maxwell stress tensor and parallel-plate capacitor [Archive]

Ken Graham

Jun12-04, 07:11 AM

<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>a friend recently asked me about the meaning of the stress tensor\nbetween the plates of a parallel plate capacitor. If we assume the\nplates are infinitely-thin and that the voltage between the plates is\nconstant, then the electric field points in the -z direction (we\nignore the fringe field). The stress tensor T_{ij} is then diagonal.\nWhy are there contributions to the stress in the x and y directions\nwhen there\'s no electric field in those directions?\n\nThanks,\n\n-Ken\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>a friend recently asked me about the meaning of the stress tensor
between the plates of a parallel plate capacitor. If we assume the
plates are infinitely-thin and that the voltage between the plates is
constant, then the electric field points in the -z direction (we
ignore the fringe field). The stress tensor T_{ij} is then diagonal.
Why are there contributions to the stress in the x and y directions
when there's no electric field in those directions?

Thanks,

-Ken

Rob Woodside

Jun16-04, 05:26 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>klg207@psu.edu (Ken Graham) wrote in message news:<c461282d.0406071852.a5617cd@posting.google.c om>...\n> a friend recently asked me about the meaning of the stress tensor\n> between the plates of a parallel plate capacitor. If we assume the\n> plates are infinitely-thin and that the voltage between the plates is\n> constant, then the electric field points in the -z direction (we\n> ignore the fringe field). The stress tensor T_{ij} is then diagonal.\n> Why are there contributions to the stress in the x and y directions\n> when there\'s no electric field in those directions?\n>\n> Thanks,\n>\n> -Ken\n\nThe electromagnetic field is a mechanically strange fluid whose\nproperties are given by the Maxwell stress-energy tensor. At a point\non a field line where the energy density is E, there is a tension E\nalong the field line and a pressure E in all directions perpendicular\nto the field line.\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>klg207@psu.edu (Ken Graham) wrote in message news:<c461282d.0406071852.a5617cd@posting.google.com>...
> a friend recently asked me about the meaning of the stress tensor
> between the plates of a parallel plate capacitor. If we assume the
> plates are infinitely-thin and that the voltage between the plates is
> constant, then the electric field points in the -z direction (we
> ignore the fringe field). The stress tensor T_{ij} is then diagonal.
> Why are there contributions to the stress in the x and y directions
> when there's no electric field in those directions?
>
> Thanks,
>
> -Ken

The electromagnetic field is a mechanically strange fluid whose
properties are given by the Maxwell stress-energy tensor. At a point
on a field line where the energy density is E, there is a tension E
along the field line and a pressure E in all directions perpendicular
to the field line.

J. J. Lodder

Jun17-04, 04:11 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>Rob Woodside <rwmw@telus.net> wrote:\n\n> klg207@psu.edu (Ken Graham) wrote in message\nnews:<c461282d.0406071852.a5617cd@posting .google.com>...\n> > a friend recently asked me about the meaning of the stress tensor\n> > between the plates of a parallel plate capacitor. If we assume the\n> > plates are infinitely-thin and that the voltage between the plates is\n> > constant, then the electric field points in the -z direction (we\n> > ignore the fringe field). The stress tensor T_{ij} is then diagonal.\n> > Why are there contributions to the stress in the x and y directions\n> > when there\'s no electric field in those directions?\n> >\n> > Thanks,\n> >\n> > -Ken\n>\n> The electromagnetic field is a mechanically strange fluid whose\n> properties are given by the Maxwell stress-energy tensor. At a point\n> on a field line where the energy density is E, there is a tension E\n> along the field line and a pressure E in all directions perpendicular\n> to the field line.\n\nYou can see it (in your minds eye) by making the plates finite:\nthe field lines bulge out at the edges so the tension along them\nbalances the pressure gradient from inside.\n\nThis kind of thinking has gone out of fashion,\nbut as far as we know Maxwell himself visualised\nhis equations in these terms.\n\nJan\n\n--\n"Maxwell\'s theory is nothing but Maxwell\'s equations." (Hertz)\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>Rob Woodside <rwmw@telus.net> wrote:

> klg207@psu.edu (Ken Graham) wrote in message
news:<c461282d.0406071852.a5617cd@posting.google.com>...
> > a friend recently asked me about the meaning of the stress tensor
> > between the plates of a parallel plate capacitor. If we assume the
> > plates are infinitely-thin and that the voltage between the plates is
> > constant, then the electric field points in the -z direction (we
> > ignore the fringe field). The stress tensor T_{ij} is then diagonal.
> > Why are there contributions to the stress in the x and y directions
> > when there's no electric field in those directions?
> >
> > Thanks,
> >
> > -Ken
>
> The electromagnetic field is a mechanically strange fluid whose
> properties are given by the Maxwell stress-energy tensor. At a point
> on a field line where the energy density is E, there is a tension E
> along the field line and a pressure E in all directions perpendicular
> to the field line.

You can see it (in your minds eye) by making the plates finite:
the field lines bulge out at the edges so the tension along them
balances the pressure gradient from inside.

This kind of thinking has gone out of fashion,
but as far as we know Maxwell himself visualised
his equations in these terms.

Jan

--
"Maxwell's theory is nothing but Maxwell's equations." (Hertz)

Neil

Jun27-04, 05:58 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>"Ken Graham" <klg207@psu.edu> wrote in message\nnews:c461282d.0406071852.a5617cd@posting. google.com...\n> a friend recently asked me about the meaning of the stress tensor\n> between the plates of a parallel plate capacitor. If we assume the\n> plates are infinitely-thin and that the voltage between the plates is\n> constant, then the electric field points in the -z direction (we\n> ignore the fringe field). The stress tensor T_{ij} is then diagonal.\n> Why are there contributions to the stress in the x and y directions\n> when there\'s no electric field in those directions?\n>\n> Thanks,\n>\n> -Ken\n>\nThis isn\'t a direct answer to your question, but: Think about what happens if\nthe capacitor is in motion: Whatever the orientation, the transformation of E\n(to E\' and B\',) over the new volume, by themselves won\'t give the correct\nincrease in field energy to gamma times the original value. We do get the\ncorrect value if we take the "stress-corrected" energy and momentum of the plate\nmaterial. (Some answers in this thread allude to this, but don\'t seem to ID it\ndirectly.) This is all related to the so-called "4/3 problem" of excess momentum\nof the field of a moving charge. For more, see the thread I started some months\nago in sci.physics.electromag: "Anomaly: incorrect field energy not always\ncompensated by constraining forces" The problem I stated there is solved (even\nin the case of motion parallel to plate orientation) by the stress in the plates\n(Feynman\'s "rubber bands" in reference to the electron.) The Maxwell stress\ntensor supposedly (?) tries to get this all wrapped up in one piece, IIUC.\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>"Ken Graham" <klg207@psu.edu> wrote in message
news:c461282d.0406071852.a5617cd@posting.google.co m...
> a friend recently asked me about the meaning of the stress tensor
> between the plates of a parallel plate capacitor. If we assume the
> plates are infinitely-thin and that the voltage between the plates is
> constant, then the electric field points in the -z direction (we
> ignore the fringe field). The stress tensor T_{ij} is then diagonal.
> Why are there contributions to the stress in the x and y directions
> when there's no electric field in those directions?
>
> Thanks,
>
> -Ken
>
This isn't a direct answer to your question, but: Think about what happens if
the capacitor is in motion: Whatever the orientation, the transformation of E
(to E' and B',) over the new volume, by themselves won't give the correct
increase in field energy to \gamma times the original value. We do get the
correct value if we take the "stress-corrected" energy and momentum of the plate
material. (Some answers in this thread allude to this, but don't seem to ID it
directly.) This is all related to the so-called "4/3 problem" of excess momentum
of the field of a moving charge. For more, see the thread I started some months
ago in sci.physics.electromag: "Anomaly: incorrect field energy not always
compensated by constraining forces" The problem I stated there is solved (even
in the case of motion parallel to plate orientation) by the stress in the plates
(Feynman's "rubber bands" in reference to the electron.) The Maxwell stress
tensor supposedly (?) tries to get this all wrapped up in one piece, IIUC.