The numerical approximation in the spreadsheet over n rows is shown as a Riemann sum below.

\lim_{x \to \infty} \sum_{n=0}^x (1/x)/(1/(d-r cos(2 \pi n/ x))^2)Which is the integral;

\int_{0}^{1} d\theta/(d-r cos(2 \pi \theta))^2)The integral resolves to the following corrected equation for the force between a point and a sphere;

d/(d^2-r^2)^{3/2}Figure 5 – The corrected electrostatic force equation between a point and a hollow sphere.

**Results;**

- Force calculations from the “corrected equation” match the numerical approximation over various separation distances and radii.
- As few as 11 points on the circle in the numerical approximation are needed to match the derived formula to 10 digits.