homework and exercises - Calculating the field of an infinite flat sheet of charge using the superposition principle

I am trying to calculate the field of an infinite flat sheet of charge (a plain with uniform charge density $\sigma$) using the superposition principle.

I know that the field of an infinite line charged with uniform charge density $\lambda$ is $E(r)=\frac{2\lambda}{r}$.

I want to consider the plain as infinite number of lines from $-\infty$ to $\infty$.

I wrote that $$ dE=\frac{2\lambda}{r} $$

and since $\lambda=\sigma dl$ we get $$ dE=\frac{2\sigma dl}{r} $$

$$ E(r)=\int_{-\infty}^{\infty}\frac{2\sigma dl}{r} $$

Which is clearly wrong: First, I still have $r$, secondly I get that the above equals to $$ \frac{2\sigma}{r}\int_{-\infty}^{\infty}dl $$

which does not converge.

What are my mistakes, and how do I get the correct result: $2\pi\sigma$ ?

Answer

This integral:

$$E(r)=\int_{-\infty}^{\infty}\frac{\sigma dl}{2\pi\epsilon_0 r}$$

is correct.

However, you make a mistake in the next step. You cannot equate this integral to $$E(r)=\frac{\sigma }{2\pi\epsilon_0r}\int_{-\infty}^{\infty}dl$$ simply because $r$ is not constant for every infinite line element you consider, and you cannot take it out of the integral.

Lets say that the point at which you want your electric field is $d$ distance above the infinite sheet, and it is directly above some line element $L_o$. The field at this point due to line $L$ which is perpendicular distance $l$ away from $L_o$ will be given by $$dE(l)=\frac{\sigma dl}{2\pi\epsilon_0\sqrt{l^2+d^2}},$$ where $\sqrt{l^2+d^2}$ is the perpendicular distance(otherwise written as $r$) of the point from this line.
This is incomplete too. Notice that the field due to every line element is not in the same direction. So you'll have to vectorially add them, and you simply cannot directly integrate the above equation.

An easy method to do this would be to consider the net field of two line elements placed symmetrically about $L_0$. Their net field will be perpendicular to the infinite plane sheet, and given by $$dE(l)=2\frac{\sigma dl}{2\pi\epsilon_0\sqrt{l^2+d^2}} \frac{d}{\sqrt{l^2+d^2}}.$$ Now this equation you can integrate because this field has a direction perpendicular to the plane for all $l$. So the answer simply is $$E=\frac{\sigma d}{\pi\epsilon_0} \int^\infty_{0}\frac{dl}{l^2+d^2}$$

Notice that I only integrate from $0$ to $\infty$ because we have considered the net field of both lines at $l$ and $-l$ together in the above equation. So going from $0$ to $\infty$, you also include all the elements from $-\infty$ to $0$.

This integral then correctly gives $E=\frac{\sigma}{2\epsilon_0}$