Physical Chemistry Third Edition

1004 24 Magnetic Resonance Spectroscopy

(two electric monopoles), but it is not yet certain whether magnetic monopoles can

exist separately.^1 If they do exist, they are not commonly observed.

A magnetic dipole can be produced by an electric current flowing in a closed circular

loop, as shown in Figure 24.1a. The magnetic dipole is denoted byμand is a vector

with magnitude given by the product of the current,I, and the area of the loop,A:

|μ|μIA (24.1-5)

Do not confuse the symbolμfor the magnitude of a magnetic dipole with the same

letter used for the permeability. The magnetic dipole is perpendicular to the loop and

its direction is given by a right-hand rule: If the curled fingers of the right hand point

in the direction of the current around the loop, the thumb points in the direction of the

magnetic dipole.

Area

Q

CurrentI

A current in a ring

(a)

An orbiting charge, Q

(b)





Figure 24.1 A Magnetic Dipole.
(a) From a current. (b) From a
moving charge. An orbiting charge
is equivalent to a current, and
produces a magnetic dipole just
like any other current.

EXAMPLE24.3

A current of 1.00 A is flowing counterclockwise in a horizontal circular loop 0.0500 m in

radius. Find the magnitude and direction of the magnetic moment.

Solution

μ(1.00 A)π(0.0500 m)^2  0 .0025 A m^2

By the right-hand rule the direction is upward.

Exercise 24.3

A current of 145 mA (milliampere) is flowing in a circular loop 1.45 cm in radius. Find the

magnitude of the magnetic moment.

The potential energy of a magnetic dipole in a magnetic field is given by

Emag−μ•B−|μ||B|cos(α)−μBcos(α) (24.1-6)

where the dot (•) stands for the scalar product (dot product) of the two vectors. The

scalar produce is discussed in Appendix B, and is defined by the second equality in

Eq. (24.1-6), whereαis the angle between the dipole and the field. The units of a

magnetic dipole can be given as A m^2 or as J T−^1 ( joules per tesla). The energy is at a

minimum if the dipole and the field are parallel, and is at a maximum if the dipole and

the field are antiparallel. It vanishes if they are perpendicular to each other. A compass

needle approximates a magnetic dipole and tends to orient itself in the direction of the

magnetic field.

EXAMPLE24.4

a.If the dipole of Example 24.3 points vertically upward and is located 0.200 m from the

wire of Example 24.1, find the energy of the dipole.

(^1) J. E. Dodd,The Ideas of Particle Physics, Cambridge University Press, New York, 1984, p. 169ff;
J. S. Song,J. Undergrad. Sci., 3 , 47 (1996).