Draft:Center of charge

In physics, center of charge is the unique point within a system of charged particles where the weighted relative position of the spatially distributed electric charge sums to zero. This concept is analogous to the center of mass in mechanics, where the mass of a system may be assumed to be concentrated at a single point. Computations in electromagnetism, electrostatics, and particle physics are often simplified when formulated with respect to the center of charge behavior of electric fields and the interactions between charged particles.

Definition
In a system of discrete charged particles, each particle exerts an electric force on every other particle. The center of charge is the spatial point where the total electric force acting on the system can be assumed to originate. Mathematically, it is defined as the weighted average of the positions of the individual charges, where each position is weighted by its magnitude.

For a continuous charge distribution, the center of charge is computed with integrals, similar to the calculation of the center of mass for continuous mass distributions.

A system of charged particles
For a system of discrete charged particles $Q_{i}, i = 1, ..., n$, each with charge $q_{i}$ that are located in space with coordinates $r_{i}, i = 1, ..., n$, the coordinates R of the center of charge satisfies the condition $$ \sum_{i=1}^n q_i(\mathbf{r}_i - \mathbf{R}) = \mathbf{0}.$$

Solving this equation for R yields the formula $$\mathbf{R}={\sum_{i=1}^n q_i \mathbf{r}_i\over\sum_{i=1}^n q_i }.$$

A continuous volume
If the charge distribution is continuous with the volume charge density ρ(r) within a solid S, then the integral of the weighted position coordinates of the points in this volume relative to the center of charge R over the volume V is zero, that is $$\iiint_{S} \rho(\mathbf{r}) \left(\mathbf{r} - \mathbf{R}\right) dV = 0.$$

Solve this equation for the coordinates R to obtain $$\mathbf R = \frac 1 Q \iiint_{S}\rho(\mathbf{r}) \mathbf{r} \, dV,$$ where Q is the total charge in the volume.

If a continuous charge distribution has uniform charge density, which means that ρ is constant, then the center of charge is the same as the centroid of the volume.

Properties

 * 1) Conservation of charge: The total charge of an isolated system remains constant over time. Therefore, the center of charge of a closed system also remains constant.
 * 2) Influence on electric fields: The electric field produced by a system of charges behaves as if all the charges were concentrated at the center of charge. This simplifies the analysis of electric fields in complex systems.
 * 3) Relation to center of mass: In many cases, the center of charge coincides with the center of mass of the system, especially when the charges are distributed symmetrically.

Applications

 * 1) Electric fields: Understanding the center of charge helps in simplifying computations involving the interaction of charged objects and the electric field produced by a distribution of charges. It allows for the treatment of complex charge distributions as if they were concentrated at a single point.
 * 2) Charge distributions: In complex systems with distributed charge, such as atoms, molecules, or macroscopic objects, determining the center of charge aids in understanding their behavior under the influence of electric fields.
 * 3) Electrostatic equilibrium: For a system to be in electrostatic equilibrium, the center of charge must coincide with the center of mass. This condition ensures that the system remains stable without any net force or torque acting on it due to electric interactions.
 * 4) Particle physics: In particle accelerators and high-energy physics experiments, precise knowledge of the center of charge is crucial for designing and analyzing the trajectories of charged particles.
 * 5) Stability analysis: The center of charge is used in analyzing the stability of charged systems, such as atoms and molecules. It helps determine the equilibrium positions of charges within these systems.
 * 6) Molecular chemistry: In molecular chemistry, understanding the distribution of charge within molecules is crucial for predicting their behavior and properties. The concept of center of charge is employed in various computational methods used in molecular modeling and simulations.