22.10 : Continuous Charge Distributions

Imagine a bucket of water. It contains many molecules, of the order of 10²⁶ molecules. Thus, although it contains discrete elements (molecules) at the microscopic level, macroscopically, it can be considered continuous. Small volume elements of water, infinitesimal compared to the bulk of the bucket's volume, still contain many molecules. Under this framework, quantized matter is approximated as continuous for practical purposes.

The electric charge can also be subjected to an analogical treatment. Charges are indeed quantized, and electrons and protons carry the fundamental unit of charge. But macroscopic objects contain many molecules, each containing protons and electrons. Hence, the total charge of a system can be considered a continuous charge distribution while keeping in mind that it's a suitable approximation and not the actual reality.

This kind of approximation lends the consideration of line charges, surface charges, and volume charges. For example, a charged rod can be expressed via its line charge density. Although the other two dimensions, breadth and height, are very much present, they can be ignored if there is no reason to believe that there is a significant gradient of charge along these two dimensions. Thankfully, nature follows the principle of superposition for Coulomb's law, and hence, for the electric field. Each line element of charge can then be thought to be creating its unique field, and the electric fields of all the line elements can be vectorially summed to calculate the rod's total electric field. Instead of a summation, the expression is an integral.

Similarly, for a surface charge distribution, for example, a plane or the outer surface of a spherical conductor, the description is via the surface charge density or the charge per unit surface area. The principle of superposition ensures that its total electric field is then given by a surface integral, that is, an integral over the coordinates describing this surface.

Similarly, if a particular charged body contains charge in bulk, for example, a charged insulating sphere, it is described by a volume charge density. The integral is over the coordinates describing its volume.