Centre of positive charge

Question

I know its not possible to find a centre of charge for neutral bodies as net charge on them =0, which will make the quantity undefined. But my textbook says,

"centre of collection of positive charges is defined much the same way as centre of mass."

Is this correct?

So if if we say there is a ring with half ring as negatively charged and the other half positively charged, can we assume 2 centres of charges (1 each for positive and negative charge distribution) for calculation of electric field at a point, say the centre of the ring?

Note: I don't have any idea about quadrupole and such concepts, and I'd really appreciate if the argument could be kept at the basic level.

Why isn't there a centre of charge?

This Link also discusses this aspect, but i don't understand most of the answers.

Answer 1

If you can take a decomposition of the charge density as $$ \rho(\mathbf r) = \rho_+(\mathbf r) - \rho_-(\mathbf r), $$ where $\rho_\pm(\mathbf r)>0$ are strictly positive densities of positive and negative electric charge, then yes, the centres of positive and negative charge, $$ \mathbf r_\pm = \frac{1}{Q_\pm} \int \mathbf r \: \rho_\pm(\mathbf r)\mathrm d\mathbf r \quad \text{for} \quad Q_\pm = \int \rho_\pm(\mathbf r)\mathrm d\mathbf r $$ can readily be defined, and they can be used e.g. to get the distribution's electric dipole moment as $$ \mathbf d = Q_+\mathbf r_+ - Q_- \mathbf r_-, $$ and this electric dipole moment can be used to calculate the total electric field far away from the distribution. On the other hand, these centers of charge cannot be used to say anything meaningful about the electric field closer to the distribution (like the inside of your ring) in general cases. If you want to say something useful about the field inside the distribution, then you'll need to get it via numerical integration, or apply some additional approximations.

However, the decomposition of the total charge density $\rho(\mathbf r)$ into a positive and a negative component is never unique, and it is never well-defined unless you make an additional reference to a definite physical model of the underlying charge layouts. Given that additional physical model, then everything works, but without it, it can't.