The field due to a continuous charge distribution can be obtained in much the same way as for a system of discrete charges. Suppose a continuous charge distribution in space has a charge density ρ. Choose any convenient origin O and let the position vector of any point in the charge distribution be ρ. The charge density ρ may vary from point to point, i.e., it is a function of ρ. Divide the charge distribution into small volume elements of size ∆V. The charge in a volume element ∆V is ρ∆V.
Now, consider any general point P (inside or outside the distribution) with position vector R (Fig.). Electric field due to the charge ρ∆V is given by Coulomb’* law:
where r' is the distance between the charge element and P, and ˆr' is a unit vector in the direction from the charge element to P. By the superposition principle, the total electric field due to the charge distribution is obtained by summing over electric fields due to different volume elements:
Note that r, r', ˆr' all can vary from point to point.
Hence, using Coulomb’* law and the superposition principle, electric field can be determined for any charge distribution, discrete or continuous or part discrete and part continuous.
