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)
- Delivered-to: l-mathgroup@mail-archive0.wolfram.com
- Delivered-to: l-mathgroup@wolfram.com
- Delivered-to: mathgroup-newout@smc.vnet.net
- Delivered-to: mathgroup-newsend@smc.vnet.net
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!