Electric field due to continuous charge distribution

Although electric charge is fundamentally quantised at the microscopic (atomic) level, most everyday charged objects -- a charged rod, ring, disc, or sphere -- contain such an enormous number of elementary charges that the charge can be treated as if it were spread continuously over the object's length, surface, or volume. The point-charge expressions developed so far (E = kq/r^2 r-hat) apply only to point charges, so they cannot be used directly on an extended object. Instead, the standard technique is to imagine dividing the continuous charge distribution into a very large number of infinitesimally small charge elements dq, each of which is small enough to be treated as an ordinary point charge. Every element dq produces its own tiny point-charge field dE = k(dq/r^2) r-hat at the field point of interest, and the total field due to the whole distribution is obtained by summing (integrating) all these elemental contributions, dE, over the entire charge distribution: E = integral of k(dq/r^2) r-hat. This integration technique -- outlined in the textbook's appendix on continuous charge distributions -- is the general method that later sections use implicitly to build up th …