Magnetic Force: Lorentz Force, Right-Hand Rule and Circular Motion

A magnetic field exerts a force only on moving charges. The force is perpendicular to both the velocity and the field, so it changes the direction of motion but never the speed. Most MCAT questions are a right-hand rule plus a circular motion radius.

Core formulas

θ is the angle between v (or the current) and B. Force is zero when they are parallel and largest when perpendicular.

Right-hand rule

Point fingers along v (or the current), curl them toward B, and the thumb gives the force on a positive charge. For a negative charge, reverse the result. For the field around a straight wire, point the thumb along the current and the fingers curl in the field direction.

Why the path is a circle

The force is always perpendicular to v, so it acts as a centripetal force: qvB = mv²/r. Solving gives r = mv/(qB). A faster or heavier particle makes a bigger circle. A stronger field or larger charge makes a smaller one. If v has a component along B, the path is a helix.

Worked example

A proton (m = 1.7 x 10^-27 kg, q = 1.6 x 10^-19 C) moves at 2 x 10⁶ m/s perpendicular to a 0.5 T field.

• F = qvB = (1.6 x 10^-19)(2 x 10⁶)(0.5) = 1.6 x 10^-13 N.
• r = mv/(qB) = (1.7 x 10^-27)(2 x 10⁶) / ((1.6 x 10^-19)(0.5)) = about 0.043 m.
• Its kinetic energy does not change.

Common traps

• Expecting a magnetic force on a stationary charge. There is none.
• Saying the field does work. It does not, since F is perpendicular to motion.
• Forgetting to flip the direction for electrons.
• Using sinθ with the wrong angle. It is the angle to B, not to the normal.

One-minute recap

F = qvB sinθ, perpendicular to v and B, no work done. Right-hand rule for positive charge, reverse for negative. r = mv/(qB). Velocity selector v = E/B.