Difficulties with dipoles & point charges

[Lisa...]

Difficulties with dipoles & point charges...

Help would be appreciated a lot with the following problems:

~ A positive point charge +Q is at the origin, and a dipole of moment p is a distance r away (r>>L) and in the radial direction as shown below:

http://img462.imageshack.us/img462/1482/dipole6fp.th.gif

a) Show that the force exerted on the dipole by the point charge is attractive and has a magnitude ~ 2kQp/r³ (see previous problem).

[the previous problem was: an electric dipole consists of two charges +q and -q separated by a very small distance 2a. Its center is on the x-axis at x=x₁ and it points along the x-axis in the positive x direction. The dipole is in a nonuniform electric field, which is also in the x direction, given by E=Cxi where C is a constant.

- Find the force on the positive charge and that on the negative charge and show that the net force on the dipole is Cpi
- Show that, in general if a dipole of moment p lies along the x-axis in an electric field in the x direction the net force on the dipole is given approximately by (dE_x/dx)pi].


I know the force on the negative side of the dipole is -kQq/r² and the force on the positve side of the dipole is kQq/(r+L)², but this leads to nothing... I can derive an expression for the electric field of the dipole as k2p/r³ and conclude the force of the dipole on the point charge= the force on the dipole of the point charge (Newtons 3rd law)= QE, therefore F= Qk2p/r³, but that would ruin the second question below (because I need to use the formula derived in this question in order to prove E= k2p/r³ for a dipole. So how should I tackle this problem?

b) Now assume that the dipole is centered at the origin and that a point charge Q is a distance r away along the line of the dipole. Using Newton's third law and your result for part (a), show that at the location of the positive point charge the electric field E due to the dipole is toward the dipole and has a magnitude of ~k2p/r³.

I think I can handle this one if I get part (a) correct

~ Two neutral polar molecules attract each other. Suppose that each molecule has a dipole moment p and that these dipoles are aligned along the x-axis and separated by a distance d. Derive an expression for the force of attraction in terms of p and d.

I do need some help with this one. First of all, if a line like this - is a symbol for the dipole, are the two aligned one after another (like - -) or parallel to each other (like =)? Secondly, I don't really know how to treat a system of dipoles if the distance isn't big (in the previous question it was) so I would really appreciate a couple of hints to give me something to start off with... I totally don't have a clue...

PS Sorry for the big text!

 

[siddharth]

I guess you need to use a binomial approximation.

If you take the distance to the center of the dipole as 'r', then the distances to each charge will be r - L/2 and r+L/2, so that your field will be

\frac{1}{4\pi\epsilon_0} [\frac{-q}{(r-L/2)^2} + \frac{q}{(r+L/2)^2}]

Now take out an r from the denominator so that you get
\frac{1}{r^2} \frac{-q}{(1-\frac{L}{2r})^2)}
since L/R is <<1, you can expand the deonimator using the binomial exapnsion and ignore higher order terms. Do this for the second term also and this should give you the answer.

 

[Lisa...]

Thank you very very much! :D I managed to solve the problem! :)
Yet I'm still puzzeling on this one:

~ Two neutral polar molecules attract each other. Suppose that each molecule has a dipole moment p and that these dipoles are aligned along the x-axis and separated by a distance d. Derive an expression for the force of attraction in terms of p and d.

I do need some help with this one. First of all, if a line like this - is a symbol for the dipole, are the two aligned one after another (like - -) or parallel to each other (like =)? Secondly, I don't really know how to treat a system of dipoles if the distance isn't big (in the previous question it was) so I would really appreciate a couple of hints to give me something to start off with... I totally don't have a clue...

 

[siddharth]

For the second question, I would still assume that d>>L (In fact, this assumption is valid in most situations, because the inter-molecular distances are very small).
I know the field due to the dipole along the axis. From that, I can calculate the force on each charge of the second dipole to find the force of attraction.
Do the Binomial trick again to get it in terms of p and d.