Δn ¼ À
k
q
eλ
c
þ p
θ sinh ka exp kx
ð Þ,
ρ
P
À qΔn ¼ k
eλ
c
þ p
θ sinh ka
cε 0
e 2 þ cε
exp kx
ð Þ,
ð7:44Þ
where
k
2
¼
n 0 μ
n
D
n
q
ε T
33
:
ð7:45Þ
In the central region with |x| < a, we have
u ¼ À
e
c
eλ þ pc
ð
Þ θ
k e 2 þ cε
ð
Þ
exp Àka
ð
Þsinh kx þ
λ
c
θ x þ a
ð
Þ,
S ¼ À
e
c
eλ þ pc
ð
Þ θ
e 2 þ cε
exp Àka
ð
Þcosh kx þ
λ
c
θ,
ð7:46Þ
φ ¼
eλ þ pc
ð
Þ θ
k e 2 þ cε
ð
Þ
exp Àka
ð
Þsinh kx,
E ¼ À
eλ þ pc
ð
Þ θ
e 2 þ cε
exp Àka
ð
Þcosh kx,
D ¼ À
eλ
c
þ p
θ exp Àka
ð
Þcosh kx þ
eλ
c
þ p
θ,
P ¼ À
eλ
c
þ p
θ 1 À
cε 0
e 2 þ cε
exp Àka
ð
Þcosh kx þ
eλ
c
þ p
θ,
ρ
P
¼ k
eλ
c
þ p
θ 1 À
cε 0
e 2 þ cε
exp Àka
ð
Þsinh kx,
ð7:47Þ
Δn ¼
k
q
eλ
c
þ p
θ exp Àka
ð
Þsinh kx,
ρ
P
À qΔn ¼ Àk
eλ
c
þ p
θ
cε 0
e 2 þ cε
exp Àka
ð
Þsinh kx:
ð7:48Þ
For x > a, the fields are
u ¼ À
e
c
eλ þ pc
ð
Þ θ
k e 2 þ cε
ð
Þ
sinh ka exp Àkx
ð
Þþ2
λ
c
θL,
S ¼
e
c
eλ þ pc
ð
Þ θ
e 2 þ cε
sinh ka exp Àkx
ð
Þ,
ð7:49Þ
7.2 Effects of a Local Temperature Change
185