is a statement about charge carrier motion, and the
theorem is a statement about the electrical consequences of such motion.
If you start with Ohm's law in its most microscopic form: v =D
(mobility)(electric field), then you can use the Shockley-Ramo theorem
to sum over all the charges in a macroscopic resistor to get I=DV/R.
But, to get the drift velocity proporitonal to the electric field, you
need to consider a microscopic model of charge carrier motion which is
dominated by inelastic collisions. This is done in terms of what
solid-state physicists know as the Boltzmann equation (a broader class
of transport equations than what the chemical disciplines consider to be
the Boltzmann equation).

- Bill Frensley