How to Find the Average Dipole Force for Unpolarized Dipoles?

[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!

 

[gabbagabbahey]

To start with, your expression for $\textbf{F}$ looks very wrong to me...does [tex](\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?

 

[realcomfy]

Oops, I did make one mistake. The last term in the equation should be $$\left( p \cdot \hat d \right) p$$ 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.

 

[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?

 

[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.