Physical Chemistry , 1st ed.

magnetic field and a magnetic dipole. When a magnetic dipole is subjected

to a magnetic field B, there is a potential energy of interaction. The magnetic

potential energy,Emag, is given by

EmagBBcos (16.3)

where the between and Bspecifies the dot product (or scalar product) of

the two vectors, which is given by the second relationship in equation 16.3.

The potential energy of interaction is a scalar, not a vector. And because of the

cos term in equation 16.3, the potential of interaction is at a minimum when

the magnetic vectors are parallel, the potential is equal to zero when the magnetic

vectors are perpendicular, and the potential is at a maximum when the vectors are

directly opposed to each other (or antiparallel).

Example 16.3

Show that the units for magnetic field and magnetic dipole multiply together

to yield units of energy. Use the fact that 1 T 1 kg/(s^2 amp).

Solution

The cos term has no units; it is simply a number. The magnitudes ofand

Bhave the units ampm^2 and T, respectively, so the unit of the interaction be-

tween them is, according to equation 16.3,

amp m^2 T 

am

s

p

2





m

am

2

p

kg



kg

s



2

m^2

J

J is the SI unit of energy. This illustrates that the interaction between mag-

netic dipoles and fields imposes an energy change on the system.

We have been assuming that electrical charges at the atomic (actually, sub-

atomic) scale behave like macroscopic electrical currents. Actually, this is not

the case, although the examples above illustrate that individual electrons can

be treated like electrical currents. There is another presumption we have been

using. We have been assuming a classical,continuousmagnetic field. Because

electromagnetism combines electricity and magnetism, and electricity is quan-

tized (as electrons) and electromagnetic radiation is also quantized (as pho-

tons), we might expect that a quantum theory of magnetism would also be ap-

propriate. Technically, this is indeed the case. However, most of the ideas we

present in this chapter deal with the magnetic field as a classical phenomenon,

not a quantum one. Only in the most advanced cases will a quantized mag-

netic field be considered.

Finally, we can relate the magnetic dipole to another quantity of quantum-

mechanical importance: the angular momentum. Consider a particle with a

charge ofqmoving around in a circle. It induces a magnetic dipole. If the par-

ticle has a linear velocity vin meters per second and is traveling in a circle hav-

ing radius rmeters, then the time necessary for one circular orbit is

time 

2 

v

r



Since current is defined as charge passing a point per second, the current Iat

any point in the particle’s orbit is

I

2

Q



v

r



16.2 Magnetic Fields, Magnetic Dipoles, and Electric Charges 563

Distance of separation

NSSN

Energy of interaction

(b)

Distance of separation

NSNS

Energy of interaction

(a)

Figure 16.5 Interactions can be (a) attractive
or (b) repulsive. Attractive interactions contribute
to a lowering of the overall energy, whereas re-
pulsive interactions contribute to an increase in
the overall energy. The interactions of the two bar
magnets shown mimic the interactions of mag-
netic fields and magnetic dipoles.