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Find a function $f$ such that the vector field $\nabla f$ looks as in following picture:

this picture

I'm having some trouble with this one. I'm looking at $y = 0$, and realizing that it is negative between $-1 \le x \le 1$, but positive elsewhere. Any help PLEASE?

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  • $\begingroup$ This is how the field of two charges with opposite sign should look like. $\endgroup$
    – user228113
    Nov 8, 2017 at 16:39
  • $\begingroup$ looks like the electric field from a dipole to me. $\endgroup$
    – achille hui
    Nov 8, 2017 at 16:40
  • $\begingroup$ Yes, but I'm confused as to how to find the function of a dipole. $\endgroup$
    – Dana K
    Nov 8, 2017 at 16:44
  • $\begingroup$ Yes this is a dipole! The potential is $$f(\vec{r})=k\Big(\frac{1}{|\vec{r}-\vec{r}_0|}-\frac{1}{|\vec{r}+\vec{r}_0|}\Big)$$ and $\vec{r}_0=(x_0,0,0)$ with $x_0=1$. Here is $k$ a factor for the units. $\endgroup$
    – Fakemistake
    Nov 8, 2017 at 16:44
  • $\begingroup$ Do you have actual values for the vector field or are you supposed to come up with an approximation from the image? If the latter, it looks to me like the streamlines are a family of ellipses that pass through a fixed pair of points. The equipotential lines (level curves of $f$) are orthogonal to these streamlines, which suggests to me a family of confocal hyperbolas. $\endgroup$
    – amd
    Nov 8, 2017 at 23:12

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