Questions: Electric Current and Continuity Equation

Question 1

A wire narrows from a wide section to a thin section while carrying a steady current. What happens to the current density J in the thin section?

• A. J decreases, because fewer carriers fit in the thin section
• B. J increases, because the same charge flow must pass through a smaller cross-sectional area
• C. J stays the same, because current is conserved
• D. J doubles, because the resistance doubles

Answer: (B)

Current I = J·A is conserved along the wire in steady state. If I is constant and the cross-sectional area A decreases, then J = I/A must increase proportionally. The same charge per second is being forced through a smaller cross-section, so the charge density of flow (current density) must be higher. This is why current density is a vector field — it varies spatially even when the total current is constant.

Question 2

Kirchhoff's Current Law (the sum of currents into a junction equals the sum leaving) is best understood as which of the following?

• A. An empirical rule discovered by measuring hundreds of circuits
• B. A consequence of charge conservation (continuity equation) in steady state, where ∇·J = 0
• C. A consequence of Ohm's Law at junctions
• D. A rule that applies only to resistive circuits, not to capacitive or inductive ones

Answer: (B)

KCL follows directly from the continuity equation ∂ρ/∂t + ∇·J = 0 in steady state. Setting ∂ρ/∂t = 0 gives ∇·J = 0 — current has no sources or sinks inside the conductor. Integrating over a small volume surrounding a junction, the divergence theorem converts this to: net charge flow out = 0, which is precisely KCL. It is not empirical but derived from charge conservation, and it applies to any circuit element in steady state, not just resistors.

Question 3

In a metal wire, conventional current flows in the same direction as the drift velocity of the electrons.

• T. True
• F. False

Answer: False

Conventional current is defined as the direction positive charges would flow — historically established before the discovery that electrons (negative charges) carry current in metals. In a metal, electrons drift opposite to the electric field direction. Since conventional current is opposite to electron drift, conventional current and electron drift velocity point in opposite directions. This is a persistent source of sign errors; always distinguish between carrier motion and conventional current direction.

Question 4

The continuity equation ∂ρ/∂t + ∇·J = 0 implies that in steady state, no charge accumulates or depletes at any point inside the conductor.

• T. True
• F. False

Answer: True

In steady state, all quantities are time-independent, so ∂ρ/∂t = 0. The continuity equation then requires ∇·J = 0 everywhere inside the conductor — the current field has no divergence. Any point where more current flowed out than in would see charge depletion (∂ρ/∂t < 0), violating steady state. This confirms that charge distribution is static in a DC circuit, and any current entering a region must equal current leaving it.

Question 5

Why does the continuity equation reduce to Kirchhoff's Current Law in the context of a DC circuit junction?

Model answer: In steady state, ∂ρ/∂t = 0, so the continuity equation becomes ∇·J = 0. Integrating over a small closed volume surrounding the junction and applying the divergence theorem converts this into a surface integral: the net charge flux out of the surface is zero. This means the sum of currents entering the junction equals the sum leaving — which is KCL.

The derivation shows KCL is not an independent empirical law but a macroscopic consequence of the local charge conservation law. The continuity equation holds at every point in space; KCL is what you get when you apply it to the specific geometry of a circuit junction. Understanding this connection reveals that all of circuit theory ultimately rests on conservation laws, not just convenient approximations.