Connect force direction with constant-speed magnetic curvature

Establish the model and evidence

A moving charge in a magnetic field experiences magnitude F=|q|vB sinθ, where θ is the angle between velocity and field. The force is perpendicular to both directions. Use the right-hand orientation for positive charge and reverse it for negative charge. A stationary charge or a charge moving exactly parallel to the field has no magnetic force from that field alone.

The model chooses +x to the right, +y upward and +z out of the page. For a positive particle entering along +x with B along +z, the initial force is −y. A negative particle curves upward instead. A written statement such as “right-hand rule” is not enough: name the velocity, field, positive-charge force and any reversal caused by charge sign.

For perpendicular entry in a uniform field, the magnetic force supplies centripetal force. Equating |q|vB=mv²/r gives r=mv/(|q|B). Faster particles have larger radii, but the period 2πm/(|q|B) is unchanged in this nonrelativistic model. The force changes velocity direction, not speed; magnetic power F·v is zero, so kinetic energy remains constant.

A velocity component parallel to the field is unaffected, producing a helix rather than the planar circle shown. A moving charge itself also produces a magnetic field, whose direction is perpendicular to its velocity and the displacement toward the observation point; changing charge sign reverses that direction. Here source fields are held fixed and particle self-fields, collisions and radiation are omitted. This is not a particle-beam construction exercise.

Worked evidence

For m=2×10⁻²⁷ kg, |q|=1.60×10⁻¹⁹ C, B=0.20 T and v=2×10⁵ m/s, r=0.0125 m. Initial force magnitude is 6.4×10⁻¹⁵ N and the period is about 3.927×10⁻⁷ s. Positive charge starts curving downward; reversing charge preserves radius and energy but reverses curvature.

Predict and investigate

Will doubling speed change the radius, period and kinetic energy by the same factor?

Compare positive and negative charge at speed 2×10⁵ m/s, then increase speed to 4×10⁵ m/s. Use the common spatial axis scale and numerical ledger, not the screen width across rescaled plots. Check the neutral straight-path limit.

Transfer to changed conditions

A positive particle enters along +y in a field along +z. Find the force direction. Then describe what happens if velocity is instead entirely along +z.

Compare the worked transfer

For positive charge, +y crossed with +z gives +x force. If velocity is parallel to +z, the sine factor is zero and there is no magnetic deflection; no circular radius should be assigned to that straight parallel motion.

Explain the boundary

Explain how acceleration can occur without kinetic-energy change, and identify when this circular model fails.

Separate the field made by a charge from the force on it

The field made by a slowly moving point charge is strongest at observation directions perpendicular to its velocity and zero along that velocity axis in the elementary moving-charge model. Its magnitude depends on speed and source-observer separation; this is not the force law qvB for a second charge. No quantitative moving-point-source formula is required here.

Predict and sketch: a positive source moves along +x. At an observation point displaced along +y from it, the field points +z by the velocity-cross-position right-hand rule; at the opposite point it points −z. At a point on the +x axis it vanishes in this model. Reverse source charge to reverse the nonzero directions. Compare points at the same radius before making the perpendicular-versus-parallel magnitude claim. (Current clause 12.2.A.)

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.