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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• Delivered-to: l-mathgroup@wolfram.com
• Delivered-to: mathgroup-newout@smc.vnet.net
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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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