Hi, This is my fisrt time to use the site. So, I will givesome background about

Question

Hi, This is my fisrt time to use the site. So, I will givesome background about myself. I studied Computational FluidDynamics (CFD) at school more than 10 year ago.  Iconsider myself having decent math background and now I am teachingmyself Emag. I have only finished readingelectrostatics. And here is my first question. Suppose that I have a few conductors in a space each with atotal charge of Q1, to Qn on them respectively. I would liketo determine the electric field, E, outside the conductor. The best way, IMO, is through the potential. If Iknow the potential or the normal derivative thereof on eachconductor surface (but not both), then I can solve the Laplacianpotential equation and then obtain E from the potential. Itis probably hard to know the potential on the surfaces (but ifyou know a way, please tell me.). However, normal derivative of thepotential on the surface is related to surface chargedistribution. But unfortunately, I only know total chargeinstead of surface change distribution. So, IMO the questionboils down to how to determine surface charge density on conductorsurfaces given total charges. To make the problem evensimpler without loss of generality, suppose I have only oneconductor. So, the question seems to be how to determinesurface charge density on one conductor in a space with given totalcharge without additional charge distribution in the space? I think that surface charge density can be determined from thefact that E is zero within the conductor. But how? I canthink of a numerical way but with some uncertainties. I candiscretize the conducor surface into, say 1000, pieces. ThenI will need 999 equations along with the fact that we know thetotal charge to determine the surface charge on those discrete 1000pieces. To achieve that, I can pick up 999 points inside theconductor and use Couloumb's law plus superposition to derive an Eequation for each point. Of course, E is zero for each of the999 points. But this method obviously has a flaw. For the1000 equations that I end up with, how do I make sure that they areall independent? Physically, duing the point picking process,some subsequent points picked may already have E=0 and thus no newinformation was added to the system of equations, and thefore thesolution to the system of equations is not unique. But weknow the solution to this problem is unique from the seconduniqueness therom mentioned in Griffiths book. I have been searching Jackson's book for the answer but couldnot find one. So, thanks for your help and I look forward toyour reply. Regards, atuzhai PS. This is not a test book problem. But the field asksfor one, so I put a random number there. Hi, This is my fisrt time to use the site. So, I will givesome background about myself. I studied Computational FluidDynamics (CFD) at school more than 10 year ago.  Iconsider myself having decent math background and now I am teachingmyself Emag. I have only finished readingelectrostatics. And here is my first question. Suppose that I have a few conductors in a space each with atotal charge of Q1, to Qn on them respectively. I would liketo determine the electric field, E, outside the conductor. The best way, IMO, is through the potential. If Iknow the potential or the normal derivative thereof on eachconductor surface (but not both), then I can solve the Laplacianpotential equation and then obtain E from the potential. Itis probably hard to know the potential on the surfaces (but ifyou know a way, please tell me.). However, normal derivative of thepotential on the surface is related to surface chargedistribution. But unfortunately, I only know total chargeinstead of surface change distribution. So, IMO the questionboils down to how to determine surface charge density on conductorsurfaces given total charges. To make the problem evensimpler without loss of generality, suppose I have only oneconductor. So, the question seems to be how to determinesurface charge density on one conductor in a space with given totalcharge without additional charge distribution in the space? I think that surface charge density can be determined from thefact that E is zero within the conductor. But how? I canthink of a numerical way but with some uncertainties. I candiscretize the conducor surface into, say 1000, pieces. ThenI will need 999 equations along with the fact that we know thetotal charge to determine the surface charge on those discrete 1000pieces. To achieve that, I can pick up 999 points inside theconductor and use Couloumb's law plus superposition to derive an Eequation for each point. Of course, E is zero for each of the999 points. But this method obviously has a flaw. For the1000 equations that I end up with, how do I make sure that they areall independent? Physically, duing the point picking process,some subsequent points picked may already have E=0 and thus no newinformation was added to the system of equations, and thefore thesolution to the system of equations is not unique. But weknow the solution to this problem is unique from the seconduniqueness therom mentioned in Griffiths book. I have been searching Jackson's book for the answer but couldnot find one. So, thanks for your help and I look forward toyour reply. Regards, atuzhai PS. This is not a test book problem. But the field asksfor one, so I put a random number there.

Explanation / Answer

Through some more study, I found the answer to my ownquestion. We start by assuming that the potential is 1 on theconductor surface. Once V is solved based on that boundarycondition (Dirichlet), the surface charge distribution can becalculated. Of course, the total amout of charge will not becorrect. But the charge distribution can be scaled up/down tomatch the total amount of charge. That will give us the finalcharge distribution on the conductor surface. This ispossible because of the following. When doubling the chargeeverywhere, E will double due to superposition. If E doubles,V will double. The doubled V will still satisfy the Laplaceequation. With multiple conductors, we start with the problem of onlythe first conductor having charge and the rest is grounded. Thiswill give potential Vv1. Do the same for Vv2, ... Vvn. It is easy to verify that the final solution is :                 V = Vv1+...+Vvn Once V is know, it is trivial to calculate surface chargedistribution from E on the surface.

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