Dipole-Dipole Interaction

One strength of Intermediate Physics for Medicine and Biology is its many homework problems. The problems stress (but perhaps not enough) the ability to make general arguments about how some quantity will depend on a variable. Often getting a calculation exactly right is not as important as just knowing how something varies with something else. For instance, you could spend all day learning how to compute the volume and surface area of complicated objects, but it’s still useful simply to know that volume goes as size cubed and surface area as size squared. Below is a new homework problem that emphasizes the ability to determine a functional form.

Section 6.7

Problem 20½. Consider an electric dipole p a distance r from a small dielectric object. Calculate how the energy of interaction between the dipole and the induced dipole in the dielectric varies with r. Will the dipole be attracted to or repelled from the dielectric? Use the following facts:

1. The energy U of a dipole in an electric field E is U = – p · E,

2. The net dipole induced in a dielectric, p‘, is proportional to the electric field the dielectric experiences,

3. The electric potential produced by a dipole is given by Eq. 7.30.

Let’s take a closer look at these three facts.

1. When discussing magnetic resonance imaging in Chapter 18 of IPMB, we give the energy U of a magnetic dipole μ in a magnetic field B as U = – μ · B (Eq. 18.3). An analogous relationship holds for an electric dipole in an electric field. The energy is lowest when the dipole and the electric field are in the same direction, and varies as the cosine of the angle between them. I suggest treating the original dipole p as producing the electric field E, and the induced dipole p‘ as interacting it. 

2. Section 6.7 of IPMB discusses how an electric field polarizes a dielectric. The net dipole p‘ induced in the dielectric object will depend on the electric field and the objects shape and volume. I don’t want you to have to worry about the details, so the problem simply says that the net dipole is proportional to the electric field. You might get worried and say “wait, the electric field in the dielectric is not uniform!” That’s why I said the dielectric object is small. Assume that it’s small enough compared to the distance to the dipole that the electric field is approximately uniform over the volume of the dielectric. 

3. What is the electric field produced by a dipole? Russ Hobbie and I don’t actually calculate that, but we do give an equation for a dipole’s electrical potential, which falls off as one over the square of the distance. (It may look like the cube of the distance in Eq. 7.13, but there’s a factor of distance in the numerator that cancels one factor of distance cubed in the denominator, so it’s an inverse square falloff.) The electric field is the negative gradient of the potential. Calculating the electric field can be complicated in the general case. I suggest you assume the dipole p points toward the dielectric. Fortunately, the functional dependence of the energy on the distance r does not depend on the dipole direction.

I won’t work out all theentire solution here. When all is said and done, the energy falls off as 1/r⁶, and the dipole is attracted to the dielectric. It doesn’t matter if the dipole originally pointed toward the dielectric or away from it, the force is always attractive.

This result is significant for a couple reasons. First, van der Waals interactions are important in biology. Two dielectrics attract each other with an energy that falls as 1/r⁶. Why is there any interaction at all between two dielectrics? Because random thermal motion can create a fluctuating dipole in one dielectric, which then induces a dipole in a nearby dielectric, causing them to be attracted. These van der Waals forces play a role in how biomolecules interact, such as during protein folding.

From Photon to Neuron: Light, Imaging, Vision.

Second, there is a technique to determine the separation between two molecules called fluorescence resonance energy transfer (FRET). The fluorescence of two molecules, the donor and the acceptor, is affected by their dipole-dipole interaction. Because this energy falls off as the sixth power of the distance between them, FRET is very sensitive to distance. You can use this technique as a spectroscopic ruler. It’s not exactly the same as in the problem above, because both the donor and acceptor have permanent dipole moments, instead of one being a dielectric in which a dipole moment is induced. But nevertheless, the 1/r⁶ argument still holds, as long as the dipoles aren’t too close together. You can learn more about FRET in Philip Nelson’s book From Photon to Neuron: Light, Imaging, Vision.