The continuity equation in electromagnetic theory is a formal way of stating that charge is conserved. The equation for this in the differential form is the following:
or
The integral form of this equation is the following (obtained using the divergence theorem):
(here, a closed surface is considered)
In this equation, rho is the charge density (in C/m^3), the upside down triangle is the del operator, J is the current density in (A/m^2 - a vector quantity), and q_en is charge enclosed (in C). It is very difficult to picture this qualitatively in the differential form, so we shall attempt to relate to the integral form instead.
Consider a set of positive charges that all of a sudden disperse at a finite rate. Now, consider an imaginary sphere centered at the point source. The continuity equation states that the net outward flow (current, in C/s) through the surface of the imaginary sphere is equal to the rate of decrease of charge enclosed by the sphere.
Numerically, say, if 1 A of current flowed out of the sphere, 1 Coulomb of charge will have left the sphere in 1 second.
The continuity equation is very important in electromagnetic theory. Maxwell identified a flaw in Ampere's Law, identifying cases where charge was not conserved. By correcting this equation, he was able to complete the connection between Gauss' Law (for electricity and for magnetism), Faraday's Law, and Ampere's Law (all together now known as Maxwell's equations). This was the missing link in making the connection between electromagnetism and light.
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