Continuity Equation
Since charge is conserved, current density must satisfy a continuity equation. Here is a derivation from first principles.
The net flow out of some volume V (which can have an arbitrary shape but fixed for the calculation) must equal the net change in charge held inside the volume:
where ρ is the charge density, and dA is a surface element of the surface S enclosing the volume V. The surface integral on the left expresses the current outflow from the volume, and the negatively signed volume integral on the right expresses the decrease in the total charge inside the volume. From the divergence theorem:
Hence:
This relation is valid for any volume, independent of size or location, which implies that:
and this relation is called the continuity equation.
Read more about this topic: Current Density
Famous quotes containing the words continuity and/or equation:
“Only the family, society’s smallest unit, can change and yet maintain enough continuity to rear children who will not be “strangers in a strange land,” who will be rooted firmly enough to grow and adapt.”
—Salvador Minuchin (20th century)
“A nation fights well in proportion to the amount of men and materials it has. And the other equation is that the individual soldier in that army is a more effective soldier the poorer his standard of living has been in the past.”
—Norman Mailer (b. 1923)