EQUATION OF CONTINUITY FOR TIME VARYING FIELDS:
Statement: Equation of continuity represents the law of conservation of charge. That is the charge flowing out (i.e. current) through a closed surface in some volume is equal to the rate of decrease of charge within the volume :
I = -dq/dt (1)
where I is current flowing out through a closed surface in a volume and
-dq/dt is the rate of decrease of charge within the volume.
As I = ∫J.ds and q = ∫ρdv
Where J is the Conduction current density and ρ is the Volume charge density
Substitute the value of I and q in equation (1), it will become
∫J.ds = -∫dρ/dt dv (2)
Apply Gauss’s Divergence Theorem to L.H.S. of above equation to change
surface integral to volume integral,
∫[divergence (J)]dV = -∫(dρ/dt) dv
As two volume integrals are equal only if their integrands are equal
divergence (J) = – dρ/dt
This is equation of continuity for time varying fields.
Equation of Continuity for Steady Currents:
As ρ does not vary with time for steady currents,
that is dρ/dt = 0
divergence (J)= 0
The above equation is the equation of continuity for steady currents.