Short on time? This page is the recap. For the derivations, Figures 1 and 2 and the worked examples, use the full study guide. This guide is for the calculus-based course.
Recap
- Current is the rate at which charge crosses a cross-section of a conductor. 1 A = 1 C/s.
- Charge moves because a potential difference (from a source of emf, ℰ) sets up a field inside the conductor.
- Free electrons move randomly at about 10⁶ m/s. The field adds a tiny average drift velocity, often below 1 mm/s.
- Zero current means zero net motion of the carriers, not carriers at rest.
- Current density J is a vector pointing the way positive charge flows. For electrons it points opposite to their drift.
- Current has a direction along the wire but is a scalar: currents at a junction add as numbers.
- Conventional current is the direction positive charge would move. In metal wires, electrons move the other way.
Key relationships
| Quantity | Relationship | Notes |
|---|---|---|
| Current | I = dq/dt | Charge passed: q = ∫I dt |
| Current in terms of carriers | I = nqv_dA | n in m⁻³, q is one carrier’s charge |
| Uniform current density | J = I/A | A/m², A perpendicular to the flow |
| Current density (vector) | J = nqv_d | For electrons, q = −e, so J is opposite to v_d |
| Field in a conductor | E = ρJ | ρ is resistivity (Ω·m), not charge density |
| Non-uniform current density | I = ∫J·dA | Round wire: I = ∫₀ᴿ J(r) 2πr dr |
Assumptions behind the results
- All carriers have the same charge and the same average drift velocity.
- J is perpendicular to the cross-section you choose (otherwise use the dot product).
- The material is uniform, so n and ρ are the same everywhere in it.
Mistakes to avoid
- Treating current as a vector. 2 A and 3 A into a junction give 5 A out, not 3.6 A.
- Electron direction = current direction. They are opposite.
- I = JA for a varying J. Integrate over rings.
- Unit slips. 1 mm² = 10⁻⁶ m²; use the radius, not the diameter, in πr².
- “Charges are at rest when I = 0.” They move randomly with no net flow.
Quick self-check
- A steady 0.50 A flows for 60 s. How much charge passes? (30 C)
- The same current flows through a wire with three times the area, same material. What happens to v_d? (It falls to one third)
- How many electrons carry 1.0 C? (6.25 × 10¹⁸)
Next: practice questions.