Balance crossed-field forces without removing the fields

Establish the model and evidence

An electric field exerts force qE regardless of whether the particle is instantaneously moving. A magnetic field exerts q times the velocity-cross-field direction and has a speed dependence. These forces can oppose one another, but cancellation is a condition on the complete setup, not evidence that either field is absent.

Here velocity is initially +x, electric field +y and magnetic field +z. For positive charge, the electric force is +y and magnetic force −y. Zero net entry force requires E=vB, so the selected speed is E/B. Reversing the charge reverses both contributions, preserving that speed condition for any nonzero charge.

Below the selected speed, electric force dominates for a positive particle; above it, magnetic force dominates. A negative charge reverses those net directions. Neutral particles feel neither force in this idealization and therefore pass undeflected at every speed. Consequently an undeflected path alone does not establish the particle’s sign, mass or even that it carries charge.

The plotted force is evaluated at entry. A particle with nonzero net force changes velocity, which in turn changes its magnetic force. Extending the entry force as a constant acceleration for an entire path would be a new and generally incorrect assumption. The exact selected trajectory can remain straight because its velocity and opposing forces remain unchanged. The virtual model supplies no construction or operating instructions for beam equipment.

Worked evidence

With E=6.0×10⁴ N/C and B=0.20 T, selected speed is 3.0×10⁵ m/s. For q=+1.60×10⁻¹⁹ C at that speed, each opposing force has magnitude 9.6×10⁻¹⁵ N. The net force is zero, although neither individual force is zero.

Predict and investigate

If charge sign reverses, must selected speed reverse or change magnitude?

Capture speeds 2, 3 and 4×10⁵ m/s for positive charge, then repeat for negative charge. Retain both force components and their sum. Finally test neutral charge and explain why it defeats a charge-identification claim.

Transfer to changed conditions

A selector has E=4.5×10⁴ N/C and B=0.15 T. Compare selected speed for positive and negative particles of different masses, and explain the neutral case.

Compare the worked transfer

The selected speed is 3.0×10⁵ m/s for either nonzero charge sign and any mass under the ideal force balance. Neutral particles are undeflected at all speeds, so a straight path is not sufficient evidence of selection.

Explain the boundary

Why cannot one use the plotted off-selection entry force as an exact constant force for a full trajectory?

Use Hall separation as an internal force balance

In the Hall effect, moving carriers in a conductor are deflected toward a side until separated charge creates a transverse electric field that can balance magnetic force. It is a potential difference caused by the carrier redistribution, not evidence that the current has vanished. This is a virtual sign argument, not wiring guidance.

Predict and check: conventional current is +x, mobile electrons drift −x, and the applied magnetic field is +z. For those electrons, v cross B is +y and multiplying by negative charge makes the magnetic force −y. Electrons accumulate at the −y edge. The resulting electric field points −y, from the positive +y edge to the negative −y edge, giving electrons an opposing +y electric force. Thus the +y edge has higher potential. Reverse B and reverse the Hall polarity. The carrier sign and current convention must both be retained; a neutral test particle cannot establish this carrier balance. (Current clause 12.2.B.4.)

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.

Clause-level source/action reconciliation checked 31 October 2026 against the linked current framework and Fall 2026 clarification. Added packets are original virtual comparisons; no new practical or learner-completion claim.