Worked example: A proton (q = e, m = 1.67 × 10⁻²⁷ kg) enters a 0.5 T field at v = 2 × 10⁶ m/s perpendicular to B. (a) Find the orbital radius and frequency. (b) What changes for an alpha particle (q = 2e, m ≈ 4mp) at the same speed? Step 1: r = mv/qB = (1.67 × 10⁻²⁷ × 2 × 10⁶)/(1.6 × 10⁻¹⁹ × 0.5) = (3.34 × 10⁻²¹)/(8 × 10⁻²⁰) = 0.0418 m ≈ 4.2 cm. Step 2: f = qB/2πm = (1.6 × 10⁻¹⁹ × 0.5)/(2π × 1.67 × 10⁻²⁷) = (8 × 10⁻²⁰)/(1.049 × 10⁻²⁶) ≈ 7.6 × 10⁶ Hz ≈ 7.6 MHz. Step 3: Alpha: q/m is half (2e/4mp), so r doubles (≈ 8.4 cm) and f halves (≈ 3.8 MHz). This q/m dependence is exactly how mass spectrometers identify ions.