Physics equations/19-Electric Potential and Electric Field/Q:SurfaceIntegralsCalculus
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pe19surfaceIntegralsCalculus A
1
A cylinder of radius, r=2, and height, h=6, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
F
→
=
(
2.03
+
1.29
z
)
ρ
2
ρ
^
+
8.35
z
3
z
^
{\displaystyle {\vec {\mathfrak {F}}}=(2.03+1.29z)\rho ^{2}{\hat {\rho }}+8.35z^{3}{\hat {z}}}
Let
n
^
{\displaystyle {\hat {n}}}
be the outward unit normal to this cylinder and evaluate ,
|
∫
t
o
p
F
→
⋅
n
^
d
A
|
{\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}
over the top surface of the cylinder.
a) 1.315E+03
b) 1.593E+03
c) 1.930E+03
d) 2.338E+03
e) 2.833E+03
2
A cylinder of radius, r=2, and height, h=6, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
F
→
=
(
2.03
+
1.29
z
)
ρ
2
ρ
^
+
8.35
z
3
z
^
{\displaystyle {\vec {\mathfrak {F}}}=(2.03+1.29z)\rho ^{2}{\hat {\rho }}+8.35z^{3}{\hat {z}}}
Let
n
^
{\displaystyle {\hat {n}}}
be the outward unit normal to this cylinder and evaluate ,
|
∫
s
i
d
e
F
→
⋅
n
^
d
A
|
{\displaystyle \left|\int _{side}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}
over curved side surface of the cylinder.
a) 3.443E+02
b) 4.171E+02
c) 5.053E+02
d) 6.122E+02
e) 7.417E+02
3
A cylinder of radius, r=2, and height, h=6, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
F
→
=
(
2.03
+
1.29
z
)
ρ
2
ρ
^
+
8.35
z
3
z
^
{\displaystyle {\vec {\mathfrak {F}}}=(2.03+1.29z)\rho ^{2}{\hat {\rho }}+8.35z^{3}{\hat {z}}}
Let
n
^
{\displaystyle {\hat {n}}}
be the outward unit normal to this cylinder and evaluate ,
|
∮
F
→
⋅
n
^
d
A
|
{\displaystyle \left|\oint {\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}
over the entire surface of the cylinder.
a) 2.94E+03
b) 3.54E+03
c) 4.27E+03
d) 5.15E+03
e) 6.28E+03
pe19surfaceIntegralsCalculus B
1
A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
F
→
=
(
1.74
+
1.27
z
)
ρ
3
ρ
^
+
9.08
z
2
z
^
{\displaystyle {\vec {\mathfrak {F}}}=(1.74+1.27z)\rho ^{3}{\hat {\rho }}+9.08z^{2}{\hat {z}}}
Let
n
^
{\displaystyle {\hat {n}}}
be the outward unit normal to this cylinder and evaluate ,
|
∫
t
o
p
F
→
⋅
n
^
d
A
|
{\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}
over the top surface of the cylinder.
a) 2.118E+02
b) 2.567E+02
c) 3.109E+02
d) 3.767E+02
e) 4.564E+02
2
A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
F
→
=
(
1.74
+
1.27
z
)
ρ
3
ρ
^
+
9.08
z
2
z
^
{\displaystyle {\vec {\mathfrak {F}}}=(1.74+1.27z)\rho ^{3}{\hat {\rho }}+9.08z^{2}{\hat {z}}}
Let
n
^
{\displaystyle {\hat {n}}}
be the outward unit normal to this cylinder and evaluate ,
|
∫
s
i
d
e
F
→
⋅
n
^
d
A
|
{\displaystyle \left|\int _{side}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}
over the curved side surface of the cylinder.
a) 6.997E+02
b) 8.477E+02
c) 1.027E+03
d) 1.244E+03
e) 1.507E+03
3
A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
F
→
=
(
1.74
+
1.27
z
)
ρ
3
ρ
^
+
9.08
z
2
z
^
{\displaystyle {\vec {\mathfrak {F}}}=(1.74+1.27z)\rho ^{3}{\hat {\rho }}+9.08z^{2}{\hat {z}}}
Let
n
^
{\displaystyle {\hat {n}}}
be the outward unit normal to this cylinder and evaluate ,
|
∮
F
→
⋅
n
^
d
A
|
{\displaystyle \left|\oint {\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}
over the entire surface of the cylinder.
a) 4.77E+02
b) 5.78E+02
c) 7.00E+02
d) 8.48E+02
e) 1.03E+03
pe19surfaceIntegralsCalculus C
1
A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
F
→
=
(
2.48
+
2.38
z
)
ρ
3
ρ
^
+
8.41
z
2
z
^
{\displaystyle {\vec {\mathfrak {F}}}=(2.48+2.38z)\rho ^{3}{\hat {\rho }}+8.41z^{2}{\hat {z}}}
Let
n
^
{\displaystyle {\hat {n}}}
be the outward unit normal to this cylinder and evaluate ,
|
∫
t
o
p
F
→
⋅
n
^
d
A
|
{\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}
over the top surface of the cylinder.
a) 2.377E+02
b) 2.880E+02
c) 3.489E+02
d) 4.227E+02
e) 5.122E+02
2
A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
F
→
=
(
2.48
+
2.38
z
)
ρ
3
ρ
^
+
8.41
z
2
z
^
{\displaystyle {\vec {\mathfrak {F}}}=(2.48+2.38z)\rho ^{3}{\hat {\rho }}+8.41z^{2}{\hat {z}}}
Let
n
^
{\displaystyle {\hat {n}}}
be the outward unit normal to this cylinder and evaluate ,
|
∫
s
i
d
e
F
→
⋅
n
^
d
A
|
{\displaystyle \left|\int _{side}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}
over the curved side surface of the cylinder.
a) 9.973E+02
b) 1.208E+03
c) 1.464E+03
d) 1.773E+03
e) 2.149E+03
3
A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
F
→
=
(
2.48
+
2.38
z
)
ρ
3
ρ
^
+
8.41
z
2
z
^
{\displaystyle {\vec {\mathfrak {F}}}=(2.48+2.38z)\rho ^{3}{\hat {\rho }}+8.41z^{2}{\hat {z}}}
Let
n
^
{\displaystyle {\hat {n}}}
be the outward unit normal to this cylinder and evaluate ,
|
∮
F
→
⋅
n
^
d
A
|
{\displaystyle \left|\oint {\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}
over the entire surface of the cylinder.
a) 9.97E+02
b) 1.21E+03
c) 1.46E+03
d) 1.77E+03
e) 2.15E+03