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Chapter 4: Q4.7P (page 172)
Show that the energy of an ideal dipole p in an electric field E isgiven by
Thepotential energy for the dipole moment is .
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In a linear dielectric, the polarization is proportional to the field:
.If the material consists of atoms (or nonpolar molecules), the induced
dipole moment of each one is likewise proportional to the field . Question:
What is the relation between the atomic polarizabilityand the susceptibility ? Since P (the dipole moment per unit volume) is P (the dipole moment per atom)times N (the number of atoms per unit volume),, one's first inclination is to say that
And in fact this is not far off, if the density is low. But closer inspection reveals
a subtle problem, for the field E in Eq. 4.30 is the total macroscopicfield in the
medium, whereas the field in Eq. 4.1 is due to everything except the particular atom under consideration (polarizability was defined for an isolated atom subject to a specified external field); call this field · Imagine that the space allotted to each atom is a sphere of radius R ,and show that
Use this to conclude that
Or
Equation 4.72 is known as the Clausius-Mossottiformula, or, in its application to
optics, the Lorentz-Lorenzequation.
In Fig. 4.6,andare (perfect) dipoles a distance rapart. What is
the torque ondue to? What is the torque ondue to? [In each case, I want the torque on the dipole about its own center.If it bothers you that the answers are not equal and opposite, see Prob. 4.29.]
Two long coaxial cylindrical metal tubes (inner radius a,outer radiusb)stand vertically in a tank of dielectric oil (susceptibility ,mass density ).The inner one is maintained at potential V,and the outer one is grounded (Fig. 4.32). To what height (h) does the oil rise, in the space between the tubes?
Calculate W,using both Eq. 4.55 and Eq. 4.58, for a sphere of radius
Rwith frozen-in uniform polarization (Ex. 4.2). Comment on the discrepancy.
Which (if either) is the "true" energy of the system?
The Clausius-Mossotti equation (Prob. 4.41) tells you how to calculatethe susceptibility of a nonpolar substance, in terms of the atomic polariz-ability. The Langevin equation tells you how to calculate the susceptibility of apolar substance, in terms of the permanent molecular dipole moment p. Here's howit goes:
(a) The energy of a dipole in an external field E is
(Eq. 4.6), where is the usual polar angle, if we orient the z axis along E.
Statistical mechanics says that for a material in equilibrium at absolute temperature
T, the probability of a given molecule having energy u is proportional to
the Boltzmann factor,
The average energy of the dipoles is therefore
where , and the integration is over all orientations Use this to show that the polarization of a substance
containing N molecules per unit volume is
(4.73)
That's the Langevin formula. Sketch as a function of .
(b) Notice that for large fields/low temperatures, virtually all the molecules arelined up, and the material is nonlinear. Ordinarily, however, kT is much greaterthan p E. Show that in this regime the material is linear, and calculate its susceptibility,in terms of N, p, T, and k. Compute the susceptibility of water at 20°C,and compare the experimental value in Table 4.2. (The dipole moment of wateris ) This is rather far off, because we have again neglected thedistinction between E and Eelse· The agreement is better in low-density gases,for which the difference between E and Eelse is negligible. Try it for water vapor
at 100°C and 1 atm.
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