26.2 Finding the Electric Field from the Potential

FIGURE 26.6 shows two points i and f separated by a very small distance $Δ s$ , so small that the electric field is essentially constant over this very short distance. The work done by the electric field as a charge q moves through this small distance is $W = F_{s} Δ s = q E_{s} Δ s$ . Consequently, the potential difference between these two points is

Δ V = \frac{Δ U_{q + sources}}{q} = \frac{- W}{q} = - E_{s} Δ s

(26.8)

The figure illustrates an electric field doing work on a positive charge q.

FIGURE 26.6 Full Alternative Text

In terms of the potential, the component of the electric field in the s-direction is $E_{s} = - Δ V / Δ s$ . In the limit $Δ s \to 0$ ,

E_{s} = - \frac{d V}{d s}

(26.9)

Now we have reversed Equation 26.3 and can find the electric field from the potential. We’ll begin with examples where the field is parallel to a coordinate axis, then we’ll look at what Equation 26.9 tells us about the geometry of the field and the potential.