By Charles deAnne Most recent revision: December 16, 2019 This web page is a work in progress. Any suggestions or corrections are welcome and will be credited. email: charlesdeanne@sbytes.com ## Summary of results for the force between various distributions of electrons(repulsive force is positive) Force between point concentrations of electrons: (Y and Z are individual electron counts. Q _{Z} and Q_{Y} are electron counts in units of coulombs.)Force between a point concentration of Y electrons (Q _{Y} coulombs), r distance from a straight-line linear electron density, lambda_Z (charge density, lambda_Q):Force per unit length between two parallel straight lines of linear electron densities: Magnetic force per unit length between parallel electron currents. (Table of contents #6) Current in coulombs/second, is defined such that: Dividing both of the above equations by -2/r: Note that electron charge, e, has no physical dimensions. It is merely the reciprocal of the number of electrons in a coulomb, a purely arbitrary quantity determined by Ampere's decision to fix the magnetic force per meter between two coulomb-per-second currents separated by one meter to 2X10 ^{-7} newton.Reviewing the force between point charge concentrations: ## Potential energy between two quantum electric particlesThe absolute value of potential energy between two quantum electric particles (electrons or protons), separated by the classical electron radius, is equal to the rest mass energy of the electron. Since potential energy is inversely proportional to the distance of separation, the equation for potential energy between each unique pair of electric particles is:
The factors, plus or minus one, represent single particles at each end of r, with a plus assigned to protons and a minus assigned to electrons. For point concentrations of Y particles at one end of r and Z particles at the other end, the total number of unique quantum pairs is the product, YZ, giving the total potential energy as:
A spatial gradient of energy is equivalent to a force directed toward the region of lower energy. Taking the derivative of the above equation with respect to r gives the electric force between two point concentrations of quantum particles:
## Force between a single electron and an infinite straight line of electronsImagine a line of uniformly distributed electrons on the 'x' axis, and the subject single electron on the 'y' axis, distance, d, from the origin. Partitioning the 'x' axis into small, equal, incremental distances, dx, the number of electrons in each partition is: (lambda_Z is the linear density in electrons per unit length) For each dx at +x, there is a corresponding dx at -x. Since the horizontal components of force from each paired group of electrons are equal and opposite, they cancel each other, leaving the net force always perpendicular to the 'x' axis: (L is the distance from the electron to the dx located at +x, and phi is the acute angle between the 'y' axis and L.) Substituting and re-grouping, The differential angle subtended by dx is the component of dx perpendicular to L, divided by L: Integrating the incremental forces while phi goes from zero to PI/2, is equivalent to integrating the same incremental forces as x goes from zero to infinity. Since: The total vertical force on the electron is: ## Force between parallel lines of quantum particles |