Conservation of Electric Charge
The total electric charge of an isolated system remains constant regardless of changes within the system itself. This law is inherent to all processes known to physics and can be derived in a local form from gauge invariance of the wave function. The conservation of charge results in the charge-current continuity equation. More generally, the net change in charge density within a volume of integration is equal to the area integral over the current density through the closed surface, which is in turn equal to the net current :
Thus, the conservation of electric charge, as expressed by the continuity equation, gives the result:
The charge transferred between times and is obtained by integrating both sides:
where is the net outward current through a closed surface and is the electric charge contained within the volume defined by the surface.
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