# Centre of Mass - Centre of Charge?

## Main Question or Discussion Point

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:

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

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Doug Huffman
Gold Member
What does Coulomb's Law do?

What does Coulomb's Law do?
It gives the magnitude and direction of the force between two point charges.

Doug Huffman
Gold Member
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.

The law of superposition allows Coulomb's law to be extended to include any number of point charges. The force acting on a point charge due to a system of point charges is simply the vector addition of the individual forces acting alone on that point charge due to each one of the charges. The resulting force vector is parallel to the electric field vector at that point, with that point charge removed. https://en.wikipedia.org/wiki/Coulomb's_law
I am a sustaining contributor to The Wikimedia Foundation, I hop that you also will.

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.
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".

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.

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.
Then perhaps the "centre of mass" is more common than the "centre of charge" since mass appears in both Newton's Second Law and Newton's Law of Gravitation, right?

It is fair to say the the practical implications of "center of mass" are far more exploited than that of the "center of charge". However, both are quite useful in their contexts; they each make problems generally easier and, for some, even possible.