Foundations — Magnetic force on charge — F = qv × B

Before you can trust $\vec{F}=q\vec{v}\times \vec{B}$, every piece of it must be something you can see. This page builds each symbol from nothing, in an order where each one only uses the ones before it.

1. A number vs. an arrow — scalars and vectors

Look at the picture. The arrow's length is its size (the "how much"), and the way it points is its direction (the "which way"). A scalar is only the length; a vector keeps the pointing too.

Why the topic needs this: velocity $\vec{v}$, force $\vec{F}$, and field $\vec{B}$ are all vectors. If you only remembered their sizes and threw away the directions, you could never say the force is "sideways" — sideways is a direction statement, and only arrows carry directions.

2. The size of an arrow — magnitude bars $|\cdot |$

Why the topic needs this: the magnitude formula $|\vec{F}|=|q|vB\sin \theta$ uses both meanings at once — bars around vectors ($|\vec{F}|$) mean length, bars around the scalar charge ($|q|$) mean sign-stripped size, and the plain $v$, $B$ are shorthand for the lengths $|\vec{v}|$, $|\vec{B}|$. A force magnitude is always $\ge 0$, so every ingredient on the right must be $\ge 0$ too — that is exactly what the bars guarantee.

3. Breaking an arrow into pieces — components

In the figure, the slanted arrow is split into a horizontal shadow ($v_x$) and a vertical shadow ($v_y$). Adding those two shadow-arrows tip-to-tail rebuilds the original. That is all "components" means: writing one arrow as a sum of arrows along the reference axes.

Why the topic needs this: the foolproof direction method in the parent note is a component recipe, $(v_y{B}_z-v_z{B}_y,\dots )$. You cannot use it unless you can read $\vec{v}$ and $\vec{B}$ as lists of three numbers.

4. The angle between two arrows — $\theta$

Why the topic needs this: the magnitude $|\vec{F}|=|q|vB\sin \theta$ depends on this angle. When $\vec{v}$ points along $\vec{B}$, $\theta =0^{\circ }$; when it points across, $\theta =90^{\circ }$.

5. Why $\sin \theta$? — measuring "how sideways"

Picture the velocity arrow leaning at angle $\theta$ to the field. Drop it into two pieces: one parallel to $\vec{B}$ and one perpendicular to $\vec{B}$.

• The perpendicular (sideways) piece has length $v\sin \theta$.
• The parallel (along) piece has length $v\cos \theta$.

When $\theta =0^{\circ }$ (velocity along the field), $\sin 0^{\circ }=0$ — no sideways part — so no force. When $\theta =90^{\circ }$ (velocity fully across), $\sin 90^{\circ }=1$ — all sideways — so maximum force. That is precisely the experimental pattern the parent note reads off, and $\sin \theta$ is the tool that captures "how much of $\vec{v}$ is across $\vec{B}$."

6. The cross product — "an arrow perpendicular to two arrows"

The experiments say the force is perpendicular to $\vec{v}$ and to $\vec{B}$ at the same time, with size proportional to $\sin \theta$. There is exactly one arrow-operation that does both. It is the Cross product.

The figure shows two arrows in a flat plane and the cross-product arrow standing straight up out of that plane — the only way to be at a right angle to both at once.

Where does the component recipe come from? Start with the three unit vectors. Apply the two rules above (perpendicular to both, length $\sin \theta$) to the axis arrows themselves. Since $\hat{i},\hat{j},\hat{k}$ are each length $1$ and at $90^{\circ }$ to each other, every cross product of two different ones has length $1\cdot 1\cdot \sin 90^{\circ }=1$ and points along the third axis; the right-hand rule (our right-handed set) fixes the sign: $\hat{i}\times \hat{j}=\hat{k},\hat{j}\times \hat{k}=\hat{i},\hat{k}\times \hat{i}=\hat{j}$ (going forward in the cycle $x\to y\to z\to x$ gives $+$; going backward gives $-$, e.g. $\hat{j}\times \hat{i}=-\hat{k}$). Also any arrow crossed with itself is $\vec{0}$ (angle $0$, $\sin 0=0$): $\hat{i}\times \hat{i}=\vec{0}$.

Now just multiply out $\vec{v}\times \vec{B}$ term by term, treating $\times$ like distribution and using those nine little identities: $\vec{v}\times \vec{B}=(v_x\hat{i}+v_y\hat{j}+v_z\hat{k})\times (B_x\hat{i}+B_y\hat{j}+B_z\hat{k})$ Every "same-axis" term dies ($\hat{i}\times \hat{i}=\vec{0}$), and the surviving cross-terms collapse via the cycle rules into:

This is just the unit-vector algebra above, written out — the determinant/box pattern is only a memory aid for the same result.

Why the topic needs this: this single operation packages Observations 2, 3 and 4 of the parent note — perpendicularity and the $\sin \theta$ magnitude — into one clean symbol.

7. Charge $q$ and its sign

Why the topic needs this: $q$ scales the force (double $q$, double force) and its sign flips the direction. A negative $q$ sends the force the opposite way to the right-hand-rule thumb — this is why an electron curves opposite to a proton. In the magnitude formula we use $|q|$ (absolute value) because a length is never negative.

8. The magnetic field $\vec{B}$

Why the topic needs this: $\vec{B}$ is one of the two arrows fed into the cross product. Without a direction for the field, "sideways to the field" would be meaningless.

9. Force $\vec{F}$, and why perpendicular force turns things

Why the topic needs this: the dot product is the tool behind "no work, no speed change" — the single most important personality trait of magnetic force.

10. How these feed the topic

Read it top-down: arrows and their sizes are the raw material; the angle gives you $\sin \theta$; those combine into the cross product; charge and field feed it; the whole thing becomes the law, which then explains both "no work" and circular motion.

Equipment checklist

Self-test: can you answer each before moving on?

Connections

• ↑ Parent topic
• Cross product
• Lorentz force law
• Centripetal force and circular motion