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A rod of linear charge density a of length h, What will be the electric field at an axial point at a distance x from end of the rod (the end at which the origin is chosen for defining charge density function)?

Please solve the problem formally without using non standard analysis, neither infinitesimals as it is a part of non standard analysis?

The problem is off course trivial (using differentials).

That is :-

Consider the charge in infintesimal element $dx$ at a distance $x$ from the end, then electric field due this charge at a distance $x$ from the endpoint is $$dF= \frac{\rho}{4\pi \epsilon ((h-x)^2) }dx$$

Thus the net electric field due the full rod is given by

$$F=\int_{0}^{h}\frac{\rho}{4\pi \epsilon ((h-x)^2) }dx$$

which can easily be found out.

But this is not my question.

The sole purpose of mine to raise the question is to do the problem in a mathematically formal manner without using differentials ,in the realm of standard analysis .

Someone said to me that the problem could be done in a formal way even without using infinitesimals so being curious I posted for help?

The main need for the question is to increase the understanding of formal mathematics in physics.

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    $\begingroup$ I think a more formal argument would use limits instead of infinitesimals. You would have to write an expression for the field $\Delta F$ due to a small but finite element $\Delta x$, and then find the limit of $\frac {\Delta F}{\Delta x}$ as $\Delta x \rightarrow 0$. This would involve introducing some additional terms which tend to $0$ in the limit - this is the part that the less formal argument misses out. $\endgroup$ – gandalf61 Feb 1 at 14:25
  • $\begingroup$ Yes that is exactly the question the so called (error term )$/\Delta x=0$ as $\Delta x$ tends to zero , but how? $\endgroup$ – Bijayan Ray Feb 1 at 14:28
  • $\begingroup$ @gandalf61 I think that the question requires more assumptions which may become evident as one writes the equation? $\endgroup$ – Bijayan Ray Feb 1 at 14:42
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Do you realize that your question is not a mathematics question? So how can you ask for a formal solution? That is precisely the problem here; you have not formally defined "rod" and "linear charge density" and "electric field", besides other terms. Furthermore, you need to prove formal axioms governing the electric field. After you do so in a natural way, then it should be obvious that the solution is a trivial consequence of the axioms, because the integral is exactly how we define the net electric field at each point in euclidean space!

Think first about the far simpler physical notion of mass of an object, given the density function. There is no way to derive the mass as an integral of the density, without assuming it or something equivalent. After all, this is how we define the density function in the first place, namely the limit of mass divided by volume of a region surrounding the point whose diameter goes to zero, and this would imply the integral.

In cases such as the charge density and electric field, I think it's fair to say that the classical viewpoint would simply axiomatize electric field as the integral of the charge density, because that's the easiest way to do it.

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  • $\begingroup$ Anyway, the definition of electric field in sense of mathematics , atleast for this problem is the for a charge $q$ in 3 dimension space the electric field in the vector $$\vec F= \frac{q}{4\pi \epsilon (r^3) } \vec r$$ taking the charge to be at the origin. $\endgroup$ – Bijayan Ray Feb 2 at 10:54
  • $\begingroup$ The definition of linear charge density at a point on the rod (a line in terms of mathematics) is a integrable function $\rho (x)$ such that the $$q=\int_{0}^{x} \rho(t)dt $$ where q is the charge in present in the range of 0 to x on the line (rod) for $0<x<=h$ and $\rho (x)=0 $ for $x>h$ $\endgroup$ – Bijayan Ray Feb 2 at 10:57
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    $\begingroup$ @BijayanRay: What you're stating is the definition of electric field of a point charge, which is irrelevant to your question where you have a continuous charge density function. $\endgroup$ – user21820 Feb 2 at 10:58
  • $\begingroup$ That's why it is necessary for you to attempt to be precise, as you tried to in your above comments, so that it would be clear that you haven't provided a formal definition of "electric field given charge density function". $\endgroup$ – user21820 Feb 2 at 11:00
  • $\begingroup$ By the way, "rod" in mathematics has thickness. I know you meant line, based on the rest of your question, but don't use "rod" if you mean line. And I'm not criticizing the intention behind your question. I'm just trying to make clear that it really comes down to proper definitions, before the question becomes sensible, and likely trivial. $\endgroup$ – user21820 Feb 2 at 11:02

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