This question is about the equation describing the net dipole moment for an arbitrary number of point charges.
Here's what I understand:
For two point charges $q$ and $-q$ with positions $\mathbf{r_1}$ and $\mathbf{r_2}$, the dipole moment is
$$\mathbf{p} = q\mathbf{r_1} +- q\mathbf{r_2} = q\mathbf{d}$$
where we define $\mathbf{d}$ to point from the negative charge to the positive charge.
It would seem to me, incorrectly, that to find the net dipole moment of a set of $N$ point charges you just add up the individual dipole moments between every pair of them. I'll call this $\mathbf{p_{wrong}}$ for obvious reasons.
$$\mathbf{p_{wrong}} = \sum_{j=1}^{N-1}\left(\sum_{k=j+1}^{N} q_j\mathbf{r_j} + q_k\mathbf{r_k} \right) $$
which,for 3 charges would be
$(q_1\mathbf{r_1} + q_2\mathbf{r_2}) + (q_1\mathbf{r_1} + q_3\mathbf{r_3}) + (q_2\mathbf{r_2} + q_3\mathbf{r_3})$
My textbook and all sources I have consulted list the below as the correct answer with very little explanation.
$$\mathbf{p_{right}} = \sum_i^N q_i\mathbf{d_i} $$
I can't for the life of me see how this is right. In this context, I don't even understand what $\mathbf{d_i}$ means. Is my expression somewhow equivalent, is $\mathbf{d_i}$ just the absolute position vector of each point charge, or am I completely out in the weeds here?
Best Answer
Yes, ${\bf d}_i$ simply means the absolute position vector for the $i$th point charge. It turns out that if the total charge is zero (i.e. $\sum_i q_i = 0$), then the value of this vector quantity does not depend on your choice of origin. However, this fact is not necessarily immediately obvious to a beginning student, so your textbooks should have explained why it is true.
It's better to memorize and think about that the general formula ${\bf p} := \sum_i^N q_i {\bf d}_i$, and to consider the case of two opposite point charges as a special case, than to start from the special case and try to generalize to an arbitrary number of charges.