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3.6: Induced Dipoles and Polarizability

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  • We noted in section 1.3 that a charged rod will attract an uncharged pith ball, and at that time we left this as a little unsolved mystery. What happens is that the rod induces a dipole moment in the uncharged pith ball, and the pith ball, which now has a dipole moment, is attracted in the inhomogeneous field surrounding the charged rod.

    How may a dipole moment be induced in an uncharged body? Well, if the uncharged body is metallic (as in the gold leaf electroscope), it is quite easy. In a metal, there are numerous free electrons, not attached to any particular atoms, and they are free to wander about inside the metal. If a metal is placed in an electric field, the free electrons are attracted to one end of the metal, leaving an excess of positive charge at the other end. Thus a dipole moment is induced.

    What about a nonmetal, which doesn’t have free electrons unattached to atoms? It may be that the individual molecules in the material have permanent dipole moments. In that case, the imposition of an external electric field will exert a torque on the molecules, and will cause all their dipole moments to line up in the same direction, and thus the bulk material will acquire a dipole moment. The water molecule, for example, has a permanent dipole moment, and these dipoles will align in an external field. This is why pure water has such a large dielectric constant.

    But what if the molecules do not have a permanent dipole moment, or what if they do, but they cannot easily rotate (as may well be the case in a solid material)? The bulk material can still become polarized, because a dipole moment is induced in the individual molecules, the electrons inside the molecule tending to be pushed towards one end of the molecule. Or a molecule such as \(\text{CH}_4\), which is symmetrical in the absence of an external electric field, may become distorted from its symmetrical shape when placed in an electric field, and thereby acquire a dipole moment.

    Thus, one way or another, the imposition of an electric field may induce a dipole moment in most materials, whether they are conductors of electricity or not, or whether or not their molecules have permanent dipole moments.

    If two molecules approach each other in a gas, the electrons in one molecule repel the electrons in the other, so that each molecule induces a dipole moment in the other. The two molecules then attract each other, because each dipolar molecule finds itself in the inhomogeneous electric field of the other. This is the origin of the van der Waals forces.

    Some bodies (I am thinking about individual molecules in particular, but this is not necessary) are more easily polarized that others by the imposition of an external field. The ratio of the induced dipole moment to the applied field is called the polarizability \(α\) of the molecule (or whatever body we have in mind). Thus

    \[\textbf{p}=\alpha \textbf{E}\label{3.6.1}\]

    The SI unit for \(α\) is C m \((\text{V m}^{−1} )^{ −1}\) and the dimensions are \(\text{M}^{−1} \text{T}^ 2\text{Q}^ 2\) .

    This brief account, and the general appearance of equation \ref{3.6.1}, suggests that \(\textbf{p} \text{ and }\textbf{E}\) are in the same direction – but this is so only if the electrical properties of the molecule are isotropic. Perhaps most molecules – and, especially, long organic molecules − have anisotropic polarizability. Thus a molecule may be easy to polarize with a field in the x-direction, and much less easy in the y- or z-directions. Thus, in equation \ref{3.6.1}, the polarizability is really a symmetric tensor, \(\textbf{p} \text{ and }\textbf{E}\) are not in general parallel, and the equation, written out in full, is

    \[\label{3.6.2}\begin{pmatrix}p_x \\ p_y \\ p_z \\ \end{pmatrix}=\begin{pmatrix}\alpha_{xx} & \alpha_{xy} & \alpha_{xz} \\ \alpha_{xy} & \alpha_{yy} & \alpha_{yz} \\ \alpha_{xz} & \alpha_{yz} & \alpha_{zz} \\ \end{pmatrix} \begin{pmatrix}E_x \\ E_y \\ E_z \\ \end{pmatrix}\]

    (Unlike in equation 3.5.2, the double subscripts are not intended to indicate second partial derivatives; rather they are just the components of the polarizability tensor.) As in several analogous situations in various branches of physics (see, for example, section 2.17 of Classical Mechanics and the inertia tensor) there are three mutually orthogonal directions (the eigenvectors of the polarizability tensor) for which \(\textbf{p} \text{ and }\textbf{E}\) will be parallel.