Electrodynamics MOC

Electric field

The electric field is an abstraction which describes how electric forces will act upon a test charge.

⃗𝐅=𝑄⃗𝐄

The electric field follows the Principle of Superposition, so the effects of different charge distributions may be combined easily. It describes the force per unit charge, so its SI units are N⋅C−1 . If a charge distribution is static, i.e. the source charges do not move, then the field is irrotational and may therefore be given an Electrostatic potential. For the more general case see Electric and magnetic potentials.

Calculation

It follows from Maxwell’s equations that

⃗𝐄(⃗𝐫)=14𝜋𝜖0∭ℝ3𝜌(⃗𝐫′)𝔯2ˆ𝖗𝑑𝜏′

although historically this result is derived from consideration of Coulomb’s law.

Properties

• See Gauß’s law

• When crossing a surface charge distribution, the electric field undergoes a discontinuity of magnitude 𝜎/𝜖0 in its component normal to the surface, whereas parallel to the surface the field is continuous.¹ This gives the boundary conditions

  lim𝛼→0+⃗𝐄(⃗𝐫+𝛼ˆ𝐧)−⃗𝐄(⃗𝐫−𝛼ˆ𝐧)=𝜎𝜖0ˆ𝐧

  or more simply

  ⃗𝐄above−⃗𝐄below=𝜎𝜖0ˆ𝐧

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Footnotes

1. 2013. Introduction to electrodynamics, p. 87–89 (§2.3.5) ↩