A electrostatics of a point particle in 1d. #
In this module we study the electrostatics of a point particle of charge q
sitting at the origin of 1d space.
The charge distribution of a point particle of charge q
in 1d space sitting at the origin.
Mathematically, this corresponds to a dirac delta distribution centered at the origin.
Equations
Instances For
An electric potential of a charge distribution of a point particle. Mathematically
this corresponds to the distribution formed by the function |x|
multiplied by a
scalar.
Equations
- Electromagnetism.OneDimPointParticle.electricPotential q ε = -Distribution.ofBounded (fun (x : EuclideanSpace ℝ (Fin (Nat.succ 0))) => (q / (2 * ε)) • ‖x‖) ⋯ ⋯
Instances For
An electric field corresponding to a charge distribution of a point particle,
defined as the negative of the gradient of electricPotential q ε
.
This is the electric field which is symmetric about the origin, and in this sense does not sit in a constant electric field.
Equations
Instances For
The electric field electricField q ε
is related to the heavisideStep
function.
The electric field electricField q ε
corresponding to a charge distribution of a point
particle satisfies Gauss's law for the charge distribution of the point particle.
For the charge distribution of a point particle in 1-dimension, a static electric field
satifies Gauss's law if and only if it is the linear combination of the
electric field electricField q ε
(corresponding to the symmetric step function), plus
a constant electric field.
The if
direction of this result is easy to prove, whilst the only if
direction
is difficult.
Note: This result follows from the (as yet unproven) analgous result for the vacuum.