Current and Resistance
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 5 practice questions.
Current is the rate charge flows past a point: I = dQ/dt. Microscopically, it’s carriers drifting slowly: I = nqvdA. A conductor’s resistance comes from its shape and material: R = ρL/A, or ∫ρ dx/A if things vary along it.
1. Current from charge flow
2. Things that surprise people
Drift speeds are tiny (fractions of a millimeter per second), yet lights turn on instantly: the field travels through the wire at nearly light speed and pushes all the electrons at once.
For the same current, a thicker wire has a slower drift speed: vd ∝ 1/A.
Stretching a wire to three times its length at constant volume cuts its area to a third, so R = ρL/A becomes 9 times as large.
If current decays as I0e−t/τ, the total charge that flows is ∫I dt = I0τ.
3. Example, then try it
The charge through a wire is Q(t) = 0.5t² + 2t (C). Find the current at t = 4.0 s, and the charge that passes between t = 0 and 4.0 s.
I = dQ/dt = t + 2, so I(4) = 6.0 A.
Q(4) − Q(0) = 8 + 8 = 16 C.
A copper wire with cross-section 1.0 mm² (1.0 × 10⁻⁶ m²) carries 3.0 A. Copper has n = 8.5 × 10²⁸ free electrons per m³. Find the drift speed, and how long an electron takes to drift 1.0 m.
Rearrange: vd = I/(neA).
Numbers: vd = 3.0 ÷ [(8.5 × 10²⁸)(1.6 × 10⁻¹⁹)(1.0 × 10⁻⁶)] = 3.0 ÷ 13,600 ≈ 2.2 × 10⁻⁴ m/s.
Time for 1.0 m: 1.0 ÷ (2.2 × 10⁻⁴) ≈ 4500 s, about 75 minutes.
Answer: about 0.22 mm/s; over an hour to drift 1 m.
4. Common mistakes
The current in a wire is doubled. What happens to the electrons’ drift speed?
I = nqvdA, with n, q and A fixed for the same wire, so vd ∝ I.
Ready to practice?
5 questions on Current and Resistance, each with a hint and an explanation for every wrong answer.