## Dipole interaction

1. The problem statement, all variables and given/known data

See attachment

2. Relevant equations

3. The attempt at a solution

Thus far I believe I am supposed to use calculate the interaction with kQq/Δr? I have tried summing over interactions so that 1/4∏ε*[(-Q)(-q)/(r-d/2+D/2) + (-Q)(q)/(r+d/2+D/2) + (Q)(-q)/(r-d/2-D/2) + (Q)(q)/(r+d/2-D/2)] , but this doesn't work, could someone point me in the right direction.
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 Mentor With taylor series for those factors of 1/(r+d/2) and similar (and assuming d/2 << r), you should get the correct result.

 Quote by mfb With taylor series for those factors of 1/(r+d/2) and similar (and assuming d/2 << r), you should get the correct result.
Ok, thanks, is kQq/Δr the correct expression for the interaction energy between 2 charges, and more importantly is that what i'm supposed to be using? Because if so I don't see how I get any 1/(r-d/2) terms, I just get stuff like 1/(r-d/2+D/2) unless I neglect the D/2 because r >> D?

Mentor

## Dipole interaction

This is correct.

And right, you do not get "exactly" 1/(r+d/2), but you get similar expressions.

 Ok thanks for the help, I can't seem to figure out what point i'm supposed to be taylor expanding about? For instance for the interaction between -Q and -q, 1/4∏ε*[(-Q)(-q)/(r-d/2+D/2)] take r = 1/(r-d/2+D/2) and that r >> D & d, d/2 vanish & D/2 vanish? Then I am just left with 1/r, bit confused, even if i try to expand that without making the approximation mentioned I still am not quite sure about what point I am expanding about.
 Mentor The difference between 1/r and your terms is the point of the expansion. $$\frac{1}{r+x} = \frac{1}{r} \cdot \frac{1}{1+\frac{x}{r}} \approx \frac{1}{r} (1 \pm \dots)$$

 Quote by mfb The difference between 1/r and your terms is the point of the expansion. $$\frac{1}{r+x} = \frac{1}{r} \cdot \frac{1}{1+\frac{x}{r}} \approx \frac{1}{r} (1 \pm \dots)$$
Yeh I got that bit but I mean if f(x) = 1/r * 1/(1 + x/r) and this expands to 1/r [f(a) + f'(a)(x-a) . . .] what is "a" supposed to be?

 Mentor That is the point where you develop your taylor expansion. It depends on the definition of the f you choose, but I would expect 1 there.

 Quote by mfb That is the point where you develop your taylor expansion. It depends on the definition of the f you choose, but I would expect 1 there.
Sorry I think I'm confused because I'm not entirely sure what the point is i'm developing the taylor expansion from, because if r = (r-d/2+D/2) that's over a length not a point :S? so kQq/(r-d/2+D/2) where i take f(x) = 1/r*(1/(1-x/r)) where x = d/2-D/2, still can't see what the point i'm expanding about is? Is it a=d/2 or a=D/2 or even a=r? so f(x) ≈ 1/r*[1/(1-a/r) + 1/r*(1/(1-a/r)2)*(x-a)] but so confused about the a :(.

 Recognitions: Gold Member Homework Help Science Advisor Staff Emeritus You're expanding about x/r = 0. To be a bit more explicit, the function you're expanding is ##f(z) = \frac{1}{1-z}## about ##z=0## and then plugging in ##z=x/r##.

 Quote by vela You're expanding about x/r = 0.
Yey, thanks very much to both of you for the help.

 Quote by vela You're expanding about x/r = 0. To be a bit more explicit, the function you're expanding is ##f(z) = \frac{1}{1-z}## about ##z=0## and then plugging in ##z=x/r##.
Actually I just tried that and my expansion ≈ 1/r + (d/2 + D/2)/r^2, then i tried for the remainder terms and the final result came to (2rQq - 2rQq)/r^2 = 0, i imagine my expansion came out wrong, is it wrong?

 Recognitions: Gold Member Homework Help Science Advisor Staff Emeritus Nope, that's fine. If you look at the result you're trying to arrive at, you should notice it has r3 on the bottom, so you want the 1/r and 1/r2 terms to cancel out.

 Quote by vela Nope, that's fine. If you look at the result you're trying to arrive at, you should notice it has r3 on the bottom, so you want the 1/r and 1/r2 terms to cancel out.
Ok brilliant, thanks a bunch for the help, I'll take it from here just got to get the manipulation right :D.