Dipole-Induced Dipole Interaction

[djeikyb]

Homework Statement

Given a permanent dipole, call it p_NO, at the origin pointing along r₁ (general vector, not along any particular axis). A polarizable atom as at position r₂. What is the induced dipole moment, call it p_He, of this atom in terms of p_NO and r₂?

Homework Equations

p_NO=a₁E_par+a₂E_perp
a1 is the coefficient of polarizability in the direction parallel to the electric field, a2 is the perpendicular one.
The second part of this expression should be zero, as the induced dipole should point in the direction of the electric field.

The Attempt at a Solution

I know/understand that whatever this induced dipole is, it will point in the direction of the electric field due to the permanent dipole. Really, it's not the answer I'm after- rather, I need help with the method. Would I just say that for the permanent dipole, there is essentially a negative charge at the origin and a positive charge at p₁ and superimpose the electric fields? It seems like this should work, but it's messy/tedious/not giving me the right answer, so I am probably making a wrong assumption with this method.
I suppose the real problem I have is not solving for the dipole, but rather the electric field due to a general permanent dipole. I would vastly prefer a point in the direction of deriving it rather than someone just giving a formula.

 

[djeikyb]

Addendum- If I can just find the electric field, it would be simple to solve for the projection of it along r₂.

 

[kuruman]

Do you know what the dipole potential looks like? If so, take its negative gradient and you have the dipole electric field.

 

[djeikyb]

Also, if this affects anybody's willingness to help, this is not a graded homework assignment. It's an assignment in that I am working under an AMO professor, and this is one of the many calculations he wants me to do- not that this is going to help him in any way, but I need to have the ability/understanding of this to understand one of the terms we will be adding to our Hamiltonian.

 

[djeikyb]

Do you know what the dipole potential looks like? If so, take its negative gradient and you have the dipole electric field.

That... was a lot easier than I was making it out to be. Kudos to you, sir. Shame on me.

 

[djeikyb]

Out of curiosity, should the more complicated still yield the correct answer?

 

[kuruman]

It should, but at some point you will have to do a Taylor expansion for d <<r where d is the separation between the positive and negative charges and keep the leading term. Finding the approximate field along the direction of the dipole (the easiest case) is a standard problem in introductory physics.