Questions: Current Density and Current Distribution

Question 1

A wire carrying current I narrows so that its cross-sectional area halves (from A to A/2). What happens to the current density J in the narrower section?

• A. J halves — less area means less current can flow
• B. J doubles — the same current passes through half the area
• C. J stays the same — current is conserved through the wire
• D. J becomes undefined because the cross-section changed

Answer: (B)

Current I is conserved — the same charge per second must pass through every cross-section of a series conductor (Kirchhoff's current law). Since J = I/A and I is fixed, halving the area doubles the current density. This higher J also means a larger local electric field (via J = σE) and greater power dissipation per unit volume (P/V = J·E = J²/σ), which is why thin wires heat up more than thick ones carrying the same total current.

Question 2

In a metal conductor, the direction of the current density vector J points:

• A. In the same direction as the drift velocity of the electrons
• B. Opposite to the drift velocity of the electrons
• C. Perpendicular to the drift velocity of the electrons
• D. J has no direction — it is a scalar quantity

Answer: (B)

J is defined as the direction of positive charge flow (conventional current direction). In a metal, the charge carriers are electrons, which carry negative charge and drift opposite to the electric field. Since conventional current flows in the direction of positive charge flow, J points opposite to the electron drift. The formula J = nqv_d accounts for this: with q = −e (negative) and v_d pointing left (say), the product gives J pointing right. Tracking signs carefully here prevents errors in all subsequent electromagnetic calculations.

Question 3

If current I is conserved along a conductor, then the current density J is expected to also be the same at nearly every cross-section.

• T. True
• F. False

Answer: False

Current I is conserved (the same total charge per second passes each cross-section), but current density J = I/A depends on the local cross-sectional area. Where the conductor is narrow, J is large; where it is wide, J is small. This is why current density is a more informative quantity than current alone — it captures the spatial distribution of charge flow and directly governs local effects like heating (P/V = J²/σ) and the driving electric field (J = σE).

Question 4

The continuity equation ∂ρ/∂t + ∇·J = 0 is a mathematical expression of the conservation of electric charge.

• T. True
• F. False

Answer: True

The continuity equation states that any net outflow of current from a region (∇·J > 0) must be accompanied by a decrease in the local charge density (∂ρ/∂t < 0), and vice versa. Charge is neither created nor destroyed — it can only move. In steady-state circuits where charge density doesn't change (∂ρ/∂t = 0), this reduces to ∇·J = 0: as much current enters any volume as leaves it. This is the field-theoretic version of Kirchhoff's current law.

Question 5

Explain why current density J, rather than current I, is the more fundamental quantity for describing how current flows through a conductor with varying cross-sectional area.

Model answer: I is a scalar giving total charge flow per second through a cross-section, but it says nothing about how that flow is distributed spatially. J is a vector field that captures the local magnitude and direction of current at every point. In a conductor with varying cross-section, I is conserved but J varies inversely with area. J connects directly to local physical effects: J = σE relates current density to the local electric field; P/V = J²/σ gives local heating. I can be recovered from J by surface integration, making J the more complete and fundamental description.

The distinction between I and J mirrors the distinction between total flux and flux density in other areas of physics. Knowing total water flow in a pipe doesn't tell you the local flow speed; that requires dividing by local cross-sectional area. Similarly, J = I/A (for uniform cross-sections) gives the local intensity of charge flow. In inhomogeneous conductors or complex geometries, only J fully describes what's happening — I alone cannot distinguish a thin, hot wire from a thick, cool one carrying the same total current.