Practical Way of Calculating Electric Field

Beyond the fundamental point-charge formula, there is a very practical, everyday way of both calculating AND measuring an electric field, using a pair of charged parallel plates. Consider two parallel plates A and B, separated by a distance d, with a potential difference V applied across them from an external source; the electric field in the region between the plates is UNIFORM, directed from the higher-potential plate (A) to the lower-potential plate (B), as shown in Fig. 10.10.

A positive charge +q placed between the plates experiences a force F due to this uniform field. To move this charge AGAINST the direction of the field -- that is, from the negative plate B toward the positive plate A -- external work must be done on it. If the charge +q is moved the full separation distance d, from B to A, the work done against the field is simply $W=Fd.$ On the other hand, by the very definition of potential difference, the work done in moving a charge q through a potential difference V is $W=Vq.$ Equating these two expressions for the same work done, $Vq=Fd\quad\Rightarrow\quad\frac{F}{q}=\frac{V}{d}.$ But $F/q$ is exactly the definition of electric field E, so this gives the widely-used PRACTICAL formula for the field between parallel plates: $E=\frac{V}{d}.$ This E = V/d relation is, in practice, the most commonly used way of both defining and actually m …