Week 1: Discrete Charge and the Electrostatic Field 43
for a linear charge distribution on a particular lineL.
Because one has to integrate the vector components independently, and since their contribution
and geometry can vary as one moves~rabout in space, this integral is remarkably difficult to integrate
in the general case for most charge density distributions. We will manage to find a few examples
(below) where the difficulty of the integration process is reduced due to thesymmetryof the charge
distribution, which may allow us to cancel (and hence avoid having to do) particular parts of the
integrals from symmetry alone, but the methodology overall will be very cumbersome and is rarely
used in real physics problems.
Instead in a few chapters we’ll derive a similar form, but far more tractable integral form for the
electrostaticpotential, a scalar quantity, and obtain the field (if it is desired at all) by taking the
negative gradient of the potential, since vector calculus differentiation is often easier algebraically
than vector calculus integration. Even here, however, from a purely practical point of view only very
simple and symmetric charge distributions can be solved algebraically,and for most “real world”
problems one must resort to using a computer tonumerically integratethe expressions above, by (for
example) computing a direct sum of the fields or potentials in the
∑
iform where eachqi=ρ∆Vi
for some suitable partitioning of the distribution into a finite number of indexed chunks of size ∆Vi.