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How many points exist at which electrostatic field due to a point charge is zero? Clearly we have, due to a point charge Q,
$$\textbf E(r,\theta)=\frac{Q}{4\pi\epsilon_{o}r^2}(\cos(\theta)\textbf i+\sin(\theta)\textbf j)$$
and
$$\lim_{r\to\infty}\textbf E(r,\theta)=\textbf0$$
$$\forall\theta\in[0,2\pi]$$ Thus can we conclude there exists infinitely many points at an infinite radial distance from a point charge where the electric field is zero? Is this okay?

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    $\begingroup$ You need to be careful using infinity in calculations. The limit of the field as $r \rightarrow \infty$ is indeed zero, but that's just a limit. If you consider a single charge in its own universe there is nowhere in the universe where the field is zero. $\endgroup$ – John Rennie Jul 21 '17 at 7:12
  • $\begingroup$ So we can't say what happens at $\infty$? and we can only talk about as $\infty$ is approached? $\endgroup$ – Nishant Garg Jul 21 '17 at 7:17
  • $\begingroup$ We can say that as $r$ approaches $\infty$, $\textbf E$ approaches $\textbf 0$, but exactly what happens at $\infty$ is undefined, is that right? $\endgroup$ – Nishant Garg Jul 21 '17 at 7:19
  • $\begingroup$ Yes, or at least that is the position I would take. I suspect most physicists believe no real quantity is ever infinite, with the possible exception of the size of the universe though even that is contentious. $\endgroup$ – John Rennie Jul 21 '17 at 7:19
  • $\begingroup$ I don't think it makes sense to say what happens at $\infty$ is undefined. The field has a well defined limit - it's just that you could never reach that limit. $\endgroup$ – John Rennie Jul 21 '17 at 7:21
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Infinitely distant points in three-dimensional space can be considered as none (i.e. "such concepts are not points at all"), one, or infinitely many.

The first interpretation is the most straightforward. From John Rennie's comment:

The limit of the field as $r→∞$ is indeed zero, but that's just a limit. If you consider a single charge in its own universe there is nowhere in the universe where the field is zero.

But there are some theoretical concepts in mathematics that could apply:

  • Projectively extended real line for one dimension, Riemann sphere for two; this can be generalized for more dimensions. They treat infinity as a single point. Also compare: To put a point at infinity.
  • Extended real number line tells apart $-∞$ from $∞$, so there are two points at infinity in this case. Multi-dimensional generalization would be something like projective geometry where points at infinity are considered different according to the directions of straight lines they belong to. This leads to infinitely many (uncountably many) points at infinity. I guess you have intuitively applied something similar in your conclusion.

My conclusion: the answer to your question depends on the theoretical concept of space you choose.

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  • $\begingroup$ I thought about it this way: r can either take a large finite value or an infinite value. For a large finite value, E is non zero and assigning an infinite value to r is not possible as infinity is not a number. So E is non zero for finite values and not really defined at infinity. We can only say E approaches 0 as r approaches infinity. If E was 0 at infinity, we never would've needed a concept like limit. So there exists no points in the finite space, and since we can't evaluate for infinite distance, we don't talk about that. $\endgroup$ – Nishant Garg Jul 21 '17 at 8:02
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I now believe the answer to be no points. E is non zero for finite r>0 and since r can never reach infinity, we cannot talk about points at infinity. It would be just wrong to say we have infinitely many points where E is 0.

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