# Centre of Mass - Centre of Charge?

1. Dec 28, 2014

We define the position of the centre of mass of a system of $n$ masses as:

$$\vec{r_{cm}} = \frac{\sum_{i=1}^n m_i \vec{r_i}}{\sum_{i=1}^n m_i}$$

Why is there no such thing as "centre of charge", defined for $n$ point charges:

$$\vec{r_{cq}} = \frac{\sum_{i=1}^n q_i \vec{r_i}}{\sum_{i=1}^n q_i}$$

2. Dec 28, 2014

### Doug Huffman

What does Coulomb's Law do?

3. Dec 28, 2014

It gives the magnitude and direction of the force between two point charges.

4. Dec 28, 2014

### Doug Huffman

Hmm, I always thought it analogous to Newton's Law of Universal Gravitation.

All y'all's struggle with fancy notation/typography is moot without some small understanding of how things work, the theoretical minimum.

I am a sustaining contributor to The Wikimedia Foundation, I hop that you also will.

5. Dec 29, 2014

I am familiar with superposition. What I meant to ask was whether the notion of an average position weighted by charge is of any use in physics, since I never came across the term "centre of charge".

6. Dec 29, 2014

### Joshua L

Of course there is, even though the term "center of charge" is uncommonly used.

Coincidentally, I recently browsed through my new electromagnetism textbook and read something very interesting:

The net force on a charged particle outside of a charged hollow sphere with an equal charge distribution is

$\vec F = \frac{kQq}{\|r\| ^2} \hat r$ $\\\ \text{if}\ \ r>R$

Q is the net charge of the sphere,
q is the charge of the particle located outside of the sphere,
k is Coulomb's constant,
||r|| is the distance between the particle and the center of the sphere, and
R is the radius of the sphere.

One can see that as long as the particle is outside the sphere, the force from the sphere applied on the particle will act as though the force came from a particle, with net charge Q, located at the center of the sphere. This realization implies the practicality of the idea of "center of charge" without specifically stating such a phrase.

7. Dec 29, 2014