# Mutual energy of two dipoles.

1. May 26, 2007

### MathematicalPhysicist

i need to show that the mutual energy between two dioples p1 and p2 (not necessarily parallel to eachother) is $$U=-\frac{p_1\cdot p_2}{|r|^3}-3\frac{(p_1\cdot r)(p_2\cdot r)}{|r|^5}$$
where r is the vector from p_1 to p_2. (the p's are moments of diople).

i tried using this equation: $$U=\int dV \rho_2 * \phi_1$$
and also this :$$\phi=\frac{p\cdot r}{|r|^3}$$
(phi is the potential and rho is the density).
$$\rho_2=-1/4\pi\nabla^2\phi_2=-1/4\pi[\frac{1}{|r|^2}@/@r(r^2@\phi/@r)+\frac{1}{|r|^2*sin(\theta)}@/@\theta(sin(\theta)@\phi/@\theta)]$$
where @ stands for peratial derivative, but i didnt get to the desired answer.
any pointers?

2. May 26, 2007

### Pythagorean

$$U=-{m\cdot B}$$

where m is the vector for one of the dipoles in field B from the other dipole.

3. May 26, 2007

### Pythagorean

you'll have to dig the equation for the field (B) of dipole, too, I reckon.

4. May 26, 2007

### MathematicalPhysicist

by B, you mean the magnetic field, well i havent learned it yet (i mean we havent touched it in class as of yet, i myself read it form purcell), i pretty much sure i don't need here to use B, perhaps something else?

5. May 26, 2007

### Pythagorean

So these are electric dipoles then? I guess p usually denotes electric dipole and m is for magnetic dipole.

6. May 26, 2007

### Pythagorean

anyway, same equation, just:

U = -p.E

and dig up the E field for an electric dipole

7. May 26, 2007

### MathematicalPhysicist

well for E i found already, shouldn't i prove that it equals p.E? or in other words how to derive it?

8. May 26, 2007

### Pythagorean

If you want to find a derivation for U = -p.E, you might also remember that potential Energy is

the integral of F.dl

and the Force from a dipole is