# How to do this problem without using infinitesimal?

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.

• 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. – gandalf61 Feb 1 at 14:25
• Yes that is exactly the question the so called (error term )$/\Delta x=0$ as $\Delta x$ tends to zero , but how? – Bijayan Ray Feb 1 at 14:28
• @gandalf61 I think that the question requires more assumptions which may become evident as one writes the equation? – Bijayan Ray Feb 1 at 14:42

• 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. – Bijayan Ray Feb 2 at 10:54
• 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 $x>h$ – Bijayan Ray Feb 2 at 10:57