À2Ge
1
ð Þ
33 u 2,333 À b εφ ,33 ¼ q Δp À Δn
ð
Þ A
2
ð Þ
:
ð6:93Þ
Equations (6.89), (6.88), and the current boundary conditions in Eq. (6.90) lead to
Àqp 0 μ
p
33 φ ,3 À qD
p
33 Δp
ð Þ ,3 ¼ 0,
Àqn 0 μ
n
3 φ ,3 þ qD
n
33 Δn
ð Þ ,3 ¼ 0:
ð6:94Þ
The integration of Eq. (6.94) gives
qΔp ¼ À
qp 0 μ
p
33
D
p
33
φ þ C 1 ,
ÀqΔn ¼ À
qn 0 μ
n
33
D
n
33
φ þ C 2 ,
ð6:95Þ
where C 1 and C 2 are undetermined constants. The substitution of Eqs. (6.92) and
(6.95) into Eq. (6.93) yields
φ ,33 À k
2
φ ¼
2Ge
1
ð Þ
33
e εD
F À
1
e ε
C 1 þ C 2
ð
Þ A
2
ð Þ ,
ð6:96Þ
where
k
2
¼
1
e ε
qp 0 μ
p
33
D
p
33
þ
qn 0 μ
n
33
D
n
33
A
2
ð Þ ,
e ε ¼ b ε þ
2Ge
1
ð Þ
33
2
D
:
ð6:97Þ
The general solution of Eq. (6.96) is
φ ¼ kC 3 sinh kx 3 þ kC 4 sinh k x 3 À L
ð
Þ
À
2Ge
1
ð Þ
33
k
2
e εD
F þ
1
k
2
e ε
C 1 þ C 2
ð
Þ A
2
ð Þ ,
ð6:98Þ
where C 3 and C 4 are undetermined constants. Then, from Eq. (6.92),
u 2 ¼
2Ge
1
ð Þ
33
D
C 3 cosh kx 3 þ C 4 cosh k x 3 À L
ð
Þ
½
Š
À
F
6D
x
3
3 þ C 5 x
2
3 þ C 6 x 3 þ C 7 ,
ð6:99Þ
6.3 Bending of Beams with e 33
159