This preview shows pages 1–2. Sign up to view the full content.
H
*
Here we solve the partial differential equation for steady state temperature
distribution in the semi

infinite slab with the Laplace equation
“
2
T
@
x,y
D
=
0 with the Boundary Conditions
H
BC
L
T
H
x,0
L
=
T
0
=
100, T
@
x,
¶
D
=
0, T
@
0,y
D
=
0,
T
@
L,y
D
=
0 where L
=
10. Assuming a solution of the form T
@
x,y
D
=
X
@
x
D
*
Y
@
y
D
and using the 2D Laplacian
“
2
=
∂
2
∂
x
2
+
∂
2
∂
y
2
which gives X"
@
x
D
Y
@
y
D
+
X
@
x
D
Y"
@
y
D
=
0. Dividing by X
@
x
D
Y
@
y
D
≠
0 we get X"
@
x
Dê
X
@
x
D
+
Y"
@
y
Dê
Y
@
y
D
=
0,
which can only be satisfied if both terms are the same constant or X"
@
x
Dê
X
@
x
D
=
Y"
@
y
Dê
Y
@
y
D
=

k
2
, where k is the separation constant. We can solve these two diffy

Q's using DSolve.
*
L
DSolve
@
X''
@
x
D
+
k^2
*
X
@
x
D
ã
0, X
@
x
D
, x
D
88
X
@
x
D
Ø
C
@
1
D
Cos
@
k x
D
+
C
@
2
D
Sin
@
k x
D<<
DSolve
@
Y''
@
y
D

k^2
*
Y
@
y
D
ã
0, Y
@
y
D
, y
D
99
Y
@
y
D
Ø ‰
k y
C
@
1
D
+ ‰

k y
C
@
2
D==
H
*
That is X
@
x
D
=+
A
*
Sin
@
k
*
x
D
+
B
*
Cos
@
k
*
x
D
and Y
@
y
D
=
C
*
Exp
@
k
*
y
D
+
D
*
Exp
@

k
*
y
D
,
as in the notes. Demanding the BC's X
@
0
D
=
0,
X
@
L
D
=
0 gives B
=
0 and k
=
k
n
=
n
p
ê
L
H
the "quantization condition"
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '09
 DOWLING
 Physics

Click to edit the document details