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I actually got a problem in understanding while doing physics. It is said that the maximum value of electric field on axis of a charged ring of radius a is at a distance $\frac{a}{\sqrt2}$. This corresponds to when $\frac{\mathrm{d}}{\mathrm{d}x}$ of expression for E = 0. What does $\frac{\mathrm{d}E}{\mathrm{d}x}=0$ mean exactly here? If rate of change of electric field with distance is zero, does it mean that if an object is placed at $\frac{a}{\sqrt2}$ and moved slightly by a distance $\mathrm{d}x$, the field remains the same? In that case when will it begin to change? Would someone please give an easy intuitive explanation of $\frac{\mathrm{d}}{\mathrm{d}x}$ being zero at a point?

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    $\begingroup$ You might want to take a look at critical points. $\endgroup$
    – EuYu
    Aug 25, 2018 at 11:28

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Not sure about the problem but the strength of the electrical field, $E$, depends on your distance from it, which I assume is $x$. $$\frac{dE}{dx}$$ then, is how much the strength of the field changes as you move along the the axis. Now suppose that we're at a point, $P$, where $\frac{dE}{dx}=0$, what would happen to the field if me moved backwards, and what would happen if we moved forward? We got a lot of possibilities:

  • Nothing happens, $\frac{dE}{dx}=0$ when going forward and backward. This would mean that the field doesn't change at all, so $E$ must be constant.
  • When going backward $\frac{dE}{dx}<0$, that is $E$ is getting smaller, and when going forward $\frac{dE}{dx}<0$ also, that is $E$ is again getting smaller. So we're at a point $P$, and no matter which direction we move the field is getting smaller, this would mean that the field is strongest at $P$.
  • With the case $\frac{dE}{dx}>0$ in any direction from $P$ we would get the opposite conclusion, no matter which way we move the field, $E$, is increasing so at $P$ the field has a minimium value.

There are also some other cases, which are more complicated, but the upshot is that when you're at a point $P$ where $\frac{dE}{dx}=0$ you're at a point where the field is at is strongets or weakest (you should also look up global/local maximum/minimum).

KhanAcademy has a lot of helpful videos on this subject.

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