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Thread: Non-linear problem

  1. #1
    Junior Member
    Jun 2010

    Non-linear problem

    Assume we have a straight piece of wire with two end points $\displaystyle A$ and
    $\displaystyle B$ and with length $\displaystyle L$ where $\displaystyle x_{A}=0$ and $\displaystyle x_{B}=L$. The wire
    has non-ohmic resistance and hence the current is not proportional
    to the potential difference, i.e. $\displaystyle \left(V_{A}-V_{B}\right)$. In
    fact the current is a function of the voltage at $\displaystyle A$ and $\displaystyle B$, that
    is $\displaystyle I=f\left(V_{A},V_{B}\right)$.

    I know $\displaystyle f$ and hence I know the current. However, I do not know $\displaystyle V$
    as a function of $\displaystyle x$ $\displaystyle \left(0<x<L\right)$. I tried several mathematical
    tricks, mainly from the calculus of variation, trying to find $\displaystyle V\left(x\right)$
    but I did not get a sensible result. Can any one suggest a method
    (whether from the calculus of variation or other branches of mathematics)
    to solve this problem and obtain $\displaystyle V\left(x\right)$.
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  2. #2
    Senior Member
    Aug 2011

    Re: Non-linear problem

    Hi !
    The general solution, only based on the first wording, is :
    V(x) = Va +(f(x)-f(xa))(Vb-Va)/(f(xb)-f(xa)) or V(x) = Va +(f(x)-f(0))(Vb-Va)/(f(L)-f(0))
    where f(x) is any continuous function.
    You cannot determine what kind of function f(x) is without a descriptive physical model for the electrical behaviour from A to B.
    Last edited by JJacquelin; Apr 12th 2013 at 10:51 PM.
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