24.6: Calculations of Electric Potential I
Consider a ring of radius R with a uniform charge density λ. What will the electric potential be at point M, which is located on the axis of the ring at a distance x from the center of the ring?
The ring is divided into infinitesimal small arcs such that point M is equidistant from all the arcs. Here, the cylindrical coordinate system is used to calculate the electric potential at point M. A general element of the arc between angles θ and θ + dθ is of the length Rdθ and has a charge of λRdθ.
The distance between this element of the ring and the point M is
The potential at point M due to the charge on the whole ring is
Integrating the above equation over the limits gives,
Here, 2πRλ accounts for the whole charge on the ring; therefore, the above equation can be written as
This result is expected because point M is equidistant from all the infinitesimal ring elements, and the total potential will be similar to if the total charge were positioned at a common distance from point M.