Solving Non-linear Problem for x between 0 and L

[JulieK]

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

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

 

[mfb]

From the mathematical side, your given functions/values say nothing about V(x) apart from x=a and x=b. Physics limits V(x) to be between V(a) and V(b) (unless you have some external connections to the wire), but nothing else. You need more assumptions about the potential to get V(x). Do you expect a linear shape of the voltage?

 

[JulieK]

I assume the following hypothetical situation hoping this may bring us closer to the solution although it is not realistic.

Let's assume that the radius of this exceptional wire at any point x is proportional to the voltage at that point, that is the radius expands proportionally to the voltage. I am thinking of a possible balance relation that could be exploited to find the optimal V(x) which satisfies this relation and the two boundary conditions, as well as f. Can we find a solution from this extra condition?

 

[mfb]

If you change the voltage, the wire changes its diameter? ;)

that is the radius expands proportionally to the voltage

The voltage relative to what? Voltage needs a reference point to be meaningful for a point in the circuit.

You can set up a differential equation for basically any assumption you like, and solve it (if it is not too complicated), sure.

 

[JulieK]

"The voltage relative to what?"
To the same reference voltage to which $V_A$ and $V_B$ are defined.

"You can set up a differential equation for basically any assumption you like, and solve it (if it is not too complicated), sure. "
Can you suggest a form for this differential equation?

 

[JJacquelin]

Hi !
The general solution, only based on the first wording, is :
V(x) = Va +(f(x)-f(xa))(Vb-Va)/(f(xb)-f(xa))
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.

 

[mfb]

To the same reference voltage to which $V_A$ and $V_B$ are defined.

The problem is that this definition is arbitrary as well.

Can you suggest a form for this differential equation?

As soon as I understand what you actually want, sure.

As I is constant, the equation should look like $\frac{dV}{dx}=I R(V,x)$ where R(V,x) is the differential resistance (resistance per length) of the wire.