scbv
@@
3
DD
.zhat
0
so zhat = Cos[
Θ
] rhat 
Θ
hat Sin[
Θ
]
2. Compute
S
v·dS ,
where S is the surface of the unit sphere and
v= x
3
x
‘
+
3
x
y z
2
y
‘
+
a
y z
z
‘
.
Convert this to a volume integral and do the integration in spherical coordinates.
vcart
=
8
x^3,3xyz^2, ayz
<
9
x
3
,3xyz
2
,ayz
=
SetCoordinates
@
Cartesian
@
x,y,z
DD
Cartesian
@
x,y,z
D
div
=
Div
@
vcart
D
3x
2
+
ay
+
3xz
2
Thread
@8
x,y,z
<
>
8
rCos
@
Φ
D
Sin
@
Θ
D
,rSin
@
Θ
D
Sin
@
Φ
D
,rCos
@
Θ
D<D
8
x
fi
rCos
@
Φ
D
Sin
@
Θ
D
,y
fi
rSin
@
Θ
D
Sin
@
Φ
D
,z
fi
rCos
@
Θ
D<
divsp
=
div .Thread
@8
x,y,z
<
>
8
rCos
@
Φ
D
Sin
@
Θ
D
,rSin
@
Θ
D
Sin
@
Φ
D
,rCos
@
Θ
D<D
Simplify
rSin
@
Θ
D I
3r
2
Cos
@
Θ
D
2
Cos
@
Φ
D
+
3rCos
@
Φ
D
2
Sin
@
Θ
D
+
aSin
@
Φ
DM
Integrate
@
divspr^2Sin
@
Θ
D
,
8
Φ
,0,2
Π
<
,
8
Θ
,0,
Π
<
,
8
r,0,1
<D
4
Π
5
3. A plane wave polarized in the x direction and propagating in the z direction has an electric field E1={ex,
0, 0}
ª
H
k
.
x
Ω
t
L
, where ex is the magnitude of the field and k is a constant vector
k={0, 0, km}
.
According to Maxwell's equations, this wave has a magnetic field associated with it which obeys the
equation ×E=

¶
B
¶
t
.
a) Assume B={bx, by, bz}
ª
H
k
.
x
Ω
t
L
, and determine bx, by, and bz.
Ef
=
8
ex,0,0
<
E^
H
I
H8
0,0,km
<
.
8
x,y,z
<
 Ω
t
LL
9
ª
H
kmz

t
Ω
L
ex,0,0
=
SetCoordinates
@
Cartesian
@
x,y,z
DD
Cartesian
@
x,y,z
D
Curl
@
Ef
D
9
0,
ª
H
kmz

t
Ω
L
exkm,0
=
Bf
=
8
bx, by, bz
<
E^
H
I
H8
0,0,km
<
.
8
x,y,z
<
 Ω
t
L L
9
bx
ª
H
kmz

t
Ω
L
,by
ª
H
kmz

t
Ω
L
,bz
ª
H
kmz

t
Ω
L
=
2
212HomeWork6sol.nb