Foundations — Magnetic field of straight wire, circular loop, solenoid, toroid

Before you can read a single formula in the parent note, every squiggle in it must mean something to you. This page builds each one from nothing.

0. What is a "current" and what is a "field"?

Why the topic needs it: magnetic fields are made by currents. No $I$, no $\vec{B}$. Every formula in the parent has $I$ sitting in the numerator — double the flow, double the field.

In figure s01 the blue circles are the field wrapping the wire. Notice: no beginning, no end — magnetic field lines close on themselves. That circling is the whole personality of magnetism.

1. Vectors and the little hat: $\vec{a}$, $\hat{r}$

Why the topic needs it: in Biot–Savart, $\hat{r}$ answers "in what direction does this little piece of wire aim its influence?" Separating direction ($\hat{r}$) from distance ($r$) is what lets the $1/r^2$ weakening and the swirl-direction be handled independently.

2. Distance $r$ and perpendicular distance

Why the topic needs it: magnetic influence fades with distance. The parent's straight-wire law says the field halves when you double $r$ — so $r$ must be crisply defined before that sentence makes sense.

3. Tiny pieces of wire: $d\vec{l}$ and its length $dl$

Why the topic needs it: Biot–Savart adds up the contribution of every $d\vec{l}$. Whenever the direction has already been dealt with, only the plain length $dl$ survives in the arithmetic — so you must know that $dl$ is nothing more mysterious than $|d\vec{l}|$.

4. The cross product $\times$ — the swirl machine

This is the symbol most people meet for the first time here, so we build it slowly.

Reading the size formula: the $\sin \theta$ is the key. When the two arrows are parallel ($\theta =0$, $\sin 0=0$) the cross product is zero — a wire-piece sends no field straight along its own direction. When they are perpendicular ($\theta =90^{\circ }$, $\sin 90^{\circ }=1$) the field is strongest. Look at the pink dot in s03: the result is largest exactly when the two black arrows make an L-shape.

5. The dot product $\cdot$ — the "how-much-lines-up" machine

Ampère's law is built on a different product, so we need it too.

Look at s04: the dot product keeps only the shadow of $\vec{b}$ cast onto $\vec{a}$ (the blue projection). When $\vec{b}$ leans fully along $\vec{a}$ the shadow is full-length; when $\vec{b}$ stands straight up, the shadow shrinks to nothing.

Why the topic needs it: in $\oint \vec{B}\cdot d\vec{l}$, on a well-chosen loop the field is parallel to the walk so $\cos \theta =1$ and $\vec{B}\cdot d\vec{l}=Bdl$. This single simplification is what turns Ampère's law into an easy $B(2\pi r)$. The perpendicular sides of the solenoid's rectangle contribute nothing precisely because there $\cos 90^{\circ }=0$.

6. The integral signs: $\int$ and $\oint$

Why the topic needs it: Biot–Savart is defined as an integral (add the contributions of every wire-piece). Ampère's law is defined as a closed-loop integral of the field. You cannot read either law without knowing these two symbols mean "add tiny pieces."

7. Angles $\theta$, $\alpha$, ${\theta }_1$, ${\theta }_2$ — keeping them straight

There are only two jobs an angle does in this topic; naming them once removes all confusion.

Why the topic needs it: every trig factor in the parent ($\sin {\theta }_1$, $\cos \alpha$, the $\sin \theta$ hidden in each cross product) is one of these. Read the subscript or letter and you instantly know which job the angle is doing.

8. The constants: ${\mu }_0$, $\pi$, and turn counts $N$, $n$

Why the topic needs it: these turn a bare single-wire result into a real coil's field. Forgetting to multiply by $N$ is the parent's flagged classic mistake.

Prerequisite map

Read this as a chain, top to bottom — each idea below rests on the one above it:

1. Current $I$ flows → it makes a magnetic field $\vec{B}$ that circles it.
2. Vectors and the unit vector $\hat{r}$ let us give every arrow a direction.
3. Chopping the wire gives tiny pieces $d\vec{l}$ (length $dl$).
4. The cross product ($\sin \theta$) makes the new perpendicular field direction → powers Biot-Savart Law.
5. The dot product ($\cos \theta$) measures how much lines up → powers Ampere's Law.
6. Integrals $\int ,\oint$ add all the pieces up; constants ${\mu }_0,\pi$ and turn counts $N,n$ supply the numbers.
7. Out come the fields of the straight wire, loop, solenoid, and toroid of the parent topic.

Where these tools reappear

Once you have these symbols, the same vocabulary powers neighbouring topics: the cross product also gives the force on a current, the loop's field is really that of a magnetic dipole, and adding up $\vec{B}$ over an area gives Magnetic flux, the ingredient of Faraday's law of induction.

Equipment checklist

Test yourself — reveal only after answering out loud.