Electric dipole moment

The electric dipole moment p\mathbf{p} is a measure of how separated positive and negative electric charges are within a system. A dipole moment is defined for any pair of charges within a system, that is, for each electric dipole, and the total moment of the system is the sum of each dipole moment. For a system of point charges, it is defined as

p=∑i=1nqiri′[C m]\mathbf{p}=\sum_{i=1}^{n} q_{i}\mathbf{r}_{i}'\qquad[\text{C m}]

where qiq_{i} is ii-th charge and ri′\mathbf{r}'_{i} is the position vector of that charge. For a physical dipole where charges are ±q\pm q it is

p=qd\mathbf{p}=q\mathbf{d}

where d\mathbf{d} is the vector going from the negative charge to the positive charge.

Origin dependency

Notice the important fact that the dependency is on the position vector, not explicitly the distance between any two charges. Of course, it can become the distance, as seen in the physical dipole, but the only strict necessity to have a dipole is for the charge to not be at the origin point. In fact, a sole point charge qq (a monopole) can have a dipole moment too

p=qr′\mathbf{p}=q\mathbf{r}'

where r′\mathbf{r}' is its position with respect to the origin. This means that the choice of origin can drastically alter the dipole moment of a system. If the total charge is zero, however, dipole moment is origin-independent.

Suppose that we displace the origin by an amount a\mathbf{a}, the new dipole moment is

p~=∫r~′ρ(r′) dτ′=∫(r′−a)ρ(r′) dτ′=∫r′ρ(r′) dτ′−a∫ρ(r′) dτ′=p−Qa\tilde{\mathbf{p}}=\int \tilde{\mathbf{r}}'\rho(\mathbf{r}')\ d\tau'=\int(\mathbf{r'}-\mathbf{a})\rho(\mathbf{r}')\ d\tau'=\int \mathbf{r}'\rho(\mathbf{r'})\ d\tau'-\mathbf{a}\int \rho(\mathbf{r}')\ d\tau'=\mathbf{p}-Q\mathbf{a}

If Q=0Q=0, then the moment does not change.