Coloumb’s law says that the electric field intensity at x of a charge q located at point x0 is given by
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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 x0 if q is positive and points toward x0 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,
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The other thing which is significant about this is that in three dimensions and for x≠x0,
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This is left as an exercise for you to verify.
These observations will be used to derive a very important formula for the integral
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where E is the electric field intensity due to a charge, q located at the point x0 ∈ U, a bounded open set for which the divergence theorem holds.
Let Uε denote the open set obtained by removing the open ball centered at x0 which has radius ε where ε is small enough that the following picture is a correct representation of the situation.
Then on the boundary of Bε the unit outer normal to Uε is −
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It follows that 4πkq = ∫ ∂UE ⋅ ndS. If there are several charges located inside U, say q1,q2,
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This is known as Gauss’s law and it is the fundamental result in electrostatics.