dQdt

dQ/dt denotes the time derivative of the electric charge Q(t). In electromagnetism, the instantaneous current through a surface is defined as I(t) = dQ/dt, so dQ/dt is commonly interpreted as the current entering or leaving a region, depending on sign conventions. If Q(t) represents the total charge within a fixed region, a positive dQ/dt indicates a net addition of positive charge to that region.

The quantity dQ/dt has units of coulombs per second, i.e., amperes. In circuit contexts, I(t) is often used instead of dQ/dt, with dQ/dt and I(t) having the same numerical value.

Conservation and locality: Total charge is conserved in isolated systems, so the net dQ/dt for the entire system is zero. Locally, charge density ρ and current density J are linked by the continuity equation: partial rho / partial t plus the divergence of J equals zero. The rate of change of the total charge in a fixed volume equals minus the net current flowing outward across the volume’s boundary.

Practical interpretations: In a resistor or conductor with a steady current, dQ/dt is constant. In a charging capacitor, dQ/dt equals the charging current, increasing the charge on the capacitor plates; during this process, conduction current is complemented by displacement current to satisfy continuity.

Notational notes: Some authors use the dot notation Q̇ for dQ/dt. In many texts, the symbol I(t) is used to denote the same quantity, with context determining the convention.