This approximation again uses a simple spreadsheet, basic trigonometry and basic algebra to determine the correct electrostatic force between two spheres **using the corrected equation** derived above for the point to sphere force calculations.

For the purposes of this approximation the symmetry of a circle is used to approximate a sphere. Unit charge is also assumed for simplicity.

The illustration below demonstrates the approach of calculating the force between a sphere 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 the corrected equation F_{e}=q_{1}q_{2}d/(d^2-r^2)^{3/2}) , not Coulomb’s Law, for the point to sphere force calculation.

Using a Cartesian plane to calculate the forces, Point 1, the centroid of Circle 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 using the corrected equation for the point to point force calculation (Point 1 is a sphere). 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 summed individual forces of the numerical approximation of a sphere to sphere do not equal the force calculated by Coulomb’s Law where two spheres are approximated as a points.

Note r_{1} is used in the force equation and used in the calculation of distance for the force.

Using the cosine law c=(r_{2}^2 + d^2 -2r_{2}d \cdot cos(\alpha))^{0.5}

The force along c is F_{e}=q_{1}q_{2}d/((c^{2}-r_{1}^{2})^{(3/2)})

Using basic trigonometry determine the X axis component of the force for each point as follows;

\beta =\pi-\alpha h=r_{2}\sin(\beta) \theta=a\sin(h/c)Resultant force along X axis = =F_{e} \cos(\theta))

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

For simplicity assume unit force on the point and the circle.

360 degrees are converted to rads purely to explain the concept in degrees.

Create a spreadsheet with nine columns;

1 | Degrees | Populate column one with the numbers 0 to 359 (one row for each degree) although this approximation is reproducible with as few as 11 rows. |

2 | Rads | The 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(α) | The cos(α) column is the cosine of the radians in the Rads column. |

4 | Separation distance c | Determine the separation distance from the rotating point to the centroid of Circle 1 c=(r_{2}^2 + d^2 -2r_{2}d \cdot cos(\alpha))^{0.5} as alpha increases, moving point 2 around Circle 1, one degree at a time. |

5 | β | π –α |

6 | h | r2 sin(β) |

7 | θ | = asin(h/c) |

8 | Corrected force calculation | F_{e}=q_{1}q_{2}d/((c^{2}-r_{1}^{2})^{(3/2)}) |

9 | Force along X axis | Fe cos(θ) |

Summing column nine gives the total force between the two spheres using the corrected equation in a numerical approximation.

A sample spreadsheet table is below. The full table is in Appendix 3.

**Results;**

- Using unit charge on both spheres, with a radius of 1 and a distance of 5r the numerical approximation yields a force of k0.044220897 N. Coulomb’s Law, treating the circle as a point, yields a force of 0.04. Increasing the number of points on the circle improves the precision of the result but a difference remains.
- The numerical approximation produces a force between two spheres that is different from the previous force calculation between a point and a sphere.