Section 02
The Idea, Step by Step
A fridge magnet sticks to steel but does nothing to a marble rolling past it. Yet that same kind of magnet can grab a beam of moving electrons in an old TV tube and bend it sideways, and it bends the charged particles streaming from the Sun into the glowing curtains we call the aurora. Here is the whole secret in one line: a magnet pushes on electric charge only when the charge is moving, and the push is always sideways. Sitting still, a charge feels nothing. Moving, it gets shoved at a right angle to where it is going.
To make that into numbers we need three things: how much charge is moving ($q$), how fast it moves ($v$), and how strong the field is ($B$, measured in tesla). When the motion is square-on to the field, the force is simply
Try a real number. A proton ($q = 1.6\times10^{-19}$ C) flying at $v = 1\times10^{6}$ m/s across a $B = 0.5$ T field feels $F = (1.6\times10^{-19})(10^{6})(0.5) \approx 8\times10^{-14}$ N. That sounds laughably small, but a proton weighs almost nothing, so this tiny sideways nudge is enough to curl its path into a tidy circle.
The precise picture
The full law is a cross product, $\vec{F} = q\,\vec{v}\times\vec{B}$, with magnitude $F = qvB\sin\theta$. "Cross product" is just shorthand for "perpendicular to both": the force points at right angles to the velocity and to the field at once. Two consequences fall straight out. First, because the push is always sideways to the motion, it can never speed the particle up or slow it down — it does zero work and only steers. Second, if the charge moves straight along the field ($\theta = 0$), $\sin\theta = 0$ and there is no force at all; move at an angle and the path becomes a corkscrew helix.
Since the sideways force is exactly a centripetal force, set $qvB = mv^2/r$ and solve: the orbit radius is $r = mv/(qB)$ and the time for one loop is $T = 2\pi m/(qB)$ — which, remarkably, does not depend on speed. In the sim, the B field slider sets $B$, velocity sets $v$, charge sets $q$ and mass sets $m$, and you watch the radius $r_c$ respond.
Try this in the sim above: (1) Slide $B$ up and watch the circle tighten — radius scales as $1/B$. (2) Increase $v$: the loop grows wider, yet the time to go around stays the same (that is the trick behind cyclotron accelerators). (3) Switch to Wire in Field mode and raise the current $I$ to see the force on a whole wire grow as $F = BIL$.
Section 03
Equations & Derivation
Lorentz Force Law
$$\vec{F}=q(\vec{E}+\vec{v}\times\vec{B}),\quad F_B=qvB\sin\theta$$
Circular Motion of Charged Particle
$$r=\frac{mv}{|q|B},\quad T=\frac{2\pi m}{|q|B},\quad\text{(cyclotron frequency: independent of }v\text{)}$$
Force on Current-Carrying Wire
$$\vec{F}=I\vec{L}\times\vec{B},\quad F=BIL\sin\theta$$
Biot-Savart Law & Long Wire
$$d\vec{B}=\frac{\mu_0}{4\pi}\frac{Id\vec{l}\times\hat{r}}{r^2},\quad B=\frac{\mu_0 I}{2\pi r}\text{ (long straight wire)}$$
Solenoid
$$B=\mu_0 nI,\quad n=N/L,\quad\mu_0=4\pi\times10^{-7}\;\text{T\,m\,A}^{-1}$$
Symbol Definitions
| Symbol | Quantity | SI Unit |
|---|
| $\vec{B}$ | Magnetic field (flux density) | T = kg A⁻¹ s⁻² |
| $q$ | Charge | C |
| $v$ | Particle speed | m s⁻¹ |
| $\mu_0$ | Permeability of free space | 4π×10⁻⁷ T m A⁻¹ |
| $n$ | Turn density of solenoid | turns m⁻¹ |
| $r_c$ | Cyclotron radius | m |
1
Lorentz force is perpendicular to $\vec{v}$. $\vec{F}_B=q\vec{v}\times\vec{B}$ — always perpendicular to velocity, so it does zero work and cannot change speed. It only changes direction. This is why magnetic fields cannot accelerate charged particles from rest, but can steer them in circles.
2
Cyclotron motion. Magnetic force provides centripetal force: $qvB=mv^2/r_c$, giving $r_c=mv/(qB)$. Period $T=2\pi m/(qB)$ is independent of speed — the basis of cyclotron particle accelerators.
3
Right-hand rule. For $\vec{F}=q\vec{v}\times\vec{B}$: point fingers along $\vec{v}$, curl toward $\vec{B}$, thumb points along $\vec{F}$ (for positive charge). For negative charge, reverse. For wire: $\vec{F}=I\vec{L}\times\vec{B}$.
4
Solenoid field. Inside an ideal solenoid: $B=\mu_0 nI$, uniform and parallel to axis. Outside: $B\approx0$. Used in MRI machines (superconducting solenoids at 1.5–3 T), particle accelerators, and inductors in electronics.
Ref: Halliday, Resnick & Walker 10th Ed., Ch. 28–29; Griffiths — Introduction to Electrodynamics (4th Ed.), Ch. 5.
Section 05
Common Misconceptions
❌ Magnetic forces can do work on charged particles.
✅ Magnetic force $\vec{F}_B=q\vec{v}\times\vec{B}$ is always perpendicular to velocity, so $\vec{F}_B\cdot\vec{v}=0$ and work = 0. Speed (and kinetic energy) is conserved. Only electric forces do work on charges. Permanent magnets attract iron — but this is because the magnetic force redistributes atomic currents inside the iron, and the electric forces between atoms do the actual work.
📖 HRW 10th Ed., §28-3.
❌ Magnetic field lines form closed loops — so a magnetic monopole must not exist.
✅ Gauss's Law for magnetism, $\nabla\cdot\vec{B}=0$ (or $\oint\vec{B}\cdot d\vec{A}=0$), implies no magnetic monopoles have ever been found. However, magnetic monopoles are not theoretically forbidden — they appear naturally in some grand unified theories. Their observed absence is still an open question in physics.
📖 HRW 10th Ed., §29-4; Griffiths — Introduction to Electrodynamics §5.3.
❌ A charged particle moving parallel to a magnetic field experiences a force.
✅ If $\vec{v}\parallel\vec{B}$, then $\sin\theta=0$ and $F_B=qvB\sin\theta=0$. No force. The particle continues in a straight line. Only the component of velocity perpendicular to $\vec{B}$ is affected. A particle with velocity at angle $\theta$ to $\vec{B}$ follows a helix: circular motion in the plane perpendicular to $\vec{B}$, uniform motion along $\vec{B}$.
📖 HRW 10th Ed., §28-2.
Misconception research: McDermott — Tutorials in Introductory Physics; Maloney et al. (2001) Am. J. Phys.