From the Maxwell’s law we could easily derive the Gauss’ theorem:
(16)
(17)
Φ E = q-
ε
<∫∫ E ∙ ds = q
ε
(s)
where ΦE is the flux of the vector of the electric intensity E through the closed
surface s. It is important to note that the full electric flux ΦD could be
concentrated only in small region ∆s of the closed surface s, so we could
approximate:
(18)
Φ D = ∫∫ D ∙ ds = q
(∆s)
In other cases when the electric field is not concentrated in such small region but
we are interested to know the partial electric flux ∆ΦD through partial surface ∆s
for which is responsible electric charge ∆q it is appropriate to use the formula:
(19) ∆Φd = ∫∫ D ∙ ds = ∆q
(∆s)
Electric current
The electric current i that is the flux of (positive) physical charges could be
defined using both scalar and vector quantities.
(20)
(21)
∆qdq
i = lim —=- = —-
∆t→0 ∆tdt
i = ∫∫ J ∙ ds =φ J
(s)
—»
where J is the density ofthe electric current.
21