Magnetic Force on a Moving Charge

An electric field acts on a charge whether it moves or not, and the force qE lies along E. A magnetic field does neither. Hold a charge still in a 1 T field and nothing pushes it; set it moving and a force appears, and careful measurement pins down exactly what that force depends on. Double the charge and it doubles; double the speed and it doubles; double the field and it doubles; swing the velocity towards the field direction and it falls away as sin θ, vanishing when the two are parallel. Whatever the arrangement, the force comes out perpendicular to both v and B. Those five facts are the entire experimental content of this topic, and F = qv×B is the one line that states them together. The forces are small in absolute terms — a proton at 3.0 × 10⁶ m s⁻¹ across a 0.25 T field feels 1.2 × 10⁻¹³ N — but that is about 7 × 10¹² times the proton's own weight, so on this scale gravity is simply not in the picture.

Do the geometry in two separate steps and the sign mistakes stop. Step one is the cross product itself, which knows nothing about the charge: point the fingers of your right hand along v, curl them towards B through the smaller of the two angles, and your thumb points along v×B. Step two is the charge: if q is positive, F lies along v×B; if q is negative, F lies exactly opposite. Take an electron heading due east through a horizontal 0.40 T field pointing due north. With east = x̂, north = ŷ and up = ẑ, the rule x̂ × ŷ = ẑ says v×B points straight up, and the electron's negative charge then turns the force straight down, of size 1.60 × 10⁻¹⁹ × 2.0 × 10⁶ × 0.40 = 1.3 × 10⁻¹³ N. Order matters too: B×v = −v×B, so writing the two factors the wrong way round inverts every answer you get.

Split the velocity into a component along the field, v∥ = v cos θ, and a component across it, v⊥ = v sin θ. The parallel part contributes nothing, because parallel vectors have zero cross product. So F = |q| v⊥ B exactly, and the sin θ in |q| v B sin θ is not decoration: it is the statement that only v⊥ is doing anything. The numbers fall off faster than most students expect. At θ = 90° you get the full |q| v B; at 30° you get half of it; at 10° you get 17% of it. An electron at 1.0 × 10⁷ m s⁻¹ in a 0.050 T field feels 8.0 × 10⁻¹⁴ N when it crosses the field squarely, and only 2.7 × 10⁻¹⁴ N when its path makes 20° with the field lines. This also gives a free check on any answer: the dependence must be a sine, largest where v and B are perpendicular and zero where they are parallel — the exact opposite of what a dot product would give.

The rate at which any force does work is P = F⋅v. Here F = qv×B is perpendicular to v by construction, so P = q(v×B)⋅v = 0 at every instant, for any field, uniform or not. Integrate along any path and W = 0, so by the work–energy theorem the kinetic energy is unchanged and the speed is constant: a particle that enters a magnetic field at 2.5 × 10⁶ m s⁻¹ leaves it at 2.5 × 10⁶ m s⁻¹, however long and however curved the journey between. What is not constant is the velocity. F = dp/dt is non-zero for the whole crossing, and momentum is a vector: its magnitude holds while its direction turns. This is why no static magnetic field can give a beam energy. Every accelerator that raises energy does so with an electric field, and the magnets are there to point the beam — a job they can do precisely because they cannot do the other one.

A charge usually sits in both kinds of field at once, and the complete statement is the Lorentz force F = q(E + v×B), with both fields read where the charge is at that instant. The two terms behave quite differently. The electric term is present at any speed, including zero, and lies along E for a positive charge. The magnetic term is velocity-dependent, so the same field is barely felt by a slow charge and felt violently by a fast one. Compare them through the product vB, which carries the units of an electric field: at 3.0 × 10⁶ m s⁻¹ in 0.25 T, vB = 7.5 × 10⁵ V m⁻¹, a quarter of the field that would break down dry air, so the magnetic term dominates almost any laboratory E. Drop the speed to a walking 1.5 m s⁻¹ and the same field gives vB = 0.375 V m⁻¹, negligible against almost any E. That contrast is why magnetic forces matter enormously for beams and plasmas and hardly at all for slow-moving charge.

Four limits are worth naming now. A charge travelling exactly along B feels no magnetic force whatever the field strength, so a strong field is no guarantee of a deflection. The force is not constant even in a perfectly uniform field, because it depends on the instantaneous v, which the force itself keeps turning — so the constant-acceleration formulas from mechanics never apply to it. No potential energy belongs to it either: qv×B is not a function of position alone, so there is nothing to differentiate, and energy is conserved here because the work is zero, not because anything is stored. Finally, the split between electric and magnetic is not absolute. An observer riding along with the charge sees it at rest, and so sees no magnetic force at all, yet still sees it deflect; that observer must attribute the deflection to an electric field. What one frame calls magnetic another calls partly electric, and the Lorentz force law holds in every frame with each observer using the E and B they measure.