What would the potential by a good approximation for a distance z that is much larger than the radius of the ring R?
Added by April B.
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We are dealing with a ring of charge and want to find the electric potential at a point located at a distance \( z \) along the axis of the ring, where \( z \) is much larger than the radius of the ring \( R \) (i.e., \( z \gg R \)). Show more…
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The potential can be calculated by superposition. Choose the plane of the upper ring as $x=l / 2$ and that of the lower ring as $x=-l / 2$ For $\quad|x|>>R, \varphi \sim \frac{q l}{4 \pi x^{2}}$ The electric field is $E=-\frac{\partial \varphi}{\partial x}$ $$ =-\frac{q l}{4 \pi \varepsilon_{0}\left(R^{2}+x^{2}\right)^{3 / 2}}+\frac{3}{2} \frac{q l}{\left(R^{2}+x^{2}\right)^{5 / 2} 4 \pi \varepsilon_{0}} \times 2 x=\frac{q l\left(2 x^{2}-R^{2}\right)}{4 \pi \varepsilon_{0}\left(R^{2}+x^{2}\right)^{5 / 2}} $$ For $\quad|x|>>R, E \sim \frac{q l}{2 \pi \varepsilon-r^{3}} . \quad$ The plot is as given in the book.
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Find the potential at a point $P$, which is at a perpendicular distance $z$ from a uniformly charged ring of radius $R$ placed in the $x-y$ plane.
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Advanced Topics
Show that the on-axis electric field of a ring of charge has the expected behavior when $z \ll R$ and when $z \gg R$
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