4.2 The Biot–Savart Law and the Field of a Circular Loop

Inside an MRI scanner the patient lies in a magnetic field of about $1.5$ T, tens of thousands of times stronger than the Earth's — and so uniform that it varies by only a few parts per million across the body. No permanent magnet could do that. The field is made by a current flowing round and round in coils of superconducting wire, and engineers design those coils by adding up, piece by piece, the field that every short length of wire produces. The same calculation tells you how strong the field is inside the coil of a relay, at the centre of a loudspeaker's voice coil, or in the magnetic stimulation coils that neuroscientists hold against the scalp.

In Section 4.1 we took the magnetic field as given and studied the force it exerts. Now we ask where it comes from. Ørsted's compass showed that currents make magnetic fields; within weeks of hearing about it, Jean-Baptiste Biot and Félix Savart in Paris had measured how the field of a current depends on distance, and the rule they found — the Biot–Savart law — does for currents what Coulomb's law does for charges. We state the law, compare it with Coulomb's law, and use it to find the field on the axis of a circular current loop and at the centre of a circular arc.

The Biot–Savart Law

A current flows in a continuous wire, so there is no "point current" to play the role of a point charge. Instead we imagine the wire chopped into very short straight pieces, work out the field of each piece, and add.

Experiments (and, ultimately, the theory of electromagnetism) show that the field of a current element is proportional to $I\,dl$, falls off as the inverse square of the distance, and is greatest at right angles to the element and zero along its line.

The direction is easy to picture with the right-hand grip: grip the element with your right thumb along the current, and your fingers curl round it in the direction of $d\mathbf B$. The field of a current element circles round it — it never points towards or away from the wire.

Comparison with Coulomb's law. The two laws are close cousins. Both fields fall off as $1/r^2$, both obey superposition, and both have a constant that fixes the size of the unit ($\tfrac{1}{4\pi\varepsilon_0}$ and $\tfrac{\mu_0}{4\pi}$). But there are three important differences:

1. The source of the electric field is a scalar, the charge $q$; the source of the magnetic field is a vector, the current element $I\,d\mathbf l$.

2. The electric field of a point charge points along $\hat{\mathbf r}$ (radially); the magnetic field of an element is perpendicular to $\hat{\mathbf r}$ and to $d\mathbf l$.

3. The magnetic field depends on the angle: it is zero at every point on the line of the element ($\theta=0$ or $180^\circ$) and greatest at right angles to it. The electric field of a point charge is the same in all directions.

The two constants are linked to light: $\dfrac{1}{\sqrt{\mu_0\varepsilon_0}}$ $=\dfrac{1}{\sqrt{(4\pi\times10^{-7})(8.85\times10^{-12})}}$ $\approx3.0\times10^8$ m/s, the speed of light in vacuum. We meet this again in Chapter 8.

The Field on the Axis of a Circular Loop

Bend a wire into a circle of radius $R$ and pass a steady current $I$ round it. We want the field at a point $P$ on the axis of the loop — the line through its centre perpendicular to its plane — at distance $x$ from the centre $O$.

Take an element $d\mathbf l$ at the top of the loop. The vector $\mathbf r$ from the element to $P$ lies in the plane that contains the axis and that radius, while $d\mathbf l$ is tangent to the circle — perpendicular to that plane. So $\theta=90^\circ$ for every element, and every element is at the same distance $r=\sqrt{x^2+R^2}$ from $P$: $$dB=\frac{\mu_0}{4\pi}\,\frac{I\,dl}{x^2+R^2}.$$ $d\mathbf B$ is perpendicular to $\mathbf r$, so it is tilted away from the axis. Split it into a component along the axis, $dB\cos\alpha$, and a component perpendicular to it, $dB\sin\alpha$, where $\alpha$ is the angle between $d\mathbf B$ and the axis, with $$\cos\alpha=\frac{R}{\sqrt{x^2+R^2}}.$$ Now use symmetry. The element diametrically opposite produces a $d\mathbf B$ of the same size whose perpendicular component points the opposite way, so the perpendicular components cancel in pairs. The axial components all point the same way and add. Adding the axial components (uses integration, but only $\oint dl=2\pi R$, the circumference), $$B=\frac{\mu_0 I}{4\pi(x^2+R^2)}\cdot\frac{R}{\sqrt{x^2+R^2}}\cdot2\pi R=\frac{\mu_0IR^2}{2\left(x^2+R^2\right)^{3/2}}.$$

The field is strongest at the centre and falls off on both sides. Far away it decreases as $1/x^3$ — exactly like the field on the axis of an electric dipole (Section 1.2). Indeed, writing $m=NI\pi R^2$ (current times area), the far field is $B\approx\dfrac{\mu_0}{4\pi}\dfrac{2m}{x^3}$, the same form as $E=\dfrac{1}{4\pi\varepsilon_0}\dfrac{2p}{x^3}$. A current loop is a magnetic dipole; we develop this in Section 4.4.

The Field at the Centre of a Circular Arc

Many exam circuits are made of straight pieces and circular arcs, and ask for the field at the centre of the arcs. Two observations settle almost all of them.

Arcs. For a point at the centre of an arc of radius $R$, every element of the arc is at distance $R$ and perpendicular to the radius ($\theta=90^\circ$), and every $d\mathbf B$ points the same way — perpendicular to the plane of the arc. So the contributions simply add: $B=\dfrac{\mu_0}{4\pi}\dfrac{I}{R^2}\times(\text{length of arc})$. An arc subtending an angle $\phi$ (in radians) at the centre has length $R\phi$.

Straight pieces pointing at the centre. If a straight wire lies along a line through the point (a radial lead), then $d\mathbf l$ is parallel or antiparallel to $\mathbf r$ for every element, $d\mathbf l\times\hat{\mathbf r}=\mathbf 0$, and that wire contributes nothing at the point.

Exercises