Intuitive

The continuity equation states that the $\color{red}{\text{total amount of a quantity (like water) that is produced (or destroyed) inside some volume}}$ is proportional to the $\color{blue}{\text{change of the quantity}}$ plus the $\color{magenta}{\text{total amount that flows in minus the amount that flows out of the volume}}$.

Or formulated differently, the total $\color{blue}{\text{change of some quantity}}$ is equal to the $\color{red}{\text{amount that gets produced}}$ plus the amount that $\color{magenta}{\text{flows in minus the amount that flows out of the volume}}$.

If we are dealing with a conserved quantity, like energy or electric charge, the total amount that is produced or destroyed is exactly zero.

Concrete

• The continuity equation in hydrodynamics describes the flow of mass. Here $ρ$ is fluid density and $ \vec v$ the fluid flow velocity.
• The continuity equation in electrodynamics describes the flow of electric charge. Here $ρ$ is the charge density and $ \vec v$ the electric flow velocity, such that $ρ \vec v = j$ is the electric current.
• The continuity equation in quantum mechanics describes the flow of probability. Here $ρ = \Psi^\dagger \Psi$ is the probability density and $ \vec v$ the probability flow velocity, such that $ρ \vec v = j = \frac { \hbar } { 2m i } [ \Psi ^ { * } ( \nabla \Psi ) - \Psi ( \nabla \Psi ^ { * } )]$ is the probability current .

Derivation of the continuity equation in hydrodynamics

Derivation of the continuity equation in electrodynamics

Derivation of the continuity equation in quantum mechanics

Abstract

Why is it interesting?

A continuity equation is important whenever we are dealing with a system where some quantity is conserved. The continuity helps us to keep track how the conserved quantity moves through the system.

Important examples are

• the continuity equation in electrodynamics that encodes the conservation of electric charge,
• the continuity equation in hydrodynamics that encodes the conservation of mass.