(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

i was deriving an infinite line of charge formula by coloumb's law:

so i got stuck with this integral (since it is in the maths forum)

[tex] \vec{E}_{\rho} = \int_{-\infty}^{\infty} \frac{\rho_L \rho dz}{4\pi\epsilon_o ({\rho}^2 + z^2)^{\frac{3}{2}}} [/tex]

2. Relevant equations

where

[tex]{\rho}_L = [/tex] linear charge density

[tex]{\rho} = [/tex] direction perpendicular to z axis in cylindrical coordinates

3. The attempt at a solution

so when integrating (using trigo substitution):

[tex] \vec{E}_{\rho} = \frac{\rho_L\rho}{4\pi\epsilon_o} (\frac{1}{{\rho}^2} \frac{z}{p^2 + z^2})_{-\infty}^{\infty} [/tex]

this is where i got stuck

no matter how i use lhopitals rule in this equation:

\lim_{z \infty} \frac{z}{({\rho}^2 + z^2)^{\frac{1}{2}}

it keeps going back

because the infinite is supposed to add to equate to 2

the answer which is

[tex] \vec{E}_{\rho} = \frac{\rho_L}{2\pi\epsilon_o\rho}[/tex]

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# Homework Help: Improper integral help: coloumb's law

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