Notes 3
Fall 2004.doc
Now, to obtain the magnetic flux,
Φ
, through the whole surface, S, we perform a surface integral:
∫
Φ
=
Φ
S
d
or in the present example,
(
)
∫∫
⋅
=
Φ
y
,
z
dydz
x
ˆ
B
G
.
Example: In the figure below a rectangular surface is shown edge-on.
Let a time-independent and
spatially
constant
magnetic
flux
density
be
represented
by
2
m
Wb
y
ˆ
3
x
ˆ
2
+
=
B
G
and
let
. Calculate the flux through the rectangular surface.
4
z
0
,
3
y
0
<
<
<
<
y
ˆ
3
x
ˆ
B
G
θ
∫
⋅
=
Φ
S
S
d
G
G
B
dydz
x
ˆ
S
d
choose
and
dydz
dS
that
Note
+
=
=
G
∫
∫
∫
⋅
=
⋅
=
Φ
=
Φ
S
S
S
dydz
x
ˆ
S
d
d
B
B
G
G
G
Since here
B
G
does not change with position (it is not a function of y or z), we can pull the quantity
x
ˆ
⋅
B
G
out of the integral, i.e.,
(
)
(
)
dydz
x
ˆ
y
ˆ
3
x
ˆ
2
S
d
x
ˆ
3
0
4
0
S
∫
∫
∫
⋅
+
=
⋅
=
Φ
G
G
B
(
)
( )( )
( )( )( )
Wb
24
4
3
2
z
3
2
dz
y
2
4
0
3
0
4
0
=
=
=
=
Φ
∫
Note that when
B
G
is independent of position (we sometimes say spatially constant) we simply
multiply
x
ˆ
⋅
B
G
by the area.
Often, as in the case of a circle or rectangle, we can simply substitute the
area formula (that we all know, right!?) in place of the integral.
Here, we could have simply replaced
with
.
Otherwise we must perform the integration to calculate the area.
So look for
simplifcations of this sort.
∫
S
S
d
G
( )( )
12
3
4
=

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Notes 3
Fall 2004.doc
Example:
A single circular wire loop of radius
cm
1
a
=
is located in the x-y plane and is centered at
the origin.
It is immersed in a magnetic field given by
(
)
t
sin
z
ˆ
3
y
ˆ
2
ω
+
=
o
B
B
G
(this field is spatially
constant, but not temporally constant).
For
T
05
.
0
=
o
B
and an angular frequency of
s
rad
377
,
calculate the magnetic flux through the surface,
Φ
, at
ms
77
.
2
t
=
.
y
Let us
choose
and note that
dS
z
ˆ
S
d
=
G
2
S
a
z
ˆ
S
z
ˆ
S
d
π
=
=
∫
G
. Then
x
(
)
(
)
( )
(
)
(
)
(
)
Wb
7
.
40
01
.
0
10
77
.
2
377
sin
05
.
0
3
a
t
sin
3
S
t
sin
3
dS
t
sin
3
dS
t
sin
3
dS
z
ˆ
t
sin
z
ˆ
3
y
ˆ
2
S
d
2
3
2
S
S
S
S
µ
=
⋅
π
×
⋅
=
π
ω
=
ω
=
ω
=
ω
=
⋅
ω
+
=
⋅
=
Φ
−
∫
∫
∫
∫
o
o
o
o
o
B
B
B
B
B
B
G
G
Example:
A rectangular surface (seen edge-on below) is located in the x-y plane and has extent
.
It is immersed in a magnetic field that is given by
cm
5
y
0
,
cm
3
x
0
<
<
<
<
(
)
t
sin
z
ˆ
y
3
y
ˆ
2
ω
+
=
o
B
B
G
.
This field is
not
spatially (nor temporally) constant since it is a function of y.
For
and an
angular frequency of
T
04
.
0
=
o
B
s
rad
377
, calculate the magnetic flux,
Φ
, at
ms
77
.
2
t
=
.
(
)
∫
∫
∫
⋅
ω
+
=
⋅
=
Φ
03
.
0
0
05
.
0
0
S
dxdy
z
ˆ
t
sin
z
ˆ
y
3
y
ˆ
2
S
d
o
B
B
G
G
4
y
z
B
G
5
(
)
∫
∫
ω
=
Φ
03
.
0
0
05
.
0
0
dxdy
t
sin
y
3
o
B
dxdy
z
ˆ
dS
z
ˆ
S
d
choose
=
=
G
ydy
x
t
sin
3
05
.
0
0
03
.
0
0
∫
⎟
⎠
⎞
⎜
⎝
⎛
ω
=
Φ
o
B
(
)
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
ω
=
Φ
05
.
0
0
2
2
1
y
03
.
0
t
sin
3
o
B
(
)(
)
( )
(
)
(
)
(
)
3
4
2
2
1
10
77
.
2
377
sin
10
125
.
1
t
sin
05
.
0
03
.
0
04
.
0
3
−
−
×
⋅
×
=
ω
=
Φ
Wb
9
.
3
µ
=
Φ