The Timpson Lyceum · University Physics II (2026) – Electricity & Magnetism

Sources of Magnetic Field – Lecture 10 (University Physics II)

Overview

The previous chapter asked how moving charges respond to a magnetic field. This lecture asks the counterpart question: how moving charges produce one. Starting from the field of a single moving point charge, we build up the Biot–Savart law for a current element, then integrate it to get the field of a straight wire, of a circular loop, and of a tightly stacked coil. Along the way the force between two parallel current-carrying wires gives the official definition of the ampere. The lecture closes with Ampère's law, which plays the same labor-saving role for magnetic fields that Gauss's law plays for electric ones — and which is the result you will reuse most going forward.

Key Concepts

Field of a moving point charge. A charge $q$ moving with velocity $\vec v$ produces, at a point a distance $r$ away in the direction of the unit vector $\hat r$ , the field

\begin{aligned} \vec B &= \frac{\mu_0}{4\pi}\,\frac{q\,\vec v \times \hat r}{r^2} \end{aligned}

The direction comes from $\vec v \times \hat r$ — use the right hand rule. The field wraps in circles around the line of motion.
The magnitude carries a factor $\sin\theta$ , where $\theta$ is the angle between $\vec v$ and $\hat r$ . So the field is strongest perpendicular to the motion and exactly zero straight ahead of and straight behind the charge.
It falls off as $1/r^2$ , like gravity and like the electric field.

The constant $\mu_0$ . By construction $\mu_0/4\pi$ is exactly $10^{-7}\ \text{T}\cdot\text{m/A}$ . This is not a coincidence of measurement — it follows from how the ampere is defined (below). The Coulomb constant can be written $k = 10^{-7}c^2$ , which gives the useful relation

\begin{aligned} \epsilon_0 \mu_0 &= \frac{1}{c^2} \end{aligned}

Magnetic vs. electric force. For two like charges moving past one another at speed $v$ , the ratio of the magnetic to the electric force between them is $v^2/c^2$ . Everyday speeds — the drift velocity in a wire is millimeters per second — make this ratio vanishingly small. Strong magnetic fields come from piling up enormous numbers of moving charges (large currents), not from fast ones.

Biot–Savart law. Treat a short length $d\vec l$ of wire carrying current $I$ as a point charge's worth of moving charge. Its contribution to the field is

\begin{aligned} d\vec B &= \frac{\mu_0}{4\pi}\,\frac{I\,d\vec l \times \hat r}{r^2}, \qquad dB = \frac{\mu_0}{4\pi}\,\frac{I\,dl\,\sin\theta}{r^2} \end{aligned}

Integrating over the whole conductor gives the total field. As with the point charge, an element contributes nothing directly along its own direction of current.

Straight wire. For a wire of length $2a$ , at a perpendicular distance $x$ from its midpoint, the integral (via the substitution $y = x\tan\theta$ ) gives

\begin{aligned} B &= \frac{\mu_0 I}{2\pi x}\cdot\frac{a}{\sqrt{x^2+a^2}} \end{aligned}

When the wire is very long compared with the distance ( $a \gg x$ ), the last factor goes to $1$ :

\begin{aligned} B &= \frac{\mu_0 I}{2\pi r} \end{aligned}

The field of a long wire falls off linearly with distance, not as $1/r^2$ — the same behavior as the electric field of a long charged rod.

Two antiparallel wires. For equal currents in opposite directions separated by $2d$ , the fields reinforce between the wires and oppose each other outside them. Far away the net field falls off as $1/x^2$ rather than $1/x$ . This is exactly why signal and return conductors are run tightly together — the pair's external field dies off fast and interferes far less with its neighbors. Two currents in the same direction do the reverse: weaker between, stronger outside.

Force between parallel conductors. Each wire sits in the other's field, so each feels a force. Per unit length,

\begin{aligned} \frac{F}{L} &= \frac{\mu_0 I I'}{2\pi r} \end{aligned}

Currents in the same direction attract; currents in opposite directions repel.

Definition of the ampere. One ampere is the current that makes two long parallel conductors exactly $1\ \text{m}$ apart feel exactly $2 \times 10^{-7}\ \text{N/m}$ . Current, not charge, is the SI base unit here for a practical reason: forces can be measured, whereas individual charges cannot be counted. The coulomb is then defined from it — the charge one ampere carries in one second — and this choice is what fixes $\mu_0/4\pi$ at exactly $10^{-7}$ .

Circular loop. On the axis of a loop of radius $a$ , a distance $x$ from its center, the off-axis contributions cancel in pairs and only the axial components survive:

\begin{aligned} B_x &= \frac{\mu_0 I a^2}{2\left(a^2+x^2\right)^{3/2}} \end{aligned}

At the center ( $x=0$ ) this reduces to $B = \mu_0 I/2a$ , the strongest point on the axis.

Coils and solenoids. Stacking $N$ tightly packed loops multiplies the field by $N$ :

\begin{aligned} B_x &= \frac{\mu_0 N I a^2}{2\left(a^2+x^2\right)^{3/2}}, \qquad B_{\text{center}} = \frac{\mu_0 N I}{2a} \end{aligned}

This is how strong laboratory fields are made. The scaling is ideal only while the coil stays compact; stretch it out and field lines begin to leak out the sides instead of threading the whole length. In terms of the magnetic dipole moment $\mu = NIA$ , the axial field can be written in one expression covering both the single-loop and $N$ -loop cases.

Gauss's law for magnetism. The net magnetic flux through any closed surface is zero,

\begin{aligned} \oint \vec B \cdot d\vec A &= 0 \end{aligned}

because isolated magnetic poles do not exist — break a magnet and you get two smaller magnets. This statement is true but useless for computing $\vec B$ from a current, since it relates flux to enclosed poles rather than to moving charge.

Ampère's law. The useful counterpart to Gauss's law is a line integral around a closed path:

\begin{aligned} \oint \vec B \cdot d\vec l &= \mu_0 I_{\text{enc}} \end{aligned}

Walk any closed loop, dotting the field with each step. The result depends only on the current the path encloses — not on the path's shape, length, or distance from the wire. A path enclosing no current gives zero. Currents count with a sign according to which way they pierce the loop, so two equal antiparallel currents inside the path contribute nothing.

Why it holds in general: for a step $d\vec l$ subtending angle $d\theta$ at the wire, the component along the field is $dl\cos\phi = r\,d\theta$ . Substituting $B = \mu_0 I/2\pi r$ makes the $r$ cancel, leaving $\oint \mu_0 I\,d\theta/2\pi$ . Any path that encircles the wire sweeps $\theta$ through $2\pi$ , giving $\mu_0 I$ regardless of shape. A path that does not enclose the wire accumulates as much negative $\theta$ as positive on the way back, giving zero.

Worked Example

A solenoid on its axis. A tightly wound coil of $N = 100$ turns and radius $a = 0.60\ \text{m}$ carries $I = 5\ \text{A}$ . Find the field on its axis $x = 0.80\ \text{m}$ from the center.

Use the axial field of an $N$ -turn coil, then substitute:

\begin{aligned} B_x &= \frac{\mu_0 N I a^2}{2\left(a^2+x^2\right)^{3/2}} \\ &= \frac{\left(4\pi \times 10^{-7}\right)(100)(5)(0.60)^2}{2\left[(0.60)^2+(0.80)^2\right]^{3/2}} \\ &= \frac{\left(4\pi \times 10^{-7}\right)(100)(5)(0.36)}{2\left(0.36+0.64\right)^{3/2}} \\ &= \frac{\left(4\pi \times 10^{-7}\right)(180)}{2} \\ &= 1.13 \times 10^{-4}\ \text{T} \end{aligned}

The radius and distance were chosen so that $a^2 + x^2 = 1.00\ \text{m}^2$ exactly, which is why the denominator collapses to $2$ . The answer, about $1.1\ \text{gauss}$ , is a couple of times the Earth's field.

How close to a wire matches the Earth's field? A single wire carries $I = 1\ \text{A}$ . Find the distance at which its field equals $B = 0.5 \times 10^{-4}\ \text{T}$ (half a gauss, roughly the Earth's field). Solve the long-wire result for $r$ :

\begin{aligned} B &= \frac{\mu_0 I}{2\pi r} \quad\Longrightarrow\quad r = \frac{\mu_0 I}{2\pi B} \\ r &= \frac{\left(4\pi \times 10^{-7}\right)(1)}{2\pi\left(0.5 \times 10^{-4}\right)} \\ &= 4 \times 10^{-3}\ \text{m} = 4\ \text{mm} \end{aligned}

Only $4\ \text{mm}$ from a one-amp wire the field already rivals the Earth's — a useful reminder of how weak the Earth's field actually is.

Summary

A moving charge produces a field $\vec B = (\mu_0/4\pi)\,q\vec v \times \hat r / r^2$ : circling the line of motion, strongest perpendicular to it, zero straight ahead and straight behind.
The Biot–Savart law extends this to a current element, and integrating it gives the field of any conductor; $\epsilon_0\mu_0 = 1/c^2$ , and the magnetic force between two moving charges is smaller than the electric force by $v^2/c^2$ .
A long straight wire gives $B = \mu_0 I/2\pi r$ , falling off linearly with distance; a pair of antiparallel wires falls off as $1/x^2$ , which is why signal and return lines are run together.
Parallel currents attract and antiparallel ones repel, with $F/L = \mu_0 I I'/2\pi r$ — the relation that defines the ampere as $2 \times 10^{-7}\ \text{N/m}$ at one meter separation.
A loop gives $B = \mu_0 I/2a$ at its center, and $N$ stacked turns multiply this by $N$ , which is how strong fields are built in practice.
Ampère's law, $\oint \vec B \cdot d\vec l = \mu_0 I_{\text{enc}}$ , depends only on the enclosed current and not on the path — the magnetic counterpart of Gauss's law, and the tool that replaces these integrals going forward.