Force on charges stacked in evenly spaced series at a single moment in time

 

Net force on point charges stacked in evenly spaced series at a single moment in time

It is at a single moment in time when all of the charges would be an equal distance from the next nearest charge as measured in a single dimension after that point of time some of the charges might move to a new position

In a one dimensional tube uncharged tube of length L going from left to right

The distance between the two farthest point charges from each other is L

N is an integer greater than or equal to two equal to the number of point charges

The total charge of all point charges is N

The charge of each point charge is Q/N

The distance between each point charge and the nearest point charge is D

D = L / ( N -1 )

Fi is the Individual force a point charge on the farthest right experiences from another point charge listed with an index of i where the charge farthest on the left has i = 0 the charge second farthest to the right has i = n - 2

Fi = [ L - i * L / ( n - 1 ) ] ^ 2

F = sum of all Fi from i = 0 to i = n - 2

Incomplete next steps

Look up notation I used in place of standard calculus symbols when letters and numbers and askey symbols are used instead of hand drawings from previous articles

Take limit as n approaches infinity to calculate F for a unlimmitted number of point charges using an integral

Compare that for all other variables having the same value but n equalling two

Do in two and then three dimensions next for the point charge in a single corner

Note :  trig functions might be useable to estimate distance between charge as a function of angle but might give wrong results if used as an integral of force with respect to angle from one angle to another angle to calculate net total force experienced by a charge in a corner of a square or cube, so it is probably better to use indexes i, j and k and do double or triple integral using caartesian coordinates instead of single or double integral using polar or spherical coordinates because the number of point charges in each angular direction maybe different 

Copyright Carl Janssen 2022 Jan 12