14
2 Basic Electromagnetism
Fig. 2.2 a Substance in which the electric charge is uniformly distributed with density ρ 0 and
b closed parallelepiped S for x > a on which Gauss’ law is applied
is the area of the surface parallel to the y-z plane. Now we estimate the total electric
charge inside S. A simple calculation leads to Axρ 0 for 0 ≤ x ≤ a and Aaρ 0 for
x > a. Thus, we have
E x = −
aρ 0
0
; x < −a,
=
xρ 0
0
; −a ≤ x ≤ a,
=
aρ 0
0
; x > a,
(2.9)
where we have used the symmetry condition with respect to x = 0. Substitution of
this result into (2.7) leads to
∇ · E =
ρ 0
0
; −a ≤ x ≤ a,
= 0; x < −a, x > a.
(2.10)
This shows that the electric charge is distributed with density ρ 0 only inside the slab,
as assumed in the beginning.
The scalar potential, i.e., the electric potential, φ, which causes the electric field
is defined as
φ(r) = −
r
r 0
E · ds,
(2.11)
2 Basic Electromagnetism
Fig. 2.2 a Substance in which the electric charge is uniformly distributed with density ρ 0 and
b closed parallelepiped S for x > a on which Gauss’ law is applied
is the area of the surface parallel to the y-z plane. Now we estimate the total electric
charge inside S. A simple calculation leads to Axρ 0 for 0 ≤ x ≤ a and Aaρ 0 for
x > a. Thus, we have
E x = −
aρ 0
0
; x < −a,
=
xρ 0
0
; −a ≤ x ≤ a,
=
aρ 0
0
; x > a,
(2.9)
where we have used the symmetry condition with respect to x = 0. Substitution of
this result into (2.7) leads to
∇ · E =
ρ 0
0
; −a ≤ x ≤ a,
= 0; x < −a, x > a.
(2.10)
This shows that the electric charge is distributed with density ρ 0 only inside the slab,
as assumed in the beginning.
The scalar potential, i.e., the electric potential, φ, which causes the electric field
is defined as
φ(r) = −
r
r 0
E · ds,
(2.11)