Modern Engineering Thermodynamics

4.7 Nonmechanical Work Modes of Energy Transport 121
Equation (4.59) is the differential of the total work associated with changing a material’s magnetic field. The first term
corresponds to the work required just to change the field itself (in a vacuum); and the second term corresponds to the
work associated with the alignment of the molecular magnetic dipoles of the material present inside the magnetic
field and represents the work of magnetization of the material exposed to the magnetic field. Hence, we can write
ðdWÞ material
magnetization
A simple and useful equation of state for a magnetic field is
= − μ 0 HdðV MÞ (4.60)
M = χ m H (4.61)
where χ m is the magnetic susceptibility (a dimensionless number) of the material. The magnetic susceptibility is
negative for diamagnetic materials and positive for paramagnetic materials (see Table 4.4). For a constant
volume magnetization process, Eq. (4.61) can be used in Eq. (4.59) to give
ðdWÞ magnetic
ðtotalÞ
= − μ 0 Vð1 + χ m ÞHdH
and assuming a constant volume and a constant magnetic susceptibility, this can be integrated to give the total
magnetic work:
Total magnetic work
ð1W 2 Þ magnetic
ðtotalÞ
= − μ 0 Vð1 + χ m Þ H2 2 − H2 1
2
(4.62)
where the increment to the total work due to the actual magnetization of the exposed material is just the actual
magnetic work:
Actual magnetic work
H2 2 ð1W 2 Þ material
= −μ 0 Vχ −
H2 1
m
2
magnetization
(4.63)
Table 4.5 summarizes the electrical and magnetic symbols used in this section.
Table 4.4 The Magnetic Susceptibility of Various Materials
Material Temperature (°C/°F) χ m (dimensionless)
Mercury 18/26 −3.2 × 10 −5
Quartz 25/77 −1.65 × 10 −5
Ice 0/32 −0.805 × 10 −5
Nitrogen (14.7 psia) 20/68 −0.0005 × 10 −5
Oxygen (14.7 psia) 20/68 0.177 × 10 −5
Aluminum 18/64 2.21 × 10 −5
Platinum 18/64 29.7 × 10 −5
Source: Reprinted by permission of the publisher from Zemansky, M. W., Abbott, M. M., Van Ness, H. C., 1975. Basic Engineering
Thermodynamics, second ed. McGraw-Hill, New York.
Table 4.5 Summary of Electrical and Magnetic Terms
Symbol Name SI Units
E Electric field strength V/m
P Polarization A · s/m 2
ε 0 Permittivity of free space 8.85419 × 10 −12 N/V 2
χ e Electric susceptibility Dimensionless
H Magnetic field strength A/m
B Magnetic induction Tesla or V ·s/m 2
M Magnetization A/m
μ 0 Magnetic permeability 4π × 10 −7 V·s/A·m