## Potential energy of 2 dipoles

For the discussion of the sciences. Physics problems, chemistry equations, biology weirdness, it all goes here.

Moderators: gmalivuk, Moderators General, Prelates

Atre
Posts: 25
Joined: Wed Aug 06, 2008 8:36 pm UTC

### Potential energy of 2 dipoles

Firstly, yes, this is a problem from my Uni degree that is giving me a bit of a brainfuck . However, answering the thing is in no way connected to my grades. I hope asking this question is not prize twatish viewtopic.php?f=18&t=4734

Problem: 2 dipoles of equal dipole moment set in a plane, they can only spin in this plane. Calculate the potential energy of one in the field of the other.

"two coplanar identical dipoles of moment "p" are supported on pivots a large distance "d" apart. Their angles of twist (theta and phi) are measured clockwise from the line joining their centres"

The result to get is [math](p^2/8(pi)(eo)d^3)(3cos(theta+phi) + cos(theta - phi))[/math] where eo is the permittivity of free space (personally I always contract 1/4(pi)(e0) to "k" but meh)

I assume I am getting boned by an unfamiliarity with playing with vector operators in spherical polars. My approach is to break the dipole moments into perpendicular components and then try the easier task of comparing dipoles parallel/perpendicular to each other but I fall down considering how translational and rotational alterations change my nice equations and re-introduce the eliminated terms...

If I could get a poke in the right direction for how to solve the problem this would be wonderful.

Thanks
Atre

ThinkerEmeritus
Posts: 416
Joined: Sat Jan 19, 2008 11:32 pm UTC

### Re: Potential energy of 2 dipoles

OK, I'll try to give you a poke in the right direction without working it for you. I haven't tried to work it for me either, so buyer beware.

The first step in working a physics problem is to understand as much as possible about what is going on. So, what is causing the potential energy? How does the one dipole push at the other? What is the starting formula for potential energy as applied to this problem?

What coordinate system are you going to use. You imply spherical, which is OK although not what I would choose. Our terminology may differ; for me spherical is a three-dimensional coordinate system. What direction is your polar axis?

Why does it matter that the two dipoles are far apart?

This probably isn't enough of a prod, but post your answers and I will know better which direction to push.
"An expert is a person who has already made all possible mistakes." -- paraphrase of a statement by Niels Bohr
Seen on a bumper sticker: "My other vehicle is a Krebs cycle".

Atre
Posts: 25
Joined: Wed Aug 06, 2008 8:36 pm UTC

### Re: Potential energy of 2 dipoles

Thankyou for helping.

Going through what you said

1) The potential energy comes from rotating the dipole (would apply in any field) but also translational because each dipole produces a non-uniform field [perhaps a part of the question is to approximate that the field is uniform at the dipole scale... only just thought of that - useful cos implies no potential wrt to translation]

2) The interaction is electrostatic (using time-invariant EM field atm)

3) I have no starting formulae for potential, I begin with the electric field produced by each dipole [math]E(r,theta) = pk/r^3(R, 2cos(theta) + THETA, sin(theta)[/math]
Where the bold r,theta are unit vectors... bollocks, maths tool doesn't like bold. I meant capital r.theta are unit vectors

and then work from there to derive a potential energy with respect to rotation integrating over changes in dipole angle (Assume that the dipole producing the field remains stationary whilst the dipole I am "working with" rotates) giving me
[math]potential = integral[(pxE)]d(phi)[/math]

assuming I am working with the dipole whose angle wrt the line joining the centres is phi.

4) Spherical was spherical limited to the plane in which I am operating, so I only utilise r and Theta. So polar co-ords and rather like an argand diagram with zero set at -pi

5) It matters that the dipoles are far apart to that I can actually treat them like dipoles and not 2 point charges. (Assume the distance between the charges is much smaller that the distance to the test charge)

My answers were to begin with the idealised components where either the R or THETA term in the Efield eqt went to zero, but I then got confused about how to re-introduce the terms and deal with the overall relative angle between the dipoles. A rather abstract intuitive solve got 1 term correct but had no second term, which I assumed meant a basic oops.

EDIT: Also, I used spherical co-ords (centred on the dipole of interest) because it gives me the simplest form of the dipole's EM field - Which system would work better? I haven't used the dipole eqts in Cartesian yet, I assume they would be most nasty?
I'd love to know another way of looking at the problem if it's there.

ThinkerEmeritus
Posts: 416
Joined: Sat Jan 19, 2008 11:32 pm UTC

### Re: Potential energy of 2 dipoles

Atre wrote:Thankyou for helping.

Going through what you said

1) The potential energy comes from rotating the dipole (would apply in any field) but also translational because each dipole produces a non-uniform field [perhaps a part of the question is to approximate that the field is uniform at the dipole scale... only just thought of that - useful cos implies no potential wrt to translation]

Good - that kind of insight comes from thinking qualitatively, which is why you should do it. But watch out a bit -- if you rotate the position of one dipole around the other, you are changing the relative orientation of the two dipoles, which does matter.

Atre wrote: 2) The interaction is electrostatic (using time-invariant EM field atm)

3) I have no starting formulae for potential, I begin with the electric field produced by each dipole [math]E(r,theta) = pk/r^3(R, 2cos(theta) + THETA, sin(theta)[/math]
Where the bold r,theta are unit vectors... bollocks, maths tool doesn't like bold. I meant capital r.theta are unit vectors

and then work from there to derive a potential energy with respect to rotation integrating over changes in dipole angle (Assume that the dipole producing the field remains stationary whilst the dipole I am "working with" rotates) giving me
[math]potential = integral[(pxE)]d(phi)[/math]

assuming I am working with the dipole whose angle wrt the line joining the centres is phi.

4) Spherical was spherical limited to the plane in which I am operating, so I only utilise r and Theta. So polar co-ords and rather like an argand diagram with zero set at -pi

5) It matters that the dipoles are far apart to that I can actually treat them like dipoles and not 2 point charges. (Assume the distance between the charges is much smaller that the distance to the test charge)

My answers were to begin with the idealised components where either the R or THETA term in the Efield eqt went to zero, but I then got confused about how to re-introduce the terms and deal with the overall relative angle between the dipoles. A rather abstract intuitive solve got 1 term correct but had no second term, which I assumed meant a basic oops.

EDIT: Also, I used spherical co-ords (centred on the dipole of interest) because it gives me the simplest form of the dipole's EM field - Which system would work better? I haven't used the dipole eqts in Cartesian yet, I assume they would be most nasty?
I'd love to know another way of looking at the problem if it's there.

Personally I use cylindrical coordinates, but that is a matter of taste as long as the spherical coordinates have their polar axis perpendicular to the plane you are working in. I wanted to check and make sure you hadn't put the polar axis in the plane, which would make the algebra more complicated. And I agree completely that Cartesian coordinates would be a disaster for this problem.

That said, if you start translating the second dipole, it is much more convenient to center the coordinates on the source dipole, and the form of the electric potential/field is more familiar with that choice. You will have to be careful about the way phi comes in if you center the coordinates on the non-source dipole.

Now, as I see it, where you are now is you have figured out the Electric field (and eliminated the magnetic field) as a function of where you are relative to the source dipole, and you have determined that only rotation, not translation, of the second dipole contributes to the potential energy if you are far enough away. So how much work is done in this field when you rotate the second dipole? It looks like you have calculated that and don't like the answer, so you might want to check that your coordinates are oriented so that your angular variables correspond with those in the problem. Since there are two components of the electric field, you expect to get two terms in the potential like your given answer has. The key now is ORGANIZATION, ORGANIZATION, ORGANIZATION. Any little goof is going to make a major mess with the complicated geometry of this problem.

[Remember I haven't worked the problem. I had the same reaction you did to the two terms, in particular disliking the cos(theta+phi) term. I think I see where it may come from now.]

Let us know how you do from here. Don't forget you have a free choice of overall constant, if necessary.
"An expert is a person who has already made all possible mistakes." -- paraphrase of a statement by Niels Bohr
Seen on a bumper sticker: "My other vehicle is a Krebs cycle".