Motion in a magnetic field

The magnetic force is always perpendicular to the velocity, so it does no work. The speed and kinetic energy of the charge stay fixed; only the direction of motion changes. An electric force, by contrast, can change the energy.

When v is perpendicular to a uniform B, qvB supplies the centripetal force mv²/r, so the charge moves in a circle of radius r = mv/(qB) in the plane normal to B. More momentum means a bigger circle.

The angular frequency is ω = 2πν = qB/m and the period T = 2πm/(qB). Neither depends on the speed or the energy: a faster charge runs a bigger circle in the same time. ν = qB/2πm is called the cyclotron frequency.

If v has a component v∥ along B, that part is untouched while the perpendicular part still turns in a circle. The path is a helix whose radius uses v⊥ and whose pitch (advance per turn) is p = v∥T = 2πm v∥/(qB).

Example 4.3: an electron (m = 9 × 10⁻³¹ kg) at 3 × 10⁷ m/s in 6 × 10⁻⁴ T moves on a circle of r = mv/(qB) = 0.28 m, at ν = v/2πr ≈ 17 MHz, with energy ½mv² ≈ 4 × 10⁻¹⁶ J ≈ 2.5 keV.

A positive and a negative charge entering the same field with the same velocity curve in opposite senses.