Question

The centre of mass is defined as $\overrightarrow{R}=\frac{1}{M}\sum _i{m}_i\overrightarrow{r_i}$. Suppose we define "centre of charge" as ${\overrightarrow{R}}_c=\frac{1}{Q}\sum _i{q}_i\overrightarrow{r_i}$ where q_i represents the ith charge placed at ${\overrightarrow{r}}_i$ and Q is the total charge of the system.
(a) Can the centre of charge of a two-charge system be outside the line segment joining the charges?
(b) If all the charges of a system are in X-Y plane, is it necessary that the centre of charge be in X-Y plane?
(c) If all the charges of a system lie in a cube, is it necessary that the centre of charge be in the cube?

Answer 1

(a) Yes

Consider a charge distributed in X-Y plane.

$X_{cm}=\frac{-6q\times 0+q\times 5a}{-6q+q}=-a$

(b) Yes. Because the z-coordinates of all the charges are zero, the centre of charge lies in X-Y plane.

(c) No, it is not necessary that the centre of charge lies in the cube because charge can be either negative or positive.