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(This isn't homework, I'm trying to make an illustration for an article I'm writing.)

Let's say that I have a thin rectangular bar of uniform conductivity, and I have point probes at various places:

The bar has width $w$ and thickness $t$ where $t \ll w $. I am going to inject current into the bar between points $\boldsymbol A$ and $\boldsymbol D$. (let's just say 1 ampere enters at point $\boldsymbol A$ and leaves at point $\boldsymbol D$) These are centered along the bar's width and are a distance $d$ apart.

How would I figure out the equipotential lines and current density?


edit: Vague memories of college electrostatics are coming back - it's Laplace's equation that is relevant here, I need to find a solution to $\nabla^2 V = 0$, then the electric fields are just the gradient of $V$ so $J = \sigma E = \sigma \nabla V$, and I know the boundary conditions at the outside of the rectangle are that the perpendicular component of $E$ is $0$, but I'm not sure what to do next.

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    $\begingroup$ Hi Jason S. If you haven't already done so, please take a minute to read the definition of when to use the homework tag, and the Phys.SE policy for homework-like problems. $\endgroup$
    – Qmechanic
    Jul 28, 2014 at 1:25
  • $\begingroup$ REPEAT This is NOT homework. I am 40 years old, a full time electrical engineer writing an article and just trying to generate a good diagram. I can sort of picture what happens between A and D as the current spreads out from the point sources/sinks, but I'm confused what happens to the left of A and the right of D. $\endgroup$
    – Jason S
    Jul 28, 2014 at 1:40
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    $\begingroup$ My removal of your sectioning and "edit" (which you just re-added) was quite intentional, since the Physics.SE discourages revision notifications as per this meta post. Also, if you read Qmechanic's homework link, you will find that this probably qualifies as a homework-like question, since you are asking for the solution method to a specific problem. Whether or not it is homework in the sense of a student doing assigned tasks is not relevant for the policy. $\endgroup$
    – ACuriousMind
    Jul 28, 2014 at 1:54
  • $\begingroup$ Screw it then, I'm just going to use numerical methods and a diffusion equation to find equilibrium in $\nabla^2 V = 0$. $\endgroup$
    – Jason S
    Jul 28, 2014 at 2:00

1 Answer 1

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I solved my problem numerically, using the diffusion equation $\frac{\partial V}{\partial t} = -k\nabla^2 V$, with the following boundary conditions:

  • Voltage at point D is fixed at 1.0
  • Voltage along the vertical line halfway between points A and D is fixed at 0.5 (voltage at point A is 0.0, use symmetry so we don't have to simulate the left half of the material)
  • The other three boundaries have the Neumann condition where the gradient of potential has zero component perpendicular to the boundary. For a rectangular array, this just means setting the top and bottom rows and the right column equal to their nearest neighbors off the boundary.

Iterate a bunch of times until equilibrium is reached and $\frac{\partial V}{\partial t} = 0$.

Then I graphed a streamplot and contour plot in matplotlib to show the current density and equipotentials:

This seems like it would be a well-known problem, though, and I'd be happier understanding an analytic solution that uses sums of the form $A_{mn} \sinh \frac{2m\pi x}{l} \cos \frac{2n\pi y}{w}$ or whatever.

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  • $\begingroup$ Very nice! Would you share the code for the numeric solution and plot? $\endgroup$
    – Alexander
    Aug 15, 2018 at 17:00
  • $\begingroup$ I'll have to find it. This was part of a blog article but I didn't publish the code at the time. $\endgroup$
    – Jason S
    Aug 16, 2018 at 17:10
  • $\begingroup$ Thank you, that would be quite instructive and allows to test for slightly different geometries where analytical solutions might not exist. $\endgroup$
    – Alexander
    Aug 16, 2018 at 19:10

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