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By Donkey Kong
What I am confused about is whether the charge of an object is concentrated within the centre of the mass itself, or does it change depending on the situation?
Charge and mass are independent properties of a body. The center of mass and charge of a body depend solely on the way the mass or charge is distributed about the body. If both are distributed in the same way, their centers will coincide.
What will affect the distribution of charge?
A rod for example may have a linear charge density, this will mean that the center will not be at the middle. Simply bringing a magnet close to a body will also disturb the distribution of charge.