07 / CHAPTER 09
Biot–Savart and Ampère
Choose integral tools for magnetic-field geometry.
ANCHOR EQUATION
Ampère’s law
In symbols: ∮B·dl = μ₀Ienc
Current circulation creates magnetic field circulation.
Magnetic force bends moving charge
A magnetic field exerts force only on moving charge and the force is perpendicular to velocity. It changes direction and therefore momentum, but does no work on an ideal point charge.
Use the cross product
The Lorentz force is F = q(E + v × B). For magnetic-only motion with v perpendicular to B, magnitude is F = |q|vB. Equating to circular centripetal force mv²/r gives r = mv/(|q|B). The radius grows with momentum and shrinks with field strength.
A proton with v = 1.0×10⁶ m/s in B = 0.10 T has r = mv/(qB) = 1.67×10⁻²⁷(10⁶)/(1.60×10⁻¹⁹·0.10) = 0.104 m.
Parallel motion is unaffected
If v is parallel to B, v × B = 0 and the particle continues straight; a general velocity splits into parallel uniform motion and perpendicular circular motion, producing a helix.
Work it: An electron enters B = 0.20 T at 3×10⁶ m/s perpendicular to the field. Find radius and cyclotron frequency f = qB/(2πm).