Electrodynamics MOC

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.

⃗∇⋅⃗𝐉=−𝜕𝜌𝜕𝑡

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

⊂⊃∬𝜕Ω⃗𝐉⋅𝑑⃗𝐚=−𝑑𝑑𝑡∭Ω𝜌𝑑𝜏′

(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

⃗∇⋅⃗∇×⃗𝐁−𝜖0𝜇0𝜕𝜕𝑡⃗∇⋅⃗𝐄=𝜇0⃗∇⋅⃗𝐉−𝜖0𝜇0𝜕𝜕𝑡𝜌𝜖0=𝜇0⃗∇⋅⃗𝐉

giving ^eq1.

This is an example of a continuity equation.

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Footnotes

1. 2023. Advanced Mathematical Methods, p. 39 ↩