# Centre of Mass - Centre of Charge?

• PFuser1232
In summary, the centre of mass of a system of ##n## masses is defined as the weighted average of the positions of each mass, while the centre of charge for ##n## point charges is not commonly used but can be applied in certain situations. Coulomb's Law gives the magnitude and direction of the force between two point charges, and the law of superposition allows it to be extended to include any number of point charges. The concept of "center of charge" is not as commonly used as "center of mass", but it can be practically applied in certain scenarios.
PFuser1232
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:

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

What does Coulomb's Law do?

Doug Huffman said:
What does Coulomb's Law do?

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

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.

Doug Huffman said:
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.

Joshua L said:
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.

## What is the Centre of Mass?

The centre of mass refers to the point in a system where the mass of the entire system can be considered to be concentrated, and it experiences no net external force. It is the point around which an object will rotate under the influence of external forces.

## What is the Centre of Charge?

The centre of charge is the point in a system where the total charge of the system can be considered to be concentrated. It is the point at which an electric field exerts no net force on the system.

## How is the Centre of Mass and Centre of Charge calculated?

The centre of mass is calculated by taking the weighted average of the positions of all the particles in the system, where the weights are determined by the mass of each particle. The centre of charge is calculated in a similar way, but using the charges of each particle instead of their masses.

## Why is the Centre of Mass and Centre of Charge important?

The centre of mass and centre of charge are important concepts in physics as they help us understand the behaviour of systems under the influence of external forces. They also help us determine the stability and equilibrium of a system.

## Can the Centre of Mass and Centre of Charge be at different locations?

Yes, the centre of mass and centre of charge can be at different locations in a system. This is because the centre of mass is determined by the distribution of mass in a system, while the centre of charge is determined by the distribution of charge. In some systems, these two may coincide, but in others, they may be at different locations.

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