In Physics 2 the magnetic force had a size, $qvB\sin\theta$, and a direction from the right-hand rule. The cross product packs both into one operation, and adding the electric force gives the complete force on a charge in any fields.
Key idea
The Lorentz force:
$$\vec{F} = q\left(\vec{E} + \vec{v}\times\vec{B}\right),$$
$$\vec{v}\times\vec{B} = \left(v_yB_z - v_zB_y,\ \ v_zB_x - v_xB_z,\ \ v_xB_y - v_yB_x\right).$$
For a negative charge, $q < 0$ reverses the force.
Why it works
A cross product has exactly the properties the magnetic force has: its size is $vB\sin\theta$ (zero when $\vec{v}$ runs along $\vec{B}$, largest when they are perpendicular), and it points perpendicular to both $\vec{v}$ and $\vec{B}$. Because the magnetic part is always perpendicular to $\vec{v}$, it does no work: $\vec{F}\cdot\vec{v} = 0$. A magnetic field can bend a charge's path but never speed it up or slow it down.
Worked examples
Worked example: Force from components
A $+2.0\ \mu\text{C}$ charge moves with $\vec{v} = (3, 0, -1) \times 10^{3}\ \text{m/s}$ in $\vec{B} = (0, 0.2, 0.5)\ \text{T}$. Find $\vec{F}$.
- Factor out the scales: $\vec{v}\times\vec{B} = (10^3)(0.1)\,\big[(3, 0, -1)\times(0, 2, 5)\big]$.
- $(3, 0, -1)\times(0, 2, 5) = \big(0\cdot 5 - (-1)(2),\ (-1)(0) - 3\cdot 5,\ 3\cdot 2 - 0\cdot 0\big) = (2, -15, 6)$.
- $\vec{F} = (2.0 \times 10^{-6})(100)(2, -15, 6)\ \text{N} = (0.4,\ -3,\ 1.2)\ \text{mN}$.
Answer: $\vec{F} = (0.4,\ -3,\ 1.2)\ \text{mN}$
Worked example: Checking that it does no work
Show that the force in the first example is perpendicular to the velocity.
- $\vec{F}\cdot\vec{v} \propto (2, -15, 6)\cdot(3, 0, -1) = 6 + 0 - 6 = 0$.
- Zero, as it must be: the magnetic force changes only the direction of motion.
Answer: $\vec{F}\cdot\vec{v} = 0$
Worked example: An electron's direction
An electron moves along $+x$ through a field along $+z$. Which way is the force?
- $\hat{\imath}\times\hat{k} = -\hat{\jmath}$, so $\vec{v}\times\vec{B}$ points along $-y$.
- The electron's charge is negative, which flips it: the force points along $+y$.
Answer: $+y$
Nature uses this all the time. Charged particles streaming from the Sun meet Earth's magnetic field and spiral along its field lines, which funnel them toward the polar regions; where they strike the upper atmosphere, it glows as the aurora.
Common mistake
Taking $\vec{B}\times\vec{v}$ instead of $\vec{v}\times\vec{B}$, or forgetting the charge's sign. Swapping the order reverses a cross product, and so does a negative charge; doing either by accident gives a force pointing exactly backward. Two checks: the force must be perpendicular to both $\vec{v}$ and $\vec{B}$ (dot products zero), and for a proton moving along $+x$ in a field along $+y$ the force is along $+z$ ($\hat{\imath}\times\hat{\jmath} = \hat{k}$).