Lorentz force

For a classical point charge q with instantaneous velocity v in prescribed fields E(r,t) and B(r,t), the electromagnetic force is the empirical Lorentz force law:

Here v is the particle's velocity in the observer's chosen inertial reference frame, and E/B are fields evaluated at its instantaneous position. The magnetic part vanishes for a stationary charge. The formula is a FORCE law given E and B, not a method to calculate B from sources. Field transformations between moving reference frames require special relativity and are not obtained by simply switching signs of a fixed B.

If θ is the angle between v and B, then the magnitude of their cross product is vB sinθ; the sign of q changes the direction but not the magnitude:

For B=+B ẑ and v=+v x̂, x̂×ẑ=−ŷ: a POSITIVE charge initially bends toward −y while a negative one bends toward +y. Reversing B or reversing v independently reverses the magnetic force; reversing two of q, v, B returns its original direction. Never use a scalar force magnitude to decide trajectory orientation.

Mechanical power delivered to a point charge is P=F·v. A cross product is orthogonal to both factors, hence

For E=0 and no other forces, K stays constant: B can curve the motion but cannot change the speed. This instantaneous orthogonality remains true even if B itself varies with time; however a changing magnetic field may induce E, and E can do work. Collisions, radiation reaction and external forces can change particle kinetic energy, so the zero magnetic-power identity alone does not guarantee conservation in a more complete system.

Take B=B ẑ constant in space and time, E=0, no collisions and v·B=0 initially. Since the force is always perpendicular to v with magnitude |q|vB, it supplies centripetal acceleration at constant speed. Newton's second law gives

The positive scalar ωc does not encode circulation direction. For the above +ẑ field, a positive charge starting toward +x bends toward −y and rotates clockwise as seen from +z; a negative one rotates the opposite way. If v=0 or B=0 there is no circular orbit to which the radius formula can be applied.

For arbitrary initial v write v=v∥+v⊥, with v∥=(v·b̂)b̂ and b̂=B/|B|. The parallel component experiences no Lorentz magnetic force and stays constant for the stated uniform static field; the perpendicular component rotates. The resulting trajectory is a helix:

Here the pitch is a SIGNED axial displacement if v∥ is signed along the chosen b̂; its geometric length is |phelix|. An initially parallel velocity has rL=0 and travels straight. A time-dependent or spatially nonuniform B, electric acceleration, and collisions generally invalidate the simple constant-radius/constant-pitch helix.

For an electron with mass m≈9.11×10⁻³¹ kg, charge magnitude e≈1.60×10⁻¹⁹ C, B=0.100 T, v⊥=1.00×10⁶ m/s and v∥=2.00×10⁵ m/s,

The total speed ≈1.02×10⁶ m/s is about 0.0034c, consistent with the nonrelativistic approximation. The electron's circulation direction is OPPOSITE that of a positive charge with the same initial velocity and B. Frequencies and radii are metrics of the chosen particle/field, not intrinsic fixed properties of an electron alone.

Take a thin stationary wire segment of vector length dℓ oriented along conventional current. Suppose n carriers per volume with individual charge q, drift velocity vd, and cross-section A, with a locally uniform applied B. A segment of length dℓ contains nA dℓ carriers and their total magnetic force is

For electron carriers q<0, their drift direction is opposite conventional current, so qvd still points along the current direction. The magnetic carrier force is transmitted to the conductor lattice in steady state. The thin-wire approximation assumes a specified local current direction and field; for a spatially distributed current, the local force density is f=J×B.

A straight wire segment of length L=0.050 m carries I=2.00 A in +x̂ through a uniform B=0.300 T in +ẑ. Because x̂×ẑ=−ŷ,

If the wire becomes parallel to B, the force vanishes. Reversing current reverses the force. The result does not say that the entire wire accelerates freely: clamps and support forces can balance the magnetic force.

The Lorentz law predicts motion or mechanical force once E and B are KNOWN. It says nothing by itself about the field produced by a specified wire shape or current density. Biot–Savart will compute B for steady currents; Ampère's law will express its circulation and becomes a practical field calculator when sufficient source symmetry exists. Time-varying systems eventually require Maxwell's equations including displacement current.

1. A positive charge q=+1.00 nC moves with v=+1000 x̂ m/s in B=+0.200 ẑ T, E=0. Compute its magnetic force vector and the force for negative q.
2. Prove algebraically that Fmag·v=0 for arbitrary θ. Can a time-dependent magnetic field nevertheless be accompanied by acceleration of particle speed? Explain.
3. A proton has v⊥=2.00×10⁵ m/s and B=0.500 T. Estimate its radius and cyclotron frequency using mp≈1.67×10⁻²⁷ kg and e≈1.60×10⁻¹⁹ C.
4. For a particle with v∥=3.00×10⁵ m/s, B=0.100 T, mass and charge magnitude equal to an electron, find its helical pitch.
5. A straight wire segment carries I=2.00 A along +x̂, length 0.050 m, in uniform B=0.300 ẑ T. Find F with sign and units; what happens on current reversal?
6. Can the Lorentz force on a test charge alone give the magnetic field of a nonuniform current distribution? Name the additional information/law needed.

Next: derive the Biot–Savart integral for steady currents and integrate it for a line, a loop and a finite straight segment without inferring field geometry from force arrows.