PhysLean Documentation

PhysLean.Electromagnetism.Electrostatics.OneDimension.PointParticle

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.

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    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.

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      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.

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        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.