Magnetic Force on Moving Charges

A magnetic force is simply the push a magnetic field gives a charge that's moving, and the strange thing about it is that it always pushes sideways. (The word "magnet" comes from Magnesia, a region of ancient Greece where naturally magnetic lodestone was found.) It's what steers particles in accelerators, sorts ions in a mass spectrometer and lights up the auroras. The formula $\mathbf{F} = q\mathbf{v}\times\mathbf{B}$ is just three pieces: $q$ (how much charge), $\mathbf{v}$ (how fast it's moving and which way) and $\mathbf{B}$ (how strong the field is and which way it points), joined by a cross product (the part that makes the push come out sideways). It looks abstract, but that one sideways push explains everything in this lesson. Bear with it.

Sideways, always. Hold a magnet near an electron beam and the beam bends, but not toward the magnet. It bends sideways, perpendicular to both the beam and the field. Charges at rest feel nothing. Charges moving along the field lines feel nothing either. Only motion across the field counts, and the push is perpendicular to the motion. That's very different from the electric force you've been working with, which pushes along the field whether the charge moves or not.

The force law. Experiments show the force grows with the charge, the speed and the field, goes like $\sin\theta$ for the angle between $\mathbf{v}$ and $\mathbf{B}$, and points perpendicular to both. That's exactly a cross product:

Point your right hand's fingers along $\mathbf{v}$, curl them toward $\mathbf{B}$, and your thumb gives the force on a positive charge; for an electron, flip it. The equation also defines the unit of field: one tesla pushes $1$ N on $1$ C moving at $1$ m/s straight across it. Earth's field is about $50\ \mu\text{T}$, a fridge magnet about $5$ mT, an MRI scanner $1.5$ to $3$ T.

Magnetic forces do no work. The power a force delivers is $\mathbf{F}\cdot\mathbf{v}$, and here

because a cross product is perpendicular to each of its factors. A magnetic field can steer a charge anywhere, but it can never speed it up or slow it down. Changing a particle's energy takes an electric field, as in Charges Moving in Electric Fields.

Circles. Send a charge straight across a uniform field. The force has constant size $|q|vB$ and always points perpendicular to the velocity: the recipe for uniform circular motion. Newton's second law with centripetal acceleration $v^2/r$ gives

The time for one lap is $T = \frac{2\pi r}{v} = \frac{2\pi m}{|q|B}$, and notice the speed has cancelled. Think about why. A faster particle feels a bigger force, but it also needs more force to turn, so it ends up on a bigger circle, covered at a proportionally higher speed. A particle twice as fast swings a circle twice as big and gets around in the same time. That lap rate, $f_c = |q|B/(2\pi m)$, is the cyclotron frequency.

The same thing as a differential equation. With $\mathbf{B} = B\hat{\mathbf{k}}$, Newton's law $m\mathbf{v}' = q\,\mathbf{v}\times\mathbf{B}$ reads

a linear system with matrix $\begin{bmatrix}0 & \omega_c\\ -\omega_c & 0\end{bmatrix}$ and eigenvalues $\pm i\omega_c$. Purely imaginary eigenvalues mean pure rotation (Complex Eigenvalues): the velocity turns at the steady rate $\omega_c$, clockwise for a positive charge when $\mathbf{B}$ points at you, and $v_z$ never changes.

Helices. Give the charge some velocity along $\mathbf{B}$ as well: that part feels nothing and carries on, while the perpendicular part circles. The path is a helix wrapped around a field line. That's how solar-wind particles spiral down Earth's field toward the poles and light up the auroras.

Crossed fields pick a speed. Add an electric field perpendicular to $\mathbf{B}$, arranged so $q\mathbf{E}$ and $q\mathbf{v}\times\mathbf{B}$ point opposite ways. The total, the Lorentz force $q(\mathbf{E} + \mathbf{v}\times\mathbf{B})$, vanishes only when $qE = qvB$, so only particles with $v = E/B$ go straight through a slit at the far end. Feed those into a pure magnetic field and each one circles with $r = mv/(|q|B)$, so where it lands measures its mass: a mass spectrometer.

The Hall effect. The same sideways push acts on the carriers drifting in a wire. In a flat strip carrying current across a field they pile up along one edge, until the electric field of the piled-up charge balances the magnetic push: $qE_H = qv_dB$. With $v_d = I/(nqA)$ from Current, Resistance and Ohm's Law, the strip develops a sideways voltage $V_H = \frac{IB}{nqt}$, where $t$ is its thickness. Hall sensors read magnetic fields this way in brushless motors, wheel-speed sensors and the switch that wakes a laptop when its lid opens, and the sign of $V_H$ is how we know the carriers in most metals are negative.

A common mistake. Since a magnetic field clearly pushes on a moving charge, it's tempting to think it can speed the charge up or slow it down. It can't. The push is always perpendicular to the motion, so it does no work and only changes the direction. Two smaller slips to watch for: forgetting to flip the direction for electrons, and using $F = qvB$ when $\mathbf{v}$ isn't perpendicular to $\mathbf{B}$ (only $v_\perp = v\sin\theta$ counts).

Mass spectrometers sort ions by $r = mv/(qB)$ to date ancient carbon, screen luggage for explosives and identify proteins. Hospital cyclotrons spin protons at one fixed frequency to make the short-lived isotopes used in PET scans. Hall sensors measure fields in brushless motors and wheel-speed sensors, and Earth's field steers solar-wind particles into the helical paths that light up the auroras.