Continuity Equation

Current density is the total amount of current flowing per unit cross-sectional area. The total current passing through a cross-sectional area can be expressed as the surface integral of the current density. This relationship allows us to calculate total current from the distribution of current density across a surface.

Charge conservation requires that the total current flowing out of a given volume equals the rate of decrease of charge within that volume. This principle connects the outward flow of electrical charge to changes in the charge stored inside the volume, establishing a fundamental relationship between current and charge density.

Starting with charge conservation, the total charge is expressed in terms of volume charge density. Using the Leibniz rule, the time derivative moves inside the integral. Applying the divergence theorem converts the closed surface integral to a volume integral. Since this holds for any volume, the integrands must be equal, yielding the continuity equation.

The continuity equation states that the divergence of the current density equals the negative rate of change of volume charge density. This mathematical statement represents local charge conservation, showing how spatial variations in current relate to temporal changes in charge density at any point.

For steady currents, the charge density is invariant with time, meaning it does not change. Since the continuity equation states that divergence of current density equals the negative rate of change of charge density, and this rate is zero for steady conditions, the divergence of current density must also be zero.

The divergence theorem converts the closed surface integral on the left side of the charge conservation equation into a volume integral. This transformation allows both sides of the equation to be expressed as volume integrals, enabling direct comparison of integrands and leading to the local form of the continuity equation.

Volume charge density represents the charge per unit volume as a space function. Since it varies with position, the Leibniz rule allows the time derivative to move inside the integral during derivation. The continuity equation directly relates changes in volume charge density to the divergence of current density at each point in space.