Derive continuity equation

The continuity equation expresses the law of conservation of charge: charge can neither be created nor destroyed. The current leaving a closed region equals the rate of decrease of charge inside it.

Integral form

Consider a closed surface S enclosing volume v and total charge Q_in. The net current flowing out through S is the total outward flux of the current density J:

I_out = ∮_S J · dS

By conservation of charge, this equals the rate at which the enclosed charge decreases:

∮_S J · dS = −dQ_in/dt = −d/dt ∫_v ρ_v dv        ← integral form

Point form

Apply the divergence theorem to the left side: ∮ J · dS = ∫_v (∇ · J) dv. For a fixed volume, the time derivative can be taken inside the integral as a partial derivative:

∫_v (∇ · J) dv = −∫_v (∂ρ_v/∂t) dv

This must hold for any volume, however small, so the integrands must be equal:

∇ · J = −∂ρ_v/∂t                 ← point form

Remarks

• For steady (time-independent) currents ∂ρ_v/∂t = 0, so ∇ · J = 0 and ∮ J · dS = 0: current lines are continuous and closed, which is Kirchhoff's current law (net current into a node is zero).
• The point form means that if current diverges from a point, the charge density there must be decreasing with time.
• It combines with Ohm's law (J = σE) and Gauss's law to give the relaxation time T_r = ε/σ.