Permanent Dipole - Permanent Dipole Interaction Derivation

[RickD]

Homework Statement

I am trying to derive the dipole-dipole interaction derivation, which is:

U=(-p1p2/4πϵ_0) (1/z^3) ((2cosθ_1cosθ_2)− (sinθ_1sinθ_2cosζ))

Where p1 and p21 are the two dipole moments, r is the distance between two dipoles on the y axis, θ_1 and θ_2 are the angles between the z axis and dipoles, and ζ is the dihedral angle

I am using U = -p2 * E_1 to derive the equation, where E_1 is the electric field of p1 and * is the dot product

So far, I have deduced E-1 in spherical coordinates as:
E_1= (1/4πϵ_0)(1/r^3) (2cosθ r + sinθ θ)
Where the bolded r and θ are the unit vectors in spherical coordinates.

Now from this, I have problems:

1. How do I formulate p2 in spherical coordinates
2. How do I dot product two vectors of spherical coordinates

Homework Equations

See above

The Attempt at a Solution

I have tried to convert E_1 into Cartesian, but it won't solve it if I don't know how to formulate p2

 

[TSny]

Homework Statement

I am trying to derive the dipole-dipole interaction derivation, which is:

U=(-p1p2/4πϵ_0) (1/z^3) ((2cosθ_1cosθ_2)− (sinθ_1sinθ_2cosζ))

Where p1 and p21 are the two dipole moments, r is the distance between two dipoles on the y axis, θ_1 and θ_2 are the angles between the z axis and dipoles, and ζ is the dihedral angle.

Did you mean to say that θ₁ and θ₂ are angles measured relative to the y axis?

You can rotate your xyz axes about the y-axis so that p₁ lies in the yz plane.

So far, I have deduced E-1 in spherical coordinates as:
E_1= (1/4πϵ_0)(1/r^3) (2cosθ r + sinθ θ)
Where the bolded r and θ are the unit vectors in spherical coordinates.

Can you draw the vectors r and θ at the location of p₂ in the above diagram?

Then try to find expressions for the r and θ components of p₂.