also valid for point masses

1. Gauss’s Law
Benjamin Yiwen Faerber
January 13, 2017
Let X be any set and T = {Ui|i ∈ I} denote a certain collection of subsets of X. The
pair (X, T) is a topological space if T satisfies the following requirements.
1. ∅, X ∈ T
2. If T is any (maybe infinite) subcollection of I, the family {Uj|j ∈ J} satisfies ∪j∈J Uj ∈
T.
3. If K is any finite subcollection of I, the family {Uk|k ∈ K} satisfies ∩k∈KUk ∈ T.
A metric d : X × X →R is a function that satisfies the conditions:
1. d (x, y) = d (y, x)
2. d (x, y) ≥ 0 where the equality holds if and only if x = y
3. d (x, y) + d (y, z) ≥ d (x, z)
for any x, y, z ∈ X. If X is endowed with a metric d, X is made into a topological space
whose open sets are given by open discs,
U (X) = {y ∈ X|d (x, y) < }
The topology T thus defined is called the metric topology determined by d. The topological
space X, T is called a metric space. Every metric space is a Hausdorff space. For an arbitrary
pair of distinct points f1, f2 ∈ X there is = d (f1, f2) /2 > 0 and U (f1) ∩ U (f2) = ∅.
Proof:
= f1 − f2 /2 > 0
U (f1) = {g ∈ E : g − f1 < }
U (f2) = {g ∈ E : g − f2 < }
f1 − f2 ≤ f1 − g + g − f2
≤ + = 2 = f1 − f2
⇒ U (f1) ∩ U (f2) = ∅
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2. 1 Gauss’s law
∆V
δ (x − X) d3
x =
1 if ∆V contains x = X
0 if ∆V does not contain x = X
E · n da =
q
4π 0
cos θ
r2
da (1)
Since E is directed along the line from the surface element to the charge q, cos θ da = r2 dΩ,
where dΩ is the element of solid angle subtended by da at the position of the charge.
Therefore
E · n da =
q
4π 0
dΩ (2)
If we now integrate the normal component of E over the whole surface we get
S
E · n da =
q/ 0 if q lies inside S , q ∈ S
0 if q lies outside S , q /∈ S
This is Gauss’s law for a single point charge. For a discrete set of charges
S
E · n da =
1
0 i
qi (3)
where the sum is over only those charges inside the surface S. If the surface S contains
a metric space, the point charges qi can be identified with distinct points in this space
with neighbourhoods U (qi). These neighbourhoods can be seen as volumes ∆V . For two
neighbourhoods of two point charges we have Uq ∩ Uq = ∅ For a continuous charge density
ρ (x), Gauss’s law becomes:
S
E · n da =
1
0 V
ρ (x) d3
x (4)
where V is the volume enclosed by S. A discrete set of point charges can be described with
a charge density by means of delta functions.
ρ (x) =
n
i=1
qiδ (x − xi) (5)
This represents a distribution of n point charges qi, located at the points xi. Note that equa-
tion (4) depends on the inverse square law for the force between charges, the central nature
of the force, and the linear superposition of the effects of different charges. Clearly, then,
Gauss’s law holds for Newtonian gravitational force fields, with matter density replacing
charge density.
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