Reconcile motional emf, magnetic drag and mechanical work

Establish the model and evidence

A moving conductor can separate charge because its mobile charges experience magnetic force. For a straight rod of length L moving perpendicular to itself and to uniform B, the emf magnitude is BLv. In a closed rail circuit this drives current; in an open rod, charges separate until the internal electric field balances magnetic deflection, without a sustained loop current.

The virtual rod is the right edge of a rectangle and moves in +x while B points +z. The +z flux increases as area grows. Lenz’s law therefore requires clockwise current, negative relative to the +z normal’s counterclockwise convention. In the moving right edge the current is downward, and its magnetic force is toward −x, opposing the imposed motion.

To maintain constant speed, an external agent supplies equal opposite force. Combining emf=BLv, I=emf/R and force magnitude |I|LB gives external mechanical power Fv=B²L²v²/R=I²R. The same energy enters mechanically and leaves as resistor heat. Increasing speed increases both emf and drag, while power grows as speed squared for fixed resistance.

There is no contradiction with the statement that a magnetic force does no work on an individual moving charge. The externally driven conductor and its constrained charges exchange energy through the complete electromechanical system. Maintaining rod speed requires mechanical input. The model neglects friction, inductance and other contact effects; it cannot certify a real generator design, efficiency or safe apparatus procedure.

Worked evidence

With B=0.50 T, L=0.40 m, v=1.0 m/s and R=4 Ω, signed emf is −0.20 V and current −0.050 A. Magnetic drag is −0.010 N. External input is +0.010 W, matching resistor heat rate; over 5 s, 0.050 J becomes heat.

Predict and investigate

If speed doubles with resistance fixed, do induced current and heating power both double?

Compare speeds 1 and 2 m/s at 4 Ω, then compare resistances 4 and 8 Ω at 1 m/s. Retain current sign, drag, sustaining force and both powers. Use either the charge-force or flux-change route to check the emf magnitude.

Transfer to changed conditions

A 0.30 m rod moves at 2 m/s through 0.40 T with total resistance 3 Ω. Find emf magnitude, current magnitude, drag magnitude and mechanical power.

Compare the worked transfer

Emf=0.24 V, current=0.080 A, drag=0.0096 N and power=0.0192 W. Current direction must be fixed by the actual geometry and chosen normal, not inferred from unsigned magnitudes alone.

Explain the boundary

Explain which omitted energy terms would be required if the rod slowed down rather than moving at prescribed constant speed.

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.