Week 3: Potential Energy and Potential 103
Note well, to get this result you need to eliminate certain componentsin the full expansion. To
accomplish this, you will need to neglect any term that issecond orderin ∆x, ∆y, or ∆z.
This is justified by taking the differential limit: ∆x→dx, etc. Then Gauss’s Law as we have
thus far learned it becomes the following vector differential form:
∑
sidesE~·nˆdA=∇~·E~dV= ρ
ǫ 0dVor
∇~·E~= ρ
ǫ 0
(123)
Congratulations! You’ve just derived Gauss’s Law in itsvector differentialform (and, inci-
dentally, have derived the divergence theorem for vector fields if we extend the sums above back to
integrals by summing over all the little differential cubes in an extended volume with interior surface
contributions cancelling out). We won’t use this this semester, but itis very important tostartto
think about how the one (integral) form is equivalent to the other (differential) form, as the latter
turns out to be very useful!