# 3. Deriving the correct formula for point to sphere electrostatic forces

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;

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