Truncated conical magnet [Archive]

Ramez A. Elgammal

Oct8-04, 06:20 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>\nI have a question about simulating the magnetic field of a truncated\nconcial magnet. The cross-section of this device would be a trapezoid.\nAnother way to think about it would be as a cylinder with tapering such\nthat the radius of the top > radius of the bottom.\n\nI will assume that the magnet is uniformly magnetized. The goal\nis to have the magnetic field at the tapered end be effectively larger\nthan the bulk magnetization of the material, therefore forming a pole\npiece. I don\'t have much experience with E&M and I was wondering if I\ncould model this as a series of current carrying rings of decreasing\nradius and then summing over these coils. I also am not sure how this\nwould be set up. What would be an expression for the surface current\ndensity?\n\nIdeally, this would lead to an analytical expression along the symmetry\naxis and off-axis a numerical solution would suffice. Do you know if this\nproblem has been solved?\n\nThanks!!!\n____________________________ _____________\nRamez Elgammal\nWeitekamp Group\nCaltech MC 127-72\nPasadena, CA 91125\n(626) 395-6573 Office\n(626) 568-8824 Fax\n_________________________________________\n\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>I have a question about simulating the magnetic field of a truncated
concial magnet. The cross-section of this device would be a trapezoid.
Another way to think about it would be as a cylinder with tapering such
that the radius of the top > radius of the bottom.

I will assume that the magnet is uniformly magnetized. The goal
is to have the magnetic field at the tapered end be effectively larger
than the bulk magnetization of the material, therefore forming a pole
piece. I don't have much experience with E&M and I was wondering if I
could model this as a series of current carrying rings of decreasing
radius and then summing over these coils. I also am not sure how this
would be set up. What would be an expression for the surface current
density?

Ideally, this would lead to an analytical expression along the symmetry
axis and off-axis a numerical solution would suffice. Do you know if this
problem has been solved?

Thanks!!!
__{_______________________________________}
Ramez Elgammal
Weitekamp Group
Caltech MC 127-72
Pasadena, CA 91125
(626) 395-6573 Office
(626) 568-8824 Fax
__{_______________________________________}

Pierre Asselin

Oct11-04, 03:52 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>\n\nRamez A. Elgammal <ramez@its.caltech.edu> wrote:\n\n> I have a question about simulating the magnetic field of a truncated\n> concial magnet. The cross-section of this device would be a trapezoid.\n> Another way to think about it would be as a cylinder with tapering such\n> that the radius of the top > radius of the bottom.\n\n> I will assume that the magnet is uniformly magnetized.\n\nI assume magnetized parallel to the axis of rotation ?\n\n\n> The goal\n> is to have the magnetic field at the tapered end be effectively larger\n> than the bulk magnetization of the material, therefore forming a pole\n> piece. I don\'t have much experience with E&M and I was wondering if I\n> could model this as a series of current carrying rings of decreasing\n> radius and then summing over these coils. I also am not sure how this\n> would be set up. What would be an expression for the surface current\n> density?\n\nI don\'t know if you\'ll get a field larger than the magnet\'s M but you\ncan try. You can indeed replace the magnetization by an equivalent\ncurrent density J= curl(M) and compute the magnetic induction by\nintegrating the Biot-Savart law.\n\nThis curl(M) is to be taken in the sense of distributions. J is zero\ninside the magnet, since M is constant, but there is a surface current\ndensity -Mxn circulating around the curved faces (n= outward unit normal).\nThere is no surface current density on the end faces, since their\nsurface normals are parallel to M.\n\n\n> Ideally, this would lead to an analytical expression along the symmetry\n> axis and off-axis a numerical solution would suffice.\n\nThe off-axis case should be doable in terms of elliptic integrals.\n\n\n> Do you know if this problem has been solved?\n\nProbably, but I wouldn\'t know where to look.\n\n\n--\npa at panix dot com\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>Ramez A. Elgammal <ramez@its.caltech.edu> wrote:

> I have a question about simulating the magnetic field of a truncated
> concial magnet. The cross-section of this device would be a trapezoid.
> Another way to think about it would be as a cylinder with tapering such
> that the radius of the top > radius of the bottom.

> I will assume that the magnet is uniformly magnetized.

I assume magnetized parallel to the axis of rotation ?

> The goal
> is to have the magnetic field at the tapered end be effectively larger
> than the bulk magnetization of the material, therefore forming a pole
> piece. I don't have much experience with E&M and I was wondering if I
> could model this as a series of current carrying rings of decreasing
> radius and then summing over these coils. I also am not sure how this
> would be set up. What would be an expression for the surface current
> density?

I don't know if you'll get a field larger than the magnet's M but you
can try. You can indeed replace the magnetization by an equivalent
current density J= curl(M) and compute the magnetic induction by
integrating the Biot-Savart law.

This curl(M) is to be taken in the sense of distributions. J is zero
inside the magnet, since M is constant, but there is a surface current
density -Mxn circulating around the curved faces (n= outward unit normal).
There is no surface current density on the end faces, since their
surface normals are parallel to M.

> Ideally, this would lead to an analytical expression along the symmetry
> axis and off-axis a numerical solution would suffice.

The off-axis case should be doable in terms of elliptic integrals.

> Do you know if this problem has been solved?

Probably, but I wouldn't know where to look.

--
pa at panix dot com

Igor Khavkine

Oct11-04, 03:52 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>\nOn Fri, 08 Oct 2004 11:20:35 +0000, Ramez A. Elgammal wrote:\n\n>\n> I have a question about simulating the magnetic field of a truncated\n> concial magnet. The cross-section of this device would be a trapezoid.\n> Another way to think about it would be as a cylinder with tapering such\n> that the radius of the top > radius of the bottom.\n>\n> I will assume that the magnet is uniformly magnetized. The goal is to\n> have the magnetic field at the tapered end be effectively larger than the\n> bulk magnetization of the material, therefore forming a pole piece. I\n> don\'t have much experience with E&M and I was wondering if I could model\n> this as a series of current carrying rings of decreasing radius and then\n> summing over these coils. I also am not sure how this would be set up.\n> What would be an expression for the surface current density?\n\nSuppose the magnetization per unit volume in the magnet is M, then\nat (vector) position r from a given infinitesimal volume dV, the\ninfinitesimal magnetic field produced by it is\n\ndB = (mu_0/4pi) [3*(M.r)*r-M*|r|^2]/|r|^5 dV.\n\nTo get the total magnetic field you must integrate over the entire volume\nof the magnet.\n\nAlternatively, you could calculate the bound surface and volume currents\ndue to the magnetization an use the Biot-Savart law. The bound surface\ncurrent density is K_b = M x n, where n is the outward unit normal on the\nsurface, and the bound current density is J_b = curl M.\n\n> Ideally, this would lead to an analytical expression along the symmetry\n> axis and off-axis a numerical solution would suffice. Do you know if\n> this problem has been solved?\n\nI\'m sure someone somewhere has looked at this problem before. The question\nis whether it will take you more time to find the existing solution or\njust evaluate the integrals that you get.\n\nHope this helps.\n\nIgor\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>On Fri, 08 Oct 2004 11:20:35 +0000, Ramez A. Elgammal wrote:

>
> I have a question about simulating the magnetic field of a truncated
> concial magnet. The cross-section of this device would be a trapezoid.
> Another way to think about it would be as a cylinder with tapering such
> that the radius of the top > radius of the bottom.
>
> I will assume that the magnet is uniformly magnetized. The goal is to
> have the magnetic field at the tapered end be effectively larger than the
> bulk magnetization of the material, therefore forming a pole piece. I
> don't have much experience with E&M and I was wondering if I could model
> this as a series of current carrying rings of decreasing radius and then
> summing over these coils. I also am not sure how this would be set up.
> What would be an expression for the surface current density?

Suppose the magnetization per unit volume in the magnet is M, then
at (vector) position r from a given infinitesimal volume dV, the
infinitesimal magnetic field produced by it is

dB = (\mu_0/4pi) [3*(M.r)*r-M*|r|^2]/|r|^5 dV.

To get the total magnetic field you must integrate over the entire volume
of the magnet.

Alternatively, you could calculate the bound surface and volume currents
due to the magnetization an use the Biot-Savart law. The bound surface
current density is K_b = M x n, where n is the outward unit normal on the
surface, and the bound current density is J_b = curl M.

> Ideally, this would lead to an analytical expression along the symmetry
> axis and off-axis a numerical solution would suffice. Do you know if
> this problem has been solved?

I'm sure someone somewhere has looked at this problem before. The question
is whether it will take you more time to find the existing solution or
just evaluate the integrals that you get.

Hope this helps.

Igor