Magnetic Lines for a Current Loop

The axial field of a circular loop (Section 10.13)

Diagram (Fig. 10.19): a circular loop of radius $R$ carrying a steady current $I$ lies in the x-y plane with its centre at the origin O. P is a point on the z axis at a distance $z$ from the centre. An element $d\vec{l}$ on the loop is joined to P by the vector $\vec{r}$; the elementary field $d\vec{B}$ at P is drawn perpendicular to the plane containing $d\vec{l}$ and $\vec{r}$, resolved into a component $dB_z$ along the axis and a component $dB_\perp$ perpendicular to it. The diametrically opposite element is also marked, with its $dB_\perp$ pointing the opposite way.

From the geometry, $r^2 = R^2 + z^2$. The element $d\vec{l}$ lies in the x-y plane and $\vec{r}$ lies in the y-z plane, so $d\vec{l}$ is always perpendicular to $\vec{r}$ and $\lvert d\vec{l}\times\vec{r} \rvert = dl\,r$. Biot-Savart law then gives

$$dB = \dfrac{\mu_0}{4\pi}\,\dfrac{I\,dl}{r^2} = \dfrac{\mu_0}{4\pi}\,\dfrac{I\,dl}{R^2+z^2}$$

The perpendicular components $dB_\perp$ from diametrically opposite elements are equal and opposite, so they cancel when summed round the loop. Only the axial components survive, with

$$dB_z = dB\cos\theta, \qquad \cos\theta = \dfrac{R}{r} = \dfrac{R}{\sqrt{R^2+z^2}}$$

Integrating round the loop, where $\displaystyle\oint dl = 2\pi R$,

$$B_z = \dfrac{\mu_0 I}{4\pi (R^2+z^2)}\cdot\dfrac{R}{\sqrt{R^2+z^2}}\oint dl = \dfrac{\mu_0 I R}{4\pi (R^2+z^2)^{3/2}}\,(2\pi R)$$

$$\therefore \ B_z = \dfrac{\mu_0 I R^2}{2\,(R^2+z^2)^{3/2}}$$

Special case $z = 0$, the field at the centre of the loop:

$$B_0 = \dfrac{\mu_0 I}{2R}$$

and for a coil of $N$ turns,

$$B_0 = \dfrac{\mu_0 N I}{2R}$$

Magnetic field lines of a current loop

Diagram (Fig. 10.20): a circular loop seen edge-on with the field lines drawn. The lines pass through the ring, are crowded and almost straight along the axis inside, and spread out widely and curve round to close on themselves outside. The pattern is that of a short bar magnet.

Right hand thumb rule for a loop (Fig. 10.21): curl the palm of the right hand along the circular wire with the fingers pointing in the direction of the current; the stretched right thumb then gives the direction of the magnetic field on the axis.

Consequently the face of the loop out of which the field lines emerge behaves as the north pole of a bar magnet, and the other face as the south pole.

The loop at large distances: a magnetic dipole

For $z \gg R$, the term $R^2$ in $(R^2+z^2)^{3/2}$ can be dropped:

$$B_z = \dfrac{\mu_0 I R^2}{2 z^3}$$

With the area of the loop $A = \pi R^2$, this becomes

$$B_z = \dfrac{\mu_0}{4\pi}\,\dfrac{2 I A}{z^3} \qquad (z \gg R)$$

The magnetic moment of a circular loop is defined as $\vec{m} = I\vec{A}$, where $\vec{A}$ has magnitude $A$ and direction perpendicular to the plane of the loop. Hence

$$B_z = \dfrac{\mu_0}{4\pi}\,\dfrac{2 m}{z^3}, \qquad \vec{B}_z = \dfrac{\mu_0}{4\pi}\,\dfrac{2\vec{m}}{z^3}$$

$\vec{B}_z$ and $\vec{m}$ point in the same direction, perpendicular to the plane of the loop. So at large distances a current loop behaves exactly like a magnetic dipole.

The electrostatic analogue

Compare the axial field of an electric dipole:

$$\vec{E} = \dfrac{1}{4\pi\varepsilon_0}\,\dfrac{2\vec{p}}{z^3}$$

The correspondence is $\dfrac{1}{4\pi\varepsilon_0} \leftrightarrow \dfrac{\mu_0}{4\pi}$, $\vec{p} \leftrightarrow \vec{m}$, $\vec{E} \leftrightarrow \vec{B}$. Using this analogue on the equatorial (perpendicular bisector) field of an electric dipole gives, for a magnetic dipole at a distance $x \gg R$ on its perpendicular bisector,

$$\vec{B} = \dfrac{\mu_0}{4\pi}\,\dfrac{\vec{m}}{x^3}$$

Fundamental difference between an electric dipole and a magnetic dipole: an electric dipole can be separated into two isolated charges, since free charges exist; a magnetic dipole cannot be separated into isolated poles, because free magnetic poles (magnetic monopoles) do not exist. Cutting a magnet in two simply produces two smaller dipoles.

Note also that $\mu_0$ appears in every result of this chapter because all of them are evaluated in free space (vacuum); $\mu_0$ is the permeability of free space.

Standard results

• $B_z = \dfrac{\mu_0 I R^2}{2(R^2+z^2)^{3/2}}$ on the axis of a loop of radius $R$
• $B_0 = \dfrac{\mu_0 I}{2R}$ at the centre; $B_0 = \dfrac{\mu_0 N I}{2R}$ for $N$ turns
• $B_z = \dfrac{\mu_0}{4\pi}\dfrac{2m}{z^3}$ for $z \gg R$, with $m = IA = I\pi R^2$
• $B = \dfrac{\mu_0}{4\pi}\dfrac{m}{x^3}$ on the perpendicular bisector for $x \gg R$
• Field lines resemble those of a short bar magnet; the emerging face is the north pole
• At $z = R$ the axial field is $\dfrac{1}{2\sqrt2}$ times the field at the centre.