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

⃗ \nabla \cdot ⃗ 𝐄 = 𝜌 𝑞 𝜖 0 0 = 0 ⃗ \nabla \times ⃗ 𝐄 = ⃗ 𝟎 ⃗ 𝐉 = ⃗ 𝟎

whence we have integral forms

\subset \supset \iint 𝜕 Ω ⃗ 𝐄 \cdot 𝑑 ⃗ 𝐚 = 1 𝜖 0 ∭ Ω 𝜌 𝑞 𝑑 𝜏 \oint 𝜕 Σ ⃗ 𝐄 \cdot 𝑑 ⃗ ℓ = 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 = - ⃗ \nabla 𝑉 \nabla 2 𝑉 = - 𝜌 𝑞 𝜖 0

hence solving Poisson's equation for sources localized to $Ω$

𝑉 (⃗ 𝐫) = 1 4 𝜋 𝜖 0 ∭ Ω 𝜌 (⃗ 𝐫') 𝔯 𝑑 𝜏' ⃗ 𝐄 (⃗ 𝐫) = 1 4 𝜋 𝜖 0 ∭ Ω 𝜌 (⃗ 𝐫') 𝔯 2 ˆ 𝖗 𝑑 𝜏'

Further properties

Applications

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