We have a line of charge with length $L$ and charge density $\lambda$ and we want to find its Electric field on a point $p$ with distance $d$ from one end of it using potential. I drew a picture like this and for voltage evaluated this integral y

$$V =\int \frac{dq}{r} = \int_0^L (\frac{\lambda dx}{(d+x)} = \lambda ~~ln(l+d/d)$$

and for Electric field (by substituting $x$ instead of $d$) we get:

$E = \partial V /\partial x = \lambda \frac{x-(l+x)}{x^2} \frac{x}{x+l}$ which is negative. But it actually should get positive because the Electric field is in direction of $+i$. What is my mistake?

  • 3
    $\begingroup$ $E=-dV/dx$, not $+dV/dx$. $\endgroup$ – ZeroTheHero Apr 21 at 21:40
  • 1
    $\begingroup$ Is the line of charge positive or negative charge? The field from $d$ to $P$ will be positive for positive charge and negative for negative charge since, by convention, the direction of the field is the direction of the force that a positive charge would experience if placed in the field. $\endgroup$ – Bob D Apr 21 at 22:53

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