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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 hydrogen atom (with the Bohr radius of half an angstrom) is situated
between two metal plates 1 mm apart, which are connected to opposite terminals of a 500 V battery. What fraction of the atomic radius does the separation distance d amount to, roughly? Estimate the voltage you would need with this apparatus to ionize the atom. [Use the value of in Table 4.1. Moral:The displacements we're talking about are minute,even on an atomic scale.]
A short cylinder, of radius a and length L, carries a "frozen-in" uniform polarization , parallel to its axis. Find the bound charge, and sketch the electric field (i) for , (ii) for , and (iii) for . [This is known as a bar electret; it is the electrical analog to a bar magnet. In practice, only very special materials-barium titanate is the most "familiar" example-will hold a permanent electric polarization. That's why you can't buy electrets at the toy store.]
The space between the plates of a parallel-plate capacitor is filled
with dielectric material whose dielectric constant varies linearly from 1 at the
bottom plate to 2 at the top plate .The capacitor is connectedto a battery of voltage V.Find all the bound charge, and check that the totalis zero.
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掳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路 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掳C and 1 atm.
A point charge qis situated a large distance rfrom a neutral atom of
polarizability .Find the force of attraction between them.
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