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;
Columns in the Spreadsheet Title of the Column Explanation of the calculation in the column
1DegreesPopulate column one with the numbers 0 to 359 (one row for each degree in a circle) although this approximation is reproducible with as few as 11 rows.
2RadsThe Rads column is the radian equivalent of degrees and in Excel this is =RADIANS() and simply the degrees in column 1 expressed as radians.
3\cos (\theta) The \cos (\theta)  column is the cosine of the radians in the Rads column.
4      Separation distance c = d-x=d-r cos(\theta))
Determine the separation distance along the X axis as c.  c = d-x=d-r cos(\theta))  as theta increases, moving point 2 around the circle, one degree at a time. 
5Coulomb’s Law force calculation F_{e}=1/c^{2}Assuming unit charge for the point and the circle. F_{e}=1/c^{2}  is the inverse of column 4 squared, or the Coulombic Law for that point to point interaction.
6Net point force (1/360)*1/c^2)Column six is the force calculation using Coulomb’s Law for the point outside the circle to the point on the circle.  Force in the x direction between Point 1 and the point on the circle is (1/360)*1/c^2)  as point 2 moves around the circle where 360 is the number of points on the circle.

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.


  1. Using unit charge on both the point outside the circle and on the circle itself, with a unit circle radius and a distance of 5r the spreadsheet numerical approximation yields a force of k0.042525864 N.  Treating the circle conventionally as a point yields a force of k0.040000 N.  Increasing the number of points on the circle improves the precision of the result but a difference remains
  2. Differences between Coulomb’s Law and the numerical approximation remain regardless of the radius of the circle or distance between the circle and the point, with the obvious exception of a zero radius circle.  In the case of a zero radius circle the numerical approximation matches Coulomb’s Law – a circle of zero radius is a point.