# Average dipole force

1. Nov 16, 2009

### realcomfy

I just have a quick question about finding the average force of a dipole.

I am given the expression (after I derived it anyway):

$$\textbf{F} = -3 \left( \frac{e-1}{e+2} \right) \frac{R^{3}}{d^{7}} \left[4( p \bullet \hat d)^{2} \hat d + p^{2} \hat d - (p \bullet \hat d) \hat d \right]$$

where p is a vector whose direction is not specified. I am asked to average this force over all directions of p to give the average force for unpolarized dipoles. I am pretty sure this has something to do with integrating over the solid angle, but I am not sure how to treat the dipole terms in the force equation. Any help would be greatly appreciated!

2. Nov 16, 2009

To start with, your expression for $\textbf{F}$ looks very wrong to me....does $$(\textbf{p}\cdot\hat{\mathbf{d}})\hat{\mathbf{d}}[/itex] have the same units as [tex]p^2\hat{\mathbf{d}}[/itex]? Does $\textbf{F}$ really have units of force? What was the original problem? 3. Nov 16, 2009 ### realcomfy Oops, I did make one mistake. The last term in the equation should be [tex] \left( p \cdot \hat d \right) p$$ Other than that everything should be correct.

e is the dielectric constant in gaussian units. p is the dipole moment. $$\hat d$$ is a unit vector in the d-direction.

4. Nov 16, 2009

### gabbagabbahey

Okay, at least the units make sense now....but still, what was the original question?

The way you've stated the problem doesn't make much sense...are you computing the average force of a single dipole on some material? The average force of a collection of dipoles on some material?The average force of some material on a single dipole? The average force of some material on a collection of dipoles? Something else entirely?

5. Nov 17, 2009

### realcomfy

The idea is to compute the average force on the dipole from a dielectric sphere placed in a uniform electric field, also oriented in the d-direction. I think I got it figured out though. The idea is to define an angle theta with respect to the d-direction, and then integrate over the solid angle. The part I was missing before is that to get the average you then have to divide by the magnitude of the solid angle: 4 Pi.