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Electric potential and interface @y=0

  • To: mathgroup at smc.vnet.net
  • Subject: [mg127558] Electric potential and interface @y=0
  • From: "Mat' G\." <ellocomateo at free.fr>
  • Date: Sat, 4 Aug 2012 06:00:26 -0400 (EDT)
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Hello,

Given a list of charges and their position of the form
     {{q1,{x1,y1}},{q2,{x3,y3}},...,{qn,{xn,yn}}}
and a dielectric interface such that the dielectric constant
     is \epsilon_A for y>0
     is \epsilon_B for y<=0
How can I compute the electric potential everywhere?
I need to get an expression which is derivable to then compute the 
electric field.

I have an expression for an interface along x=0 (instead of y=0), but I 
fail at adapting it for my case:

\!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]\((
\*FractionBox[\((1 -
\*FractionBox[\(x\ charges[\([\)\(i, 2, 1\)\(]\)]\),
SqrtBox[
SuperscriptBox[\((x\ charges[\([\)\(i, 2,
               1\)\(]\)])\), \(2\)]]])\), \(2\)]\ \((
\*FractionBox[\(2\ charges[\([\)\(i, 1\)\(]\)]\), \(4  \[Pi]\ \((
\*SubscriptBox[\(\[Epsilon]\), \(water\)] +
\*SubscriptBox[\(\[Epsilon]\), \(air\)])\)
\*SqrtBox[\(
\*SuperscriptBox[\((x - charges[\([\)\(i, 2, 1\)\(]\)])\), \(2\)] +
\*SuperscriptBox[\((y -
              charges[\([\)\(i, 2, 2\)\(]\)])\), \(2\)]\)]\)]\ )\) + \((
\*FractionBox[\(1 +
\*FractionBox[\(x\ charges[\([\)\(i, 2, 1\)\(]\)]\),
SqrtBox[
SuperscriptBox[\((x\ charges[\([\)\(i, 2,
               1\)\(]\)])\), \(2\)]]]\), \(2\)])\)
\*FractionBox[\(charges[\([\)\(i, 1\)\(]\)]\), \(4\ \[Pi]\)]\ \((\((
\*FractionBox[\(1 -
\*FractionBox[\(x\),
SqrtBox[
SuperscriptBox[\((x)\), \(2\)]]]\), \(2
\*SubscriptBox[\(\[Epsilon]\), \(air\)]\)])\) \((
\*FractionBox[\(1\),
SqrtBox[\(
\*SuperscriptBox[\((x - charges[\([\)\(i, 2, 1\)\(]\)])\), \(2\)] +
\*SuperscriptBox[\((y -
                 charges[\([\)\(i, 2, 2\)\(]\)])\), \(2\)]\)]] - \((
\*FractionBox[\(
\*SubscriptBox[\(\[Epsilon]\), \(water\)] -
\*SubscriptBox[\(\[Epsilon]\), \(air\)]\), \(\((
\*SubscriptBox[\(\[Epsilon]\), \(water\)] +
\*SubscriptBox[\(\[Epsilon]\), \(air\)])\)
\*SqrtBox[\(
\*SuperscriptBox[\((x + charges[\([\)\(i, 2, 1\)\(]\)])\), \(2\)] +
\*SuperscriptBox[\((y -
                   charges[\([\)\(i, 2,
                    2\)\(]\)])\), \(2\)]\)]\)])\))\) + \ \((
\*FractionBox[\(1 +
\*FractionBox[\(x\),
SqrtBox[
SuperscriptBox[\((x)\), \(2\)]]]\), \(2
\*SubscriptBox[\(\[Epsilon]\), \(water\)]\)])\) \((
\*FractionBox[\(1\),
SqrtBox[\(
\*SuperscriptBox[\((x - charges[\([\)\(i, 2, 1\)\(]\)])\), \(2\)] +
\*SuperscriptBox[\((y -
                 charges[\([\)\(i, 2, 2\)\(]\)])\), \(2\)]\)]] - \((
\*FractionBox[\(
\*SubscriptBox[\(\[Epsilon]\), \(air\)] -
\*SubscriptBox[\(\[Epsilon]\), \(water\)]\), \(\((
\*SubscriptBox[\(\[Epsilon]\), \(water\)] +
\*SubscriptBox[\(\[Epsilon]\), \(air\)])\)
\*SqrtBox[\(
\*SuperscriptBox[\((x + charges[\([\)\(i, 2, 1\)\(]\)])\), \(2\)] +
\*SuperscriptBox[\((y -
                   charges[\([\)\(i, 2,
                    2\)\(]\)])\), \(2\)]\)]\)])\))\))\))\)\);


Thanks for helping!



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