Electrodynamics MOC

Electrostatics MOC

Electrostatics is a special case of electrodynamics where the electric field is time-independent and there is no Magnetic field, i.e.

⃗𝐁=⃗𝟎𝜕⃗𝐄𝜕𝑡=⃗𝟎

Electrostatics was established empirically by Coulomb’s law, but of course it is fully encoded in Maxwell’s equations whose differential form become

⃗∇⋅⃗𝐄=𝜌𝑞𝜖00=0⃗∇×⃗𝐄=⃗𝟎⃗𝐉=⃗𝟎

whence we have integral forms

⊂⊃∬𝜕Ω⃗𝐄⋅𝑑⃗𝐚=1𝜖0∭Ω𝜌𝑞𝑑𝜏∮𝜕Σ⃗𝐄⋅𝑑⃗ℓ=0

consequentially the charge continuity equation becomes

𝜕𝜌𝜕𝑡=0

Since the Poynting vector and thus momentum density vanishes, no energy nor momentum is transported by the fields and no momentum is stored by the fields.

Potential

An electrostatic system is completely describes by the electric potential

⃗𝐄=−⃗∇𝑉∇2𝑉=−𝜌𝑞𝜖0

hence solving Poisson’s equation for sources localized to Ω

𝑉(⃗𝐫)=14𝜋𝜖0∭Ω𝜌(⃗𝐫′)𝔯𝑑𝜏′⃗𝐄(⃗𝐫)=14𝜋𝜖0∭Ω𝜌(⃗𝐫′)𝔯2ˆ𝖗𝑑𝜏′

see also Multipole expansion of the electrostatic potential

Further properties

• Electrostatic energy density
• Electrostatic boundary conditions at a surface

Applications

• Conductor in electrostatic equilibrium
• Diëlectric
  • Electric dipole moment
  • Perfect electric dipole
  • Electric potential of a polarized material
  • Linear diëlectric
• Electric displacement
  • Gauß’s law for diëlectrics

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