29.5.2 The Magnetic Dipole Moment

The expression for the electric field of an electric dipole was considerably simplified when we considered the field at distances significantly larger than the size of the charge separation s. Let the axis of the dipole be the z-axis. At a point where $z ≫ s$ , we found that the on-axis field of an electric dipole is

\vec{E}_{dipole} = \frac{1}{4 π ϵ_{0}} \frac{2 \vec{p}}{z^{3}}

where the electric dipole moment $\vec{p} =$ (qs, from negative to positive charge).

The on-axis magnetic field of a current loop is

B_{loop} = \frac{μ_{0}}{2} \frac{I R^{2}}{{(z^{2} + R^{2})}^{3 / 2}}

If z is much larger than the diameter of the current loop, $z ≫ R$ we can make the approximation ${(z^{2} + R^{2})}^{3 / 2} \to z^{3}$ . Then the loop’s field is

B_{loop} \approx \frac{μ_{0}}{2} \frac{I R^{2}}{z^{3}} = \frac{μ_{0}}{4 π} \frac{2 (π R^{2}) I}{z^{3}} = \frac{μ_{0}}{4 π} \frac{2 A I}{z^{3}}

(29.9)

where $A = π R^{2}$ is the area of the loop.

A photograph of an M R I image of a human head revealing its soft tissues. — *Magnetic resonance imaging*, or MRI, images soft tissues that are difficult to see with x rays. Protons, the nuclei of hydrogen atoms, have an inherent magnetic dipole moment that is called the proton’s *spin*. The laws of quantum physics dictate that a proton’s spin must be either aligned or antialigned with an external magnetic field. If hydrogen atoms in a magnetic field are irradiated with radio waves of just the right frequency, typically around 50 MHz, some protons undergo a spin flip in which their alignment changes. Clever circuits determine the hydrogen density, a density that depends on the type of tissue, by detecting this *resonance*. Software then creates an image that displays the variation of the hydrogen density.

A more advanced treatment of current loops shows that, if z is much larger than the size of the loop, Equation 29.9 is the on-axis magnetic field of a current loop of any shape, not just a circular loop. The shape of the loop affects the nearby field, but the distant field depends only on the current I and the area A enclosed within the loop. With this in mind, let’s define the magnetic dipole moment $\vec{m}$ of a current loop enclosing area A to be

\vec{m} = (A I, from the south pole to the north pole)

The SI units of the magnetic dipole moment are $A m^{2}$ .

The magnetic dipole moment, like the electric dipole moment, is a vector. It has the same direction as the on-axis magnetic field. Thus the right-hand rule for determining the direction of $\vec{B}$ also shows the direction of $\vec{m}$ . FIGURE 29.21 shows the magnetic dipole moment of a circular current loop.

The figure illustrates the magnetic dipole moment of a circular current loop.

FIGURE 29.21 Full Alternative Text

Because the on-axis magnetic field of a current loop points in the same direction as $\vec{m}$ , we can combine Equation 29.9 and the definition of $\vec{m}$ to write the on-axis field of a magnetic dipole as

\vec{B}_{dipole} = \frac{μ_{0}}{4 π} \frac{2 \vec{m}}{z^{3}} (on the axis of a magnetic dipole)

(29.10)

If you compare $\vec{B}_{dipole}$ to $\vec{E}_{dipole}$ , you can see that the magnetic field of a magnetic dipole is very similar to the electric field of an electric dipole.

A permanent magnet also has a magnetic dipole moment, and its on-axis magnetic field is given by Equation 29.10 when z is much larger than the size of the magnet. Equation 29.10 and laboratory measurements of the on-axis magnetic field can be used to determine a permanent magnet’s dipole moment. This magnetic dipole moment cannot be interpreted as the product of a current and an area; it’s simply a property of the permanent magnet.

EXAMPLE 29.7 Measuring current in a superconducting loop

You’ll learn in Chapter 30 that a current can be induced in a closed loop of wire. If the loop happens to be made of a superconducting material, with zero resistance, the induced current will—in principle—persist forever. The current cannot be measured with an ammeter because any real ammeter has resistance that will quickly stop the current. Instead, physicists measure the persistent current in a superconducting loop by measuring its magnetic field. What is the current in a 3.0-mm-diameter superconducting loop if the axial magnetic field is $9.0 μ T$ at a distance of 2.5 cm?

MODEL The measurements are made far enough from the loop in comparison to its radius ( $z ≫ R$ ) that we can model the loop as a magnetic dipole rather than using the exact expression for the on-axis field of a current loop.

SOLVE The axial magnetic field strength of a dipole is

B = \frac{μ_{0}}{4 π} \frac{2 m}{z^{3}} = \frac{μ_{0}}{4 π} \frac{2 π R^{2} I}{z^{3}} = \frac{μ_{0} R^{2} I}{2} \frac{1}{z^{3}}

where we used $m = A I = π R^{2} I$ for the magnetic dipole moment of a circular loop of radius $R$ . Thus the current is

\begin{matrix} I & = & \frac{2 z^{3} B}{μ_{0} R^{2}} = \frac{2 (0.025 {m)}^{3} (9.0 \times 10^{- 6} T)}{(1.26 \times 10^{- 6} T m / A) (0.0015 {m)}^{2}} \\ = & 99 A \end{matrix}

REVIEW This would be a very large current for ordinary wire. An important property of superconducting wires is their ability to carry current that would melt an ordinary wire.