Integrate current in space and charge in time

Choose a signed surface normal

Current is dq/dt through a selected cross-section. It is a signed scalar associated with that orientation, not a spatial vector to add by perpendicular components at a junction. Current density J is a vector: I = integral J dot dA. For uniform carrier density n, carrier charge q and average drift velocity vd, J = nq vd. Negative electron drift can therefore produce positive conventional current.

Integrate annular strips

For a circular section of radius R, the assigned profile is Jz = J0[1 + beta(r/R)²]. An annular strip has area 2pi r dr. Integrating from 0 to R gives I = pi R² J0(1 + beta/2). Multiplying the edge value by the full area would overcount when beta is positive. Dividing by area gives the mean density, not its center or edge value.

Separate drift from random motion

A potential difference supports a local electric field, which can produce a drift superposed on random carrier motion. For a locally ohmic material, E = rho J. Zero net current can mean zero mean drift while individual carriers still move. It does not mean every carrier has stopped, nor that a slow drift equals the signal-propagation speed.

Integrate a different time packet

In a separate supplied packet, let I(t) = I0(1 + t/t0) with t0 = 1 s. Charge crossing from 0 to 2 s is integral I dt = I0[2 s + (2 s)²/(2t0)] = 4 I0 s. Mean current is that charge divided by 2 s. This time packet is not a claim that the earlier steady spatial profile evolves dynamically without a new model.

Worked original case

At R = 1 mm, beta = 1 and J0 = 2×10⁶ A/m², I = 3pi A ≈ 9.42478 A. Mean J is 3×10⁶ A/m². With n = 8×10²⁸ per m³ and electron charge −e, mean drift is approximately −2.34057×10⁻⁴ m/s. The separate 2 s ramp transfers approximately 37.6991 C.

Investigation

Compare beta = 0, 1 and 2 at fixed radius. Predict the ratio of mean to edge density and retain the annular integral. Reverse the selected normal without changing physical flow. Then integrate the supplied time ramp and distinguish total charge from mean and endpoint current.

Changed-case practice

Sources and evidence boundary

Original teaching checked 31 October 2026 against the current official framework, Unit 11 and OpenStax conduction in metals. All diagrams and numerical records are virtual, supplied or generated. No physical wiring, battery connection, short circuit, meter operation or electrical handling is requested. Public feedback and retained self-reports do not certify independent reasoning, actual laboratory participation or institutional credit.