Find the potential a distance `z` above the center of a uniform line of charge with charge density `lambda ` and length `2L` .
For a continuous potential to equation is
Where `k=1/(4pi epsilon_0)` , `dq` is an infinitesimal piece of charge, and `r` is the distance away `dq` is from the point we are measuring.
We know the charge per unit length` lambda=(dq)/(dl)` where `dl` is an infinitesimal chuck of the wire. If we let the wire lay along the x-axis then `dl=dx` . Therefore,
Now, looking at the picture we have a right triangle. So we know ` r=sqrt(x^2+z^2)` . Then the integral becomes,
`V=int_(-L)^L (k*lambda dx)/sqrt(x^2+z^2)=k*lambda int_(-L)^L dx/sqrt(x^2+z^2)`
From a table of integrals we can find that
`V=k lambda ln(x+sqrt(x^2+z^2))|_(-L)^L`
`V=lambda/(4pi epsilon_0) ln((L+sqrt(L^2+z^2))/(-L+sqrt(L^2+z^2)))`
To simplify multiply the numerator and denominator by `L+sqrt(L^2+z^2)`
This is the answer.