Charge continuity equation

The charge continuity equation is an expression of local conservation of charge relating the divergence of current to the depletion of charge.

⃗ \nabla \cdot ⃗ 𝐉 = - 𝜕 𝜌 𝜕 𝑡

Intuitively, if current is diverging from a point then that point must be losing charge. It is equivalent to the statement

\subset \supset \iint 𝜕 Ω ⃗ 𝐉 \cdot 𝑑 ⃗ 𝐚 = - 𝑑 𝑑 𝑡 ∭ Ω 𝜌 𝑑 𝜏'

(the flux of current through the surface of a solid is equal to the rate at which the solid loses charge) by applying Острогра́дский's divergence theorem¹.

Derivation from Maxwell's equations

Taking the divergence of ^Differential and applying ^Differential

⃗ \nabla \cdot ⃗ \nabla \times ⃗ 𝐁 - 𝜖 0 𝜇 0 𝜕 𝜕 𝑡 ⃗ \nabla \cdot ⃗ 𝐄 = 𝜇 0 ⃗ \nabla \cdot ⃗ 𝐉 - 𝜖 0 𝜇 0 𝜕 𝜕 𝑡 𝜌 𝜖 0 = 𝜇 0 ⃗ \nabla \cdot ⃗ 𝐉

giving ^eq1.

This is an example of a continuity equation.

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Footnotes