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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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A very long cylinder of linear dielectric material is placed in an otherwise uniform electric field .Find the resulting field within the cylinder. (The radius is a , the susceptibility . and the axis is perpendicular to.)
When you polarize a neutral dielectric, the charge moves a bit, but the total remains zero. This fact should be reflected in the bound charges and · Prove from Eqs. 4.11 and 4.12 that the total bound charge vanishes.
A dipole is a distance from a point charge , and oriented so that makes an angle with the vector from to .
(a) What is the force on ?
(b) What is the force on ?
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.
An electric dipole , pointing in the ydirection, is placed midwaybetween two large conducting plates, as shown in Fig. 4.33. Each plate makes a small angle with respect to the xaxis, and they are maintained at potentials .What is the directionof the net force on?(There's nothing to calculate,here, butdo explain your answer qualitatively.)
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