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.