Coloumb’s law says that the electric field intensity at x of a charge q located at point x_{0} is given
by

E = kq-(x-−-x0)
|x − x0|3

where the electric field intensity is defined to be the force experienced by a unit positive charge placed at
the point x. Note that this is a vector and that its direction depends on the sign of q. It points away from
x_{0} if q is positive and points toward x_{0} if q is negative. The constant k is a physical constant like the
gravitation constant. It has been computed through careful experiments similar to those used with the
calculation of the gravitation constant.

The interesting thing about Coloumb’s law is that E is the gradient of a function. In fact,

( )
E = ∇ qk---1--- .
|x− x0|

The other thing which is significant about this is that in three dimensions and for x≠x_{0},

These observations will be used to derive a very important formula for the integral

∫
E ⋅ndS
∂U

where E is the electric field intensity due to a charge, q located at the point x_{0}∈ U, a bounded open set for
which the divergence theorem holds.

Let U_{ε} denote the open set obtained by removing the open ball centered at x_{0} which has
radius ε where ε is small enough that the following picture is a correct representation of the
situation.

PICT

Then on the boundary of B_{ε} the unit outer normal to U_{ε} is −

Therefore, from the divergence theorem and observation (17.21),

∫ ∫ ∫
− 4πkq+ E ⋅ndS = E ⋅ndS = ∇ ⋅EdV = 0.
∂U ∂Uε Uε

It follows that 4πkq = ∫_{∂U}E ⋅ ndS. If there are several charges located inside U, say q_{1},q_{2},

⋅⋅⋅

,q_{n}, then
letting E_{i} denote the electric field intensity of the i^{th} charge and E denoting the total resulting electric field
intensity due to all these charges,

∫ ∑n ∫ ∑n ∑n
E ⋅ndS = Ei ⋅ndS = 4πkqi = 4πk qi.
∂U i=1 ∂U i=1 i=1

This is known as Gauss’s law and it is the fundamental result in electrostatics.