Electric Dipole Moment - Expression (general Case)

Expression (general Case)

More generally, for a continuous distribution of charge confined to a volume V, the corresponding expression for the dipole moment is:

where r locates the point of observation and d3r₀ denotes an elementary volume in V. For an array of point charges, the charge density becomes a sum of Dirac delta functions:

where each r_i is a vector from some reference point to the charge q_i. Substitution into the above integration formula provides:

This expression is equivalent to the previous expression in the case of charge neutrality and N = 2. For two opposite charges, denoting the location of the positive charge of the pair as r₊ and the location of the negative charge as r₋ :

showing that the dipole moment vector is directed from the negative charge to the positive charge because the position vector of a point is directed outward from the origin to that point.

The dipole moment is most easily understood when the system has an overall neutral charge; for example, a pair of opposite charges, or a neutral conductor in a uniform electric field. For a system of charges with no net charge, visualized as an array of paired opposite charges, the relation for electric dipole moment is:

$\begin{align} \mathbf{p}(\mathbf{r}) & = \sum_{i=1}^{N} \, \int\limits_V q_i \, (\mathbf{r}_0-\mathbf{r}) \ d^3 \mathbf{r}_0 \\ & = \sum_{i=1}^{N} \, q_i\, \\ & = \sum_{i=1}^{N} q_i\mathbf{d}_i = \sum_{i=1}^{N} \mathbf{p}_i \ , \end{align}$

which is the vector sum of the individual dipole moments of the neutral charge pairs. (Because of overall charge neutrality, the dipole moment is independent of the observer's position r.) Thus, the value of p is independent of the choice of reference point, provided the overall charge of the system is zero.

When discussing the dipole moment of a non-neutral system, such as the dipole moment of the proton, a dependence on the choice of reference point arises. In such cases it is conventional to choose the reference point to be the center of mass of the system, not some arbitrary origin. It might seem that the center of charge is more reasonable reference point than the center of mass, but it is clear that this results in a zero dipole moment. This convention ensures that the dipole moment is an intrinsic property of the system.

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