# 2. Numerical Approximation of the Point to Sphere Force Calculation

For the purposes of this exercise the symmetry of a circle is used to approximate a sphere.

Figure 3 below demonstrates the approach of calculating the force between a point and a sphere by approximating a circle with many points, then simply summing the force between the point outside the circle and each of the many points on the circle using Coulomb’s Law: F_{e}=kq_{1}q_{2}d/r^{2} .  Using a Cartesian plane to calculate the forces, Point 1 lies outside the circle, the centroid of the circle is 0,0.  Point 2 moves around the circle allowing many force calculations between Point 1 and Point 2 as Point 2 revolves around the circle.  Point 1 remains stationary.

Assuming 360 points on the circle, one for each degree, each point will have 1/360th of the charge of the entire circle.  Forces in the Y direction are ignored as the corresponding point on the opposite side of the circle (across the X axis) exerts an equal and cancelling force in the Y direction.

The individual forces summed in the numerical approximation do not equal the force calculated by Coulomb’s Law where the sphere is approximated as a single point.

Figure 3 – Cartesian Coordinates Used for Numerical Approximation of point to sphere forces using Coulomb’s Law

Steps to reproduce point to sphere numerical force approximation using Coulomb’s Law in a spreadsheet;

1. For simplicity assume unit force on the point and the circle.
2. Degrees are converted to rads purely to explain the concept in degrees.
3. Create a spreadsheet with six columns;

Summing column six gives the total force between the circle and the point outside the circle using Coulomb’s Law in a numerical approximation of a point to a sphere.

A sample spreadsheet table is below.  The full table is in Appendix 2.