Separate a wire’s field from the force it feels

Establish the model and evidence

A current-carrying segment in an external field experiences magnetic force of magnitude |I|LB sinθ, where θ is the angle between the segment’s conventional-current direction and B. The field in this formula is external to the segment; inserting a wire’s own field as though it pushed the entire wire sideways confuses source and response.

The activity measures its control angle from +x toward +z while the field is +z. Therefore the perpendicular segment projection is L cos(control angle), not L sin(control angle). At 0° the segment is perpendicular to B and the force is largest. At 90° it is parallel and the force is zero. Defining angle references is part of the physical model, not cosmetic notation.

For a positive current along +x and field +z, the force points −y. Reverse current or reverse the external field and the force reverses; reverse both and its direction returns. A wire can transmit this force to its supports or move under it, but motion and energy transfer require a mechanical model beyond the static force ledger shown.

Each of two nearby long wires feels the field of the other. Currents in the same direction attract; opposite currents repel. At separation d, force per length is μ₀I₁I₂/(2πd) in magnitude, with direction determined separately. This can be derived by combining the straight-wire field with the segment force. No high-current apparatus, strong magnet handling or real conductor-motion test is requested.

Worked evidence

For I=2 A, L=0.30 m and B=0.40 T, the 0° control gives Fy=−0.24 N. At 60° from +x toward +z, the perpendicular projection halves, giving Fy=−0.12 N. At 90° the wire is parallel to B and force is zero.

Predict and investigate

Is the control angle the same angle that appears in the usual ILB sinθ formula?

Capture angles 0°, 60°, 90° and 180° at +2 A. Draw segment and field directions for each. Reverse the current and verify which force signs reverse while magnitudes remain unchanged.

Transfer to changed conditions

Two long parallel wires carry 2 A and 5 A in the same direction, separated by 0.040 m. Find force per length using μ₀/(2π)=2×10⁻⁷, and predict the change if one current reverses.

Compare the worked transfer

Force per length is 5.0×10⁻⁵ N/m and is attractive. Reversing one current makes it repulsive with the same magnitude. Midpoint cancellation is a statement about a different observation location, not the field at either wire.

Explain the boundary

Explain why two same-direction currents attract even though their fields cancel at the midpoint between them.

Sources and authorship

Physical relationship reference. Current AP Physics 2 framework, checked 30 October 2026. Original prose, computed diagrams and synthetic/model evidence, not completed laboratory measurements. All activities are virtual. No strong-magnet handling, high-current wiring, beam equipment or moving-conductor apparatus is requested; authentic practical work requires qualified supervision and a separately approved safety procedure.