# Electric dipole moment dependency of frame of reference and meaning

The electric dipole moment for a charge distribution is: $$\vec P= \int_V \rho(\vec x) \vec xd^3x$$ If we imagine that $$\rho= const$$ on a spherical volume then it's easy to see that $$\vec P$$ depends by the origin of the frame of reference, it can be zero or something totally different. For this reason it's very difficult for me to give a physical meaning at this quantity (because it's strange that depends by the origin).

• it does not depend on the origin. When you change your coordinates, you need to recompute your bounds of integration and volume element (which in this case remains the same) also. Commented Oct 23, 2019 at 7:41
• @Umaxo That's incorrect. The dipole moment does depend on the origin - it's only translation-invariant in systems with zero total charge. Commented Oct 23, 2019 at 7:47
• oh, of course. My bad. Commented Oct 23, 2019 at 8:14

With the multipole expansion the electrostatic field is expressed in its spherical coordinate components $$E_r$$, $$E_{\phi}$$ and $$E_{\theta}$$. In the example of a single point charge. If you place it at the origin, then the multipole expansion becomes very easy, since only the monopole term survives. You get $$\vec{E}=\frac{q\vec{r}}{4\pi\epsilon_{0}|\vec{r}|^3},$$ or simply put, the components of the field are $$E_r=\frac{q}{4\pi\epsilon_{0}|\vec{r}|^2}$$, $$E_{\phi}=0$$ and $$E_{\theta}=0$$. If you on the other hand place your point charge/origin somewhere else, we are get the familiar $$\vec{E}=\frac{q(\vec{r}-\vec{r}_0)}{4\pi\epsilon_{0}|\vec{r}-\vec{r}_0|^3},$$ which is much more complicated to express in terms of spherical components, but the multipole expansion is what helps you find a series expression for those field components.