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Why does the dipole moment of an electric dipole of different charges depend on the distance from the origin? Physically, I don't understand why something that measures how much of a dipole something is should depend on how far away that thing is from the origin.

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Interesting point. Seems analogous to angular momentum; this also depends upon the choice of reference point, but intuitively (to me) seems like it ought to be more intrinsic. –  BMS Apr 4 '14 at 4:53

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The electric dipole moment is defined as

$$p = \int r \; dq$$

In the case of a pair of charges for which both charges are of the same magnitude, the choice of the origin turns out to be irrelevant:

$$ p = \mathbf{r_1} q - \mathbf{r_2} q = q(\mathbf{r_1} - \mathbf{r_2}) = q\mathbf{d}$$

where $\mathbf{d}$ is the distance between the charges. However, when the two charges are not of equal magnitudes, $q$ cannot be factored out of the expression, so the choice of origin is of significance. In general, it is the case (as you noted) that dipole moment depends on a chosen point of reference.

Physically speaking, the dipole moment - like moment of inertia - is a property of a system that's dependent on the reference point you chose. This is intuitively true if you consider the following:

$$\mathbf{\tau} = \mathbf{p} \times \mathbf{E}$$

To calculate torque requires a specified axis of rotation. For the case of a pair of charges with equal magnitude, the torque is independent of the chosen axis of rotation. However, this isn't the case for non-neutral dipole, so for $\mathbf{\tau} = \mathbf{p} \times \mathbf{E}$ to hold, $\textbf{p}$ must also be dependent on the point of reference chosen.

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Wikipedia says that if the two charges have different magnitudes, the choice of origin does matter, en.wikipedia.org/wiki/Electric_dipole_moment. –  user24082 Apr 4 '14 at 4:38
Oh, oops. I assumed both charges were of equal magnitude. I'll edit my answer. –  Shivam Sarodia Apr 4 '14 at 4:51

If the total charge of the system is zero, the dipole moment does not depend on distance.

much same like: if total momentum of a system is zero, the angular momentum does not depend on the origin of reference.

Dipole moment is the intrinsic property of a system (subtract total charge to zero first); Angular momentum is the intrinsic property of a system (the reference is the system mass center);

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