Force on a Current-Carrying Wire
Currents are pushed by fields.
Lesson at a glanceShow the key points
- dF⃗ = i dl⃗ × B⃗ (34.5); F⃗ = i l⃗ × B⃗ (34.6).
- Follows from the force on drifting free electrons.
- A closed loop in a uniform field feels no net force.
A current is moving charge, so a magnetic field pushes on a wire that carries one. That push turns every electric motor and loudspeaker.
Adding up the forces on the electrons
Take a small length dl of a wire carrying current i in a field B⃗. Its free electrons drift with velocity v⃗d opposite to the current, and i = nevdA. Each feels an average force −ev⃗d × B⃗, and the element holds nA dl of them:
where dl⃗ points along the current. Using i = nAevd:
A straight wire 50 cm long carries 4.0 A at 30° to a uniform 0.20 T field. Find the force on it.
Closed loops in a uniform field
Bend a wire into an equilateral triangle carrying current, in a field perpendicular to its plane. Each side feels ilB, directed perpendicular to that side, all pointing inward (or all outward). Three equal forces at 120° cancel. A side parallel to B feels nothing, while sides at angles feel forces into and out of the plane that again cancel. In general:
A rod on rails
Two horizontal rails 20 cm apart lie in a vertical field of 0.50 T. A 50 g rod across them carries 3.0 A. (a) What is the least friction coefficient that holds it still? (b) If μ = 0.30, find its acceleration.
- 1dF⃗ = i dl⃗ × B⃗ (34.5); straight wire F⃗ = i l⃗ × B⃗ (34.6).
- 2Derived by adding the forces on drifting electrons.
- 3A closed loop in a uniform field feels no net force.
Check yourself
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