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Screwdriver
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Homework Statement
Find an expression for the electric potential a distance z away from the center of a thin uniformly charged rod of length L on a line that bisects the rod.
Homework Equations
V=k\frac{Q}{r}
The Attempt at a Solution
Determine the electric potential due to one point charge i.
V_i=k\frac{\Delta Q}{r}
r=\sqrt{y^2+z^2}
Where y is some distance along the rod to i and z is the distance along the axis to the point away.
\Delta Q=\lambda \Delta y
Then,
V_i=k\lambda \frac{\Delta y}{\sqrt{y^2+z^2}}
Then,
V_{tot}=k\lambda \sum_{i=1}^{n}\frac{\Delta y}{\sqrt{y^2+z^2}}
Now take the limit as n goes to infinity and delta y goes to zero.
V_{tot}=k\lambda \int_{\frac{-L}{2}}^{\frac{L}{2}}\frac{d y}{\sqrt{y^2+z^2}}
V_{tot}=k\lambda \; ln({\frac{2\sqrt{{\frac{L^2}{4}}+z^2}+L}{2\sqrt{{\frac{L^2}{4}}+z^2}-L}})
Is this correct? I feel like the integration was too hard for this simple problem (I had to substitute y = ztan(x)).
