Question

A ring lies in the xy plane, centered at the origin. It has a radius of R and a uniformly distributed total charge Q.What is the potential V(z)?

Trigonometric functions
ANSWERED
asked 2021-09-05
A ring lies in the xy plane, centered at the origin. It has a radius of R and a uniformly distributed total charge Q. Due to the ring on the z-axis, as a function of z, what is the potential V(z)?

Expert Answers (1)

2021-09-06
\(\displaystyle{V}{\left({r}\right)}=\frac{{1}}{{{4}\pi{\underset{{{0}}}{{\varepsilon}}}}}\times\frac{{Q}}{{r}}\) (a)
\(\displaystyle{r}^{{2}}={R}^{{2}}+{z}^{{2}}\)
\(\displaystyle{r}=\sqrt{{{R}^{{2}}+{z}^{{2}}}}\) (b)
Substitue (b) to (a) to get the potential at the z-axis:
\(\displaystyle{V}{\left({r}\right)}=\frac{{1}}{{{4}\pi{\underset{{{0}}}{{\varepsilon}}}}}\times\frac{{Q}}{{\sqrt{{{R}^{{2}}+{z}^{{2}}}}}}\)
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