Page 341 - Intro to Tensor Calculus

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where S = S T + S B is the total surface area of the outside sphere and Σ = Σ T +Σ B is the total surface area
(1)
of the inside sphere, and n is the inward normal to the sphere Σ when the top and bottom volumes are
i
combined. Applying the Gauss divergence theorem to just the isolated small sphere Σ we find
ZZ ZZZ
i (2) i
F n dσ = F dτ (2.6.33)
i ,i
Σ V Σ
(2)
where n is the outward normal to Σ. By adding the equations (2.6.33) and (2.6.32) we find that
i
ZZ ZZ ZZZ
i (1) i (2)
i
i
F n i dσ + F n i + F n i dσ = F dτ (2.6.34)
,i
S Σ V
where V = V T + V B + V Σ . The equation (2.6.34) can also be written as
ZZ ZZZ ZZ

i (1)
i (2)
i
i
F n i dσ = F dτ − F n + F n dσ. (2.6.35)
,i i i
S V Σ
In the case that V contains a surface Σ the total electric charge inside S is
ZZZ ZZ
Q e = ρ dτ + µ dσ (2.6.36)
∗
∗
V Σ
where µ is the surface charge density on Σ and ρ is the volume charge density throughout V. The Gauss
∗
∗
theorem requires that
ZZ 1 ZZZ 1 ZZ
i
∗
∗
E n i dσ = Q e = ρ dτ + µ dσ. (2.6.37)
S 0 0 V 0 Σ
In the case of a jump discontinuity across the surface Σ we use the results of equation (2.6.34) and write
ZZ ZZZ ZZ

i (1)
i (2)
i
i
E n i dσ = E dτ − E n + E n dσ. (2.6.38)
,i i i
S V Σ
Subtracting the equation (2.6.37) from the equation (2.6.38) gives
ZZZ ZZ
ρ ∗ µ ∗
i i (1) i (2)
E − dτ − E n i + E n i + dσ =0. (2.6.39)
,i
V 0 Σ 0
For arbitrary surfaces S and Σ, this equation implies the differential form of the Gauss law
ρ ∗
i
E = . (2.6.40)
,i
0
Further, on the surface Σ, where there is a surface charge distribution we have
i (1) i (2) µ ∗
E n i + E n i + =0 (2.6.41)
0
∗
which shows the electric field undergoes a discontinuity when you cross a surface charge µ .