Charge Distribution and Charge Density

Charge Distribution and Charge Density
The electric field due to a small number of charged particles can readily be computed using the superposition principle. But what happens if we have a very large number of charges distributed in some region in space? There may be these three kind of charge distributions:

1. Volume Charge Distribution and Volume Charge Density
   Suppose we wish to find the electric field at some point P. Let's consider a small volume element V_i, which contains an amount of charge q_i. The distances between charges within the volume element V_i are much smaller as compared to r, the distance between V_i and P. In the limit where V_i becomes infinitesimally small, we may define a volume charge density  as
   
   
   
   Electric field due to a small charge element q_i at point P
   
   The dimension of  is charge/unit volume (C/m³) in SI units. The total amount of charge within the entire volume V is
   
   
   
   The concept of charge density here if analogous to mass density  _m. When a large number of atoms are tightly packed within a volume, we can also take the continuum limit and the mass of an object if given by
   
   
    
2. Surface Charge Distribution and Surface Charge Density
   In a similar manner, the charge can be distributed over a surface S of area A with a surface charge density  (lowercase Greek letter sigma). Here surface charge density can be written as
   
   
   
   The unit of   is (C/m²) in SI units. The total charge on the entire surface is
   
   
   
    
3. Linear Charge Distribution and Linear Charge Density
   If the charge is distributed over a line of length l, then the linear charge density  (lowercase Greek letter lambda) is
   
   
   
   where the unit of  is (C/m). The total charge is now an integral over the entire length
   
   

Note: If charge are uniformly distributed throughout the region, the densities () then become uniform and can be written as: