What is the continuity equation expressing charge conservation, and its mathematical form?

What is the continuity equation expressing charge conservation, and its mathematical form?

• A. ∂ρ/∂t + ∇·J = 0
• B. ∂ρ/∂t − ∇·J = 0
• C. ∇ · J = 0
• D. ∂ρ/∂t + ∇ × J = 0

Answer: (A)

The main idea is local conservation of electric charge: what accumulates at a point must come from net inflow, and what leaves must be replaced. The current density J tells us how charge moves through space, while its divergence ∇·J measures the net flow out of a point per unit volume. If more charge is leaving than entering, the density ρ at that point drops, so the time rate of change ∂ρ/∂t is negative. Putting these together, the rate of change of charge density plus the net outflow must balance to zero: ∂ρ/∂t + ∇·J = 0. This equation expresses the conservation in a local form. A steady situation where nothing changes with time would require ∂ρ/∂t = 0, which would make ∇·J = 0. An alternative form that uses the curl of J would not represent charge flow or accumulation, and a version with a minus sign would misstate the balance between change in density and outflow.

The main idea is local conservation of electric charge: what accumulates at a point must come from net inflow, and what leaves must be replaced. The current density J tells us how charge moves through space, while its divergence ∇·J measures the net flow out of a point per unit volume. If more charge is leaving than entering, the density ρ at that point drops, so the time rate of change ∂ρ/∂t is negative. Putting these together, the rate of change of charge density plus the net outflow must balance to zero: ∂ρ/∂t + ∇·J = 0.

This equation expresses the conservation in a local form. A steady situation where nothing changes with time would require ∂ρ/∂t = 0, which would make ∇·J = 0. An alternative form that uses the curl of J would not represent charge flow or accumulation, and a version with a minus sign would misstate the balance between change in density and outflow.