J 3 ¼ qn 0 μ
n
33 E 3 þ qD
n
33
∂ Δn
ð Þ
∂x 3
¼ 0, x 3 ¼ 0, L,
ð3:167Þ
Δn ¼ 0, t ¼ 0:
ð3:168Þ
To make the boundary conditions in Eq. (3.167) homogeneous, we let
Δn ¼ b n þ
n 0 μ
n
33
D
n
33
φ:
ð3:169Þ
The initial-boundary-value problem for b n is
∂b n
∂t
¼ D
n
33
∂
2 b n
∂x 2
3
À
n 0 μ
n
33
D
n
33
∂φ
∂t
,
ð3:170Þ
∂b n
∂x 3
¼ 0, x 3 ¼ 0, L,
ð3:171Þ
b n ¼ 0, t ¼ 0:
ð3:172Þ
Equations (3.170), (3.171), and (3.172) form a standard mathematical problem. Its
solution by separation of variables and Laplace transform is
b n x 3 , t
ð
Þ ¼
X 1
r¼0
cos
rπ
L
x 3
Z t
0
a r τ
ð Þ exp À
rπ
L
ffiffiffiffiffiffiffi ffi
D
n
33
p
2
t À τ
ð
Þ
!
dτ,
ð3:173Þ
where
a 0 t
ð Þ ¼
1
L
Z L
0
À
n 0 μ
n
33
D
n
33
∂φ
∂t
dx 3 , r ¼ 0,
a r t
ð Þ ¼
2
L
Z L
0
À
n 0 μ
n
33
D
n
33
∂φ
∂t
cos
rπ
L
x 3
dx 3 , r ¼ 1, 2, 3Á Á Á:
ð3:174Þ
From the series solution in Eqs. (3.173) and (3.169), we show in Fig. 3.23 the
electron concentration perturbation at the same time instants as those in Fig. 3.22.
Fundamentally different from the mechanical disturbance governed by the hyperbolic wave equation in Eq. (3.161) with a finite wave speed, the motion of the
electrons is governed by the parabolic equation in Eq. (3.170) with effectively an
infinite speed for signal propagation. Therefore, in Fig. 3.23a, while mechanically
the left part of the rod is still unperturbed, the electrons along the entire rod have felt
the disturbance and been disturbed. Since the rod is electrically isolated, the total
number of electrons is conserved. Therefore the integration of Δn over [0,L] has to
vanish, which can be qualitatively seen in the figure. The series for Δn converges
very rapidly. The results corresponding to seven and eight terms are indistinguishable when plotted in the scale of Fig. 3.23.
3.12 Transient Vibration
71
n
33 E 3 þ qD
n
33
∂ Δn
ð Þ
∂x 3
¼ 0, x 3 ¼ 0, L,
ð3:167Þ
Δn ¼ 0, t ¼ 0:
ð3:168Þ
To make the boundary conditions in Eq. (3.167) homogeneous, we let
Δn ¼ b n þ
n 0 μ
n
33
D
n
33
φ:
ð3:169Þ
The initial-boundary-value problem for b n is
∂b n
∂t
¼ D
n
33
∂
2 b n
∂x 2
3
À
n 0 μ
n
33
D
n
33
∂φ
∂t
,
ð3:170Þ
∂b n
∂x 3
¼ 0, x 3 ¼ 0, L,
ð3:171Þ
b n ¼ 0, t ¼ 0:
ð3:172Þ
Equations (3.170), (3.171), and (3.172) form a standard mathematical problem. Its
solution by separation of variables and Laplace transform is
b n x 3 , t
ð
Þ ¼
X 1
r¼0
cos
rπ
L
x 3
Z t
0
a r τ
ð Þ exp À
rπ
L
ffiffiffiffiffiffiffi ffi
D
n
33
p
2
t À τ
ð
Þ
!
dτ,
ð3:173Þ
where
a 0 t
ð Þ ¼
1
L
Z L
0
À
n 0 μ
n
33
D
n
33
∂φ
∂t
dx 3 , r ¼ 0,
a r t
ð Þ ¼
2
L
Z L
0
À
n 0 μ
n
33
D
n
33
∂φ
∂t
cos
rπ
L
x 3
dx 3 , r ¼ 1, 2, 3Á Á Á:
ð3:174Þ
From the series solution in Eqs. (3.173) and (3.169), we show in Fig. 3.23 the
electron concentration perturbation at the same time instants as those in Fig. 3.22.
Fundamentally different from the mechanical disturbance governed by the hyperbolic wave equation in Eq. (3.161) with a finite wave speed, the motion of the
electrons is governed by the parabolic equation in Eq. (3.170) with effectively an
infinite speed for signal propagation. Therefore, in Fig. 3.23a, while mechanically
the left part of the rod is still unperturbed, the electrons along the entire rod have felt
the disturbance and been disturbed. Since the rod is electrically isolated, the total
number of electrons is conserved. Therefore the integration of Δn over [0,L] has to
vanish, which can be qualitatively seen in the figure. The series for Δn converges
very rapidly. The results corresponding to seven and eight terms are indistinguishable when plotted in the scale of Fig. 3.23.
3.12 Transient Vibration
71