Physics equations/19-Electric Potential and Electric Field/Q:SurfaceIntegralsCalculus/testbank

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c19ElectricPotentialField_SurfaceIntegral_v1

<quiz display=simple points="1/1"> {A cylinder of radius, r=3, 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,
$\vec{𝔉} = (2.35 + 2.57 z) ρ^{3} \hat{ρ} + 7.45 z^{3} \hat{z}$
Let $\hat{n}$ be the outward unit normal to this cylinder and evaluate ,
$| \int_{t o p} \vec{𝔉} \cdot \hat{n} d A |$
over the top surface of the cylinder. |type="()"} -a) 1.148E+03 -b) 1.391E+03 +c) 1.685E+03 -d) 2.042E+03 -e) 2.473E+03

{A cylinder of radius, r=3, 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,
$\vec{𝔉} = (2.05 + 2.59 z) ρ^{2} \hat{ρ} + 7.4 z^{2} \hat{z}$
Let $\hat{n}$ be the outward unit normal to this cylinder and evaluate ,
$| \int_{t o p} \vec{𝔉} \cdot \hat{n} d A |$
over the top surface of the cylinder. |type="()"} -a) 6.908E+02 +b) 8.369E+02 -c) 1.014E+03 -d) 1.228E+03 -e) 1.488E+03

{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,
$\vec{𝔉} = (2.12 + 1.85 z) ρ^{3} \hat{ρ} + 8.88 z^{2} \hat{z}$
Let $\hat{n}$ be the outward unit normal to this cylinder and evaluate ,
$| \int_{t o p} \vec{𝔉} \cdot \hat{n} d A |$
over the top surface of the cylinder. |type="()"} -a) 3.041E+02 -b) 3.684E+02 +c) 4.464E+02 -d) 5.408E+02 -e) 6.552E+02

{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,
$\vec{𝔉} = (2 + 1.45 z) ρ^{2} \hat{ρ} + 8.02 z^{3} \hat{z}$
Let $\hat{n}$ be the outward unit normal to this cylinder and evaluate ,
$| \int_{t o p} \vec{𝔉} \cdot \hat{n} d A |$
over the top surface of the cylinder. |type="()"} -a) 3.742E+02 -b) 4.534E+02 -c) 5.493E+02 -d) 6.655E+02 +e) 8.063E+02

{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,
$\vec{𝔉} = (2.14 + 2.8 z) ρ^{2} \hat{ρ} + 9.94 z^{2} \hat{z}$
Let $\hat{n}$ be the outward unit normal to this cylinder and evaluate ,
$| \int_{t o p} \vec{𝔉} \cdot \hat{n} d A |$
over the top surface of the cylinder. |type="()"} -a) 2.810E+02 -b) 3.404E+02 -c) 4.124E+02 +d) 4.996E+02 -e) 6.053E+02

{A cylinder of radius, r=3, 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,
$\vec{𝔉} = (1.85 + 1.33 z) ρ^{3} \hat{ρ} + 7.52 z^{2} \hat{z}$
Let $\hat{n}$ be the outward unit normal to this cylinder and evaluate ,
$| \int_{t o p} \vec{𝔉} \cdot \hat{n} d A |$
over the top surface of the cylinder. |type="()"} -a) 1.304E+03 -b) 1.579E+03 +c) 1.914E+03 -d) 2.318E+03 -e) 2.809E+03

{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,
$\vec{𝔉} = (2.07 + 2.87 z) ρ^{2} \hat{ρ} + 9.56 z^{3} \hat{z}$
Let $\hat{n}$ be the outward unit normal to this cylinder and evaluate ,
$| \int_{t o p} \vec{𝔉} \cdot \hat{n} d A |$
over the top surface of the cylinder. |type="()"} -a) 7.933E+02 +b) 9.611E+02 -c) 1.164E+03 -d) 1.411E+03 -e) 1.709E+03

{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,
$\vec{𝔉} = (2.17 + 1.5 z) ρ^{2} \hat{ρ} + 8.75 z^{2} \hat{z}$
Let $\hat{n}$ be the outward unit normal to this cylinder and evaluate ,
$| \int_{t o p} \vec{𝔉} \cdot \hat{n} d A |$
over the top surface of the cylinder. |type="()"} -a) 3.630E+02 +b) 4.398E+02 -c) 5.329E+02 -d) 6.456E+02 -e) 7.821E+02

{A cylinder of radius, r=3, 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,
$\vec{𝔉} = (2.28 + 1.72 z) ρ^{3} \hat{ρ} + 7.33 z^{3} \hat{z}$
Let $\hat{n}$ be the outward unit normal to this cylinder and evaluate ,
$| \int_{t o p} \vec{𝔉} \cdot \hat{n} d A |$
over the top surface of the cylinder. |type="()"} -a) 2.597E+03 -b) 3.147E+03 -c) 3.812E+03 -d) 4.619E+03 +e) 5.596E+03

{A cylinder of radius, r=3, 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,
$\vec{𝔉} = (2.04 + 1.66 z) ρ^{2} \hat{ρ} + 7.54 z^{2} \hat{z}$
Let $\hat{n}$ be the outward unit normal to this cylinder and evaluate ,
$| \int_{t o p} \vec{𝔉} \cdot \hat{n} d A |$
over the top surface of the cylinder. |type="()"} +a) 8.528E+02 -b) 1.033E+03 -c) 1.252E+03 -d) 1.516E+03 -e) 1.837E+03

{A cylinder of radius, r=3, 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,
$\vec{𝔉} = (2.21 + 1.16 z) ρ^{2} \hat{ρ} + 7.96 z^{3} \hat{z}$
Let $\hat{n}$ be the outward unit normal to this cylinder and evaluate ,
$| \int_{t o p} \vec{𝔉} \cdot \hat{n} d A |$
over the top surface of the cylinder. |type="()"} -a) 3.417E+03 -b) 4.140E+03 -c) 5.016E+03 +d) 6.077E+03 -e) 7.362E+03

{A cylinder of radius, r=3, 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,
$\vec{𝔉} = (2.12 + 1.68 z) ρ^{2} \hat{ρ} + 8.83 z^{3} \hat{z}$
Let $\hat{n}$ be the outward unit normal to this cylinder and evaluate ,
$| \int_{t o p} \vec{𝔉} \cdot \hat{n} d A |$
over the top surface of the cylinder. |type="()"} -a) 4.593E+03 -b) 5.564E+03 +c) 6.741E+03 -d) 8.167E+03 -e) 9.894E+03

{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,
$\vec{𝔉} = (2.05 + 2.05 z) ρ^{2} \hat{ρ} + 9.62 z^{3} \hat{z}$
Let $\hat{n}$ be the outward unit normal to this cylinder and evaluate ,
$| \int_{t o p} \vec{𝔉} \cdot \hat{n} d A |$
over the top surface of the cylinder. |type="()"} -a) 4.489E+02 -b) 5.438E+02 -c) 6.589E+02 -d) 7.983E+02 +e) 9.671E+02

{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,
$\vec{𝔉} = (1.93 + 2.31 z) ρ^{3} \hat{ρ} + 7.21 z^{2} \hat{z}$
Let $\hat{n}$ be the outward unit normal to this cylinder and evaluate ,
$| \int_{t o p} \vec{𝔉} \cdot \hat{n} d A |$
over the top surface of the cylinder. |type="()"} -a) 6.731E+02 +b) 8.154E+02 -c) 9.879E+02 -d) 1.197E+03 -e) 1.450E+03

{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,
$\vec{𝔉} = (2.24 + 1.11 z) ρ^{3} \hat{ρ} + 8.16 z^{3} \hat{z}$
Let $\hat{n}$ be the outward unit normal to this cylinder and evaluate ,
$| \int_{t o p} \vec{𝔉} \cdot \hat{n} d A |$
over the top surface of the cylinder. |type="()"} +a) 2.769E+03 -b) 3.354E+03 -c) 4.064E+03 -d) 4.923E+03 -e) 5.965E+03

{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,
$\vec{𝔉} = (1.96 + 2.52 z) ρ^{2} \hat{ρ} + 7.11 z^{2} \hat{z}$
Let $\hat{n}$ be the outward unit normal to this cylinder and evaluate ,
$| \int_{t o p} \vec{𝔉} \cdot \hat{n} d A |$
over the top surface of the cylinder. |type="()"} -a) 4.522E+02 -b) 5.478E+02 -c) 6.637E+02 +d) 8.041E+02 -e) 9.742E+02

{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,
$\vec{𝔉} = (1.86 + 2.43 z) ρ^{2} \hat{ρ} + 9.75 z^{2} \hat{z}$
Let $\hat{n}$ be the outward unit normal to this cylinder and evaluate ,
$| \int_{t o p} \vec{𝔉} \cdot \hat{n} d A |$
over the top surface of the cylinder. |type="()"} -a) 6.201E+02 -b) 7.513E+02 -c) 9.102E+02 +d) 1.103E+03 -e) 1.336E+03

{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,
$\vec{𝔉} = (2.24 + 2.08 z) ρ^{2} \hat{ρ} + 8.93 z^{3} \hat{z}$
Let $\hat{n}$ be the outward unit normal to this cylinder and evaluate ,
$| \int_{t o p} \vec{𝔉} \cdot \hat{n} d A |$
over the top surface of the cylinder. |type="()"} -a) 1.704E+03 -b) 2.064E+03 -c) 2.501E+03 +d) 3.030E+03 -e) 3.671E+03

{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,
$\vec{𝔉} = (1.89 + 1.31 z) ρ^{3} \hat{ρ} + 8.35 z^{2} \hat{z}$
Let $\hat{n}$ be the outward unit normal to this cylinder and evaluate ,
$| \int_{t o p} \vec{𝔉} \cdot \hat{n} d A |$
over the top surface of the cylinder. |type="()"} -a) 5.311E+02 -b) 6.434E+02 -c) 7.795E+02 +d) 9.444E+02 -e) 1.144E+03

{A cylinder of radius, r=3, 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,
$\vec{𝔉} = (2.37 + 2.6 z) ρ^{2} \hat{ρ} + 8.84 z^{3} \hat{z}$
Let $\hat{n}$ be the outward unit normal to this cylinder and evaluate ,
$| \int_{t o p} \vec{𝔉} \cdot \hat{n} d A |$
over the top surface of the cylinder. |type="()"} -a) 1.362E+03 -b) 1.650E+03 +c) 2.000E+03 -d) 2.423E+03 -e) 2.935E+03

{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,
$\vec{𝔉} = (2.45 + 2.26 z) ρ^{2} \hat{ρ} + 8.92 z^{3} \hat{z}$
Let $\hat{n}$ be the outward unit normal to this cylinder and evaluate ,
$| \int_{t o p} \vec{𝔉} \cdot \hat{n} d A |$
over the top surface of the cylinder. |type="()"} -a) 5.043E+02 -b) 6.109E+02 -c) 7.402E+02 +d) 8.967E+02 -e) 1.086E+03

{A cylinder of radius, r=3, 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,
$\vec{𝔉} = (1.88 + 1.29 z) ρ^{2} \hat{ρ} + 7.2 z^{2} \hat{z}$
Let $\hat{n}$ be the outward unit normal to this cylinder and evaluate ,
$| \int_{t o p} \vec{𝔉} \cdot \hat{n} d A |$
over the top surface of the cylinder. |type="()"} -a) 1.248E+03 -b) 1.512E+03 +c) 1.832E+03 -d) 2.220E+03 -e) 2.689E+03

{A cylinder of radius, r=3, 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,
$\vec{𝔉} = (2.44 + 2.86 z) ρ^{2} \hat{ρ} + 7.42 z^{3} \hat{z}$
Let $\hat{n}$ be the outward unit normal to this cylinder and evaluate ,
$| \int_{t o p} \vec{𝔉} \cdot \hat{n} d A |$
over the top surface of the cylinder. |type="()"} +a) 5.664E+03 -b) 6.863E+03 -c) 8.314E+03 -d) 1.007E+04 -e) 1.220E+04 </quiz>

c19ElectricPotentialField_SurfaceIntegral_v1

<quiz display="simple"> {A cylinder of radius, r=3, 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,
$\vec{𝔉} = (2.35 + 2.57 z) ρ^{3} \hat{ρ} + 7.45 z^{3} \hat{z}$
Let $\hat{n}$ be the outward unit normal to this cylinder and evaluate ,
$| \int_{s i d e} \vec{𝔉} \cdot \hat{n} d A |$
over the curved side surface of the cylinder.} -a) 2.221E+03 -b) 2.690E+03 -c) 3.259E+03 -d) 3.949E+03 +e) 4.784E+03 </quiz> Template:Cot

===2===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=3, 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,  <br> <math>\vec\mathfrak F = (2.05+2.59z)\rho^2\hat\rho +7.4z^2\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the curved side surface of the cylinder.}
-a) 6.457E+02  
-b) 7.823E+02  
-c) 9.477E+02  
-d) 1.148E+03  
+e) 1.391E+03  
===3===
{<!--c19ElectricPotentialField_SurfaceIntegral_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,  <br> <math>\vec\mathfrak F = (2.12+1.85z)\rho^3\hat\rho +8.88z^2\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the curved side surface of the cylinder.}
+a) 8.525E+02  
-b) 1.033E+03  
-c) 1.251E+03  
-d) 1.516E+03  
-e) 1.837E+03  
===4===
{<!--c19ElectricPotentialField_SurfaceIntegral_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,  <br> <math>\vec\mathfrak F = (2+1.45z)\rho^2\hat\rho +8.02z^3\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the curved side surface of the cylinder.}
+a) 4.021E+02  
-b) 4.872E+02  
-c) 5.902E+02  
-d) 7.151E+02  
-e) 8.663E+02  
===5===
{<!--c19ElectricPotentialField_SurfaceIntegral_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,  <br> <math>\vec\mathfrak F = (2.14+2.8z)\rho^2\hat\rho +9.94z^2\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the curved side surface of the cylinder.}
-a) 2.420E+02  
-b) 2.931E+02  
-c) 3.551E+02  
+d) 4.303E+02  
-e) 5.213E+02  
===6===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=3, 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,  <br> <math>\vec\mathfrak F = (1.85+1.33z)\rho^3\hat\rho +7.52z^2\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the curved side surface of the cylinder.}
-a) 2.622E+03  
-b) 3.177E+03  
-c) 3.849E+03  
-d) 4.663E+03  
+e) 5.649E+03  
===7===
{<!--c19ElectricPotentialField_SurfaceIntegral_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,  <br> <math>\vec\mathfrak F = (2.07+2.87z)\rho^2\hat\rho +9.56z^3\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the curved side surface of the cylinder.}
+a) 4.162E+02  
-b) 5.042E+02  
-c) 6.109E+02  
-d) 7.401E+02  
-e) 8.967E+02  
===8===
{<!--c19ElectricPotentialField_SurfaceIntegral_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,  <br> <math>\vec\mathfrak F = (2.17+1.5z)\rho^2\hat\rho +8.75z^2\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the curved side surface of the cylinder.}
-a) 2.454E+02  
-b) 2.973E+02  
-c) 3.601E+02  
+d) 4.363E+02  
-e) 5.286E+02  
===9===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=3, 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,  <br> <math>\vec\mathfrak F = (2.28+1.72z)\rho^3\hat\rho +7.33z^3\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the curved side surface of the cylinder.}
-a) 3.232E+03  
-b) 3.915E+03  
-c) 4.743E+03  
-d) 5.747E+03  
+e) 6.962E+03  
===10===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=3, 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,  <br> <math>\vec\mathfrak F = (2.04+1.66z)\rho^2\hat\rho +7.54z^2\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the curved side surface of the cylinder.}
-a) 9.431E+02  
-b) 1.143E+03  
+c) 1.384E+03  
-d) 1.677E+03  
-e) 2.032E+03  
===11===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=3, 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,  <br> <math>\vec\mathfrak F = (2.21+1.16z)\rho^2\hat\rho +7.96z^3\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the curved side surface of the cylinder.}
-a) 1.533E+03  
-b) 1.857E+03  
+c) 2.250E+03  
-d) 2.725E+03  
-e) 3.302E+03  
===12===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=3, 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,  <br> <math>\vec\mathfrak F = (2.12+1.68z)\rho^2\hat\rho +8.83z^3\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the curved side surface of the cylinder.}
+a) 2.158E+03  
-b) 2.614E+03  
-c) 3.167E+03  
-d) 3.837E+03  
-e) 4.649E+03  
===13===
{<!--c19ElectricPotentialField_SurfaceIntegral_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,  <br> <math>\vec\mathfrak F = (2.05+2.05z)\rho^2\hat\rho +9.62z^3\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the curved side surface of the cylinder.}
-a) 2.318E+02  
-b) 2.808E+02  
-c) 3.402E+02  
+d) 4.122E+02  
-e) 4.994E+02  
===14===
{<!--c19ElectricPotentialField_SurfaceIntegral_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,  <br> <math>\vec\mathfrak F = (1.93+2.31z)\rho^3\hat\rho +7.21z^2\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the curved side surface of the cylinder.}
-a) 6.546E+02  
-b) 7.931E+02  
-c) 9.609E+02  
+d) 1.164E+03  
-e) 1.410E+03  
===15===
{<!--c19ElectricPotentialField_SurfaceIntegral_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,  <br> <math>\vec\mathfrak F = (2.24+1.11z)\rho^3\hat\rho +8.16z^3\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the curved side surface of the cylinder.}
-a) 9.205E+02  
-b) 1.115E+03  
+c) 1.351E+03  
-d) 1.637E+03  
-e) 1.983E+03  
===16===
{<!--c19ElectricPotentialField_SurfaceIntegral_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,  <br> <math>\vec\mathfrak F = (1.96+2.52z)\rho^2\hat\rho +7.11z^2\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the curved side surface of the cylinder.}
-a) 4.027E+02  
-b) 4.879E+02  
+c) 5.911E+02  
-d) 7.162E+02  
-e) 8.676E+02  
===17===
{<!--c19ElectricPotentialField_SurfaceIntegral_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,  <br> <math>\vec\mathfrak F = (1.86+2.43z)\rho^2\hat\rho +9.75z^2\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the curved side surface of the cylinder.}
+a) 5.610E+02  
-b) 6.796E+02  
-c) 8.234E+02  
-d) 9.975E+02  
-e) 1.209E+03  
===18===
{<!--c19ElectricPotentialField_SurfaceIntegral_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,  <br> <math>\vec\mathfrak F = (2.24+2.08z)\rho^2\hat\rho +8.93z^3\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the curved side surface of the cylinder.}
-a) 3.799E+02  
-b) 4.603E+02  
-c) 5.576E+02  
+d) 6.756E+02  
-e) 8.185E+02  
===19===
{<!--c19ElectricPotentialField_SurfaceIntegral_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,  <br> <math>\vec\mathfrak F = (1.89+1.31z)\rho^3\hat\rho +8.35z^2\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the curved side surface of the cylinder.}
-a) 6.411E+02  
-b) 7.767E+02  
-c) 9.410E+02  
+d) 1.140E+03  
-e) 1.381E+03  
===20===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=3, 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,  <br> <math>\vec\mathfrak F = (2.37+2.6z)\rho^2\hat\rho +8.84z^3\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the curved side surface of the cylinder.}
-a) 7.465E+02  
-b) 9.044E+02  
-c) 1.096E+03  
-d) 1.327E+03  
+e) 1.608E+03  
===21===
{<!--c19ElectricPotentialField_SurfaceIntegral_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,  <br> <math>\vec\mathfrak F = (2.45+2.26z)\rho^2\hat\rho +8.92z^3\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the curved side surface of the cylinder.}
-a) 3.356E+02  
-b) 4.066E+02  
+c) 4.926E+02  
-d) 5.968E+02  
-e) 7.230E+02  
===22===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=3, 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,  <br> <math>\vec\mathfrak F = (1.88+1.29z)\rho^2\hat\rho +7.2z^2\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the curved side surface of the cylinder.}
-a) 1.579E+03  
+b) 1.914E+03  
-c) 2.318E+03  
-d) 2.809E+03  
-e) 3.403E+03  
===23===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=3, 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,  <br> <math>\vec\mathfrak F = (2.44+2.86z)\rho^2\hat\rho +7.42z^3\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the curved side surface of the cylinder.}
-a) 1.692E+03  
-b) 2.050E+03  
+c) 2.484E+03  
-d) 3.009E+03  
-e) 3.645E+03

Template:Cob

c19ElectricPotentialField_SurfaceIntegral_v1

<quiz display="simple"> {A cylinder of radius, r=3, 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,
$\vec{𝔉} = (2.35 + 2.57 z) ρ^{3} \hat{ρ} + 7.45 z^{3} \hat{z}$
Let $\hat{n}$ be the outward unit normal to this cylinder and evaluate ,
$| \oint \vec{𝔉} \cdot \hat{n} d A |$
over the entire surface of the cylinder.} -a) 4.59E+03 -b) 5.56E+03 -c) 6.73E+03 +d) 8.15E+03 -e) 9.88E+03 </quiz> Template:Cot

===2===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=3, 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,  <br> <math>\vec\mathfrak F = (2.05+2.59z)\rho^2\hat\rho +7.4z^2\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the entire surface of the cylinder.}
-a) 6.46E+02  
-b) 7.82E+02  
-c) 9.48E+02  
-d) 1.15E+03  
+e) 1.39E+03  
===3===
{<!--c19ElectricPotentialField_SurfaceIntegral_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,  <br> <math>\vec\mathfrak F = (2.12+1.85z)\rho^3\hat\rho +8.88z^2\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the entire surface of the cylinder.}
-a) 3.96E+02  
-b) 4.79E+02  
-c) 5.81E+02  
-d) 7.04E+02  
+e) 8.53E+02  
===4===
{<!--c19ElectricPotentialField_SurfaceIntegral_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,  <br> <math>\vec\mathfrak F = (2+1.45z)\rho^2\hat\rho +8.02z^3\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the entire surface of the cylinder.}
-a) 1.13E+03  
-b) 1.37E+03  
-c) 1.66E+03  
+d) 2.01E+03  
-e) 2.44E+03  
===5===
{<!--c19ElectricPotentialField_SurfaceIntegral_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,  <br> <math>\vec\mathfrak F = (2.14+2.8z)\rho^2\hat\rho +9.94z^2\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the entire surface of the cylinder.}
-a) 2.93E+02  
-b) 3.55E+02  
+c) 4.30E+02  
-d) 5.21E+02  
-e) 6.32E+02  
===6===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=3, 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,  <br> <math>\vec\mathfrak F = (1.85+1.33z)\rho^3\hat\rho +7.52z^2\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the entire surface of the cylinder.}
-a) 3.18E+03  
-b) 3.85E+03  
-c) 4.66E+03  
+d) 5.65E+03  
-e) 6.84E+03  
===7===
{<!--c19ElectricPotentialField_SurfaceIntegral_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,  <br> <math>\vec\mathfrak F = (2.07+2.87z)\rho^2\hat\rho +9.56z^3\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the entire surface of the cylinder.}
-a) 1.59E+03  
-b) 1.93E+03  
+c) 2.34E+03  
-d) 2.83E+03  
-e) 3.43E+03  
===8===
{<!--c19ElectricPotentialField_SurfaceIntegral_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,  <br> <math>\vec\mathfrak F = (2.17+1.5z)\rho^2\hat\rho +8.75z^2\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the entire surface of the cylinder.}
-a) 3.60E+02  
+b) 4.36E+02  
-c) 5.29E+02  
-d) 6.40E+02  
-e) 7.76E+02  
===9===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=3, 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,  <br> <math>\vec\mathfrak F = (2.28+1.72z)\rho^3\hat\rho +7.33z^3\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the entire surface of the cylinder.}
-a) 1.50E+04  
+b) 1.82E+04  
-c) 2.20E+04  
-d) 2.66E+04  
-e) 3.23E+04  
===10===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=3, 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,  <br> <math>\vec\mathfrak F = (2.04+1.66z)\rho^2\hat\rho +7.54z^2\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the entire surface of the cylinder.}
-a) 9.43E+02  
-b) 1.14E+03  
+c) 1.38E+03  
-d) 1.68E+03  
-e) 2.03E+03  
===11===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=3, 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,  <br> <math>\vec\mathfrak F = (2.21+1.16z)\rho^2\hat\rho +7.96z^3\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the entire surface of the cylinder.}
-a) 6.69E+03  
-b) 8.10E+03  
-c) 9.81E+03  
-d) 1.19E+04  
+e) 1.44E+04  
===12===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=3, 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,  <br> <math>\vec\mathfrak F = (2.12+1.68z)\rho^2\hat\rho +8.83z^3\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the entire surface of the cylinder.}
-a) 1.29E+04  
+b) 1.56E+04  
-c) 1.89E+04  
-d) 2.30E+04  
-e) 2.78E+04  
===13===
{<!--c19ElectricPotentialField_SurfaceIntegral_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,  <br> <math>\vec\mathfrak F = (2.05+2.05z)\rho^2\hat\rho +9.62z^3\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the entire surface of the cylinder.}
-a) 1.09E+03  
-b) 1.32E+03  
-c) 1.60E+03  
-d) 1.94E+03  
+e) 2.35E+03  
===14===
{<!--c19ElectricPotentialField_SurfaceIntegral_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,  <br> <math>\vec\mathfrak F = (1.93+2.31z)\rho^3\hat\rho +7.21z^2\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the entire surface of the cylinder.}
-a) 5.40E+02  
-b) 6.55E+02  
-c) 7.93E+02  
-d) 9.61E+02  
+e) 1.16E+03  
===15===
{<!--c19ElectricPotentialField_SurfaceIntegral_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,  <br> <math>\vec\mathfrak F = (2.24+1.11z)\rho^3\hat\rho +8.16z^3\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the entire surface of the cylinder.}
-a) 4.69E+03  
-b) 5.69E+03  
+c) 6.89E+03  
-d) 8.35E+03  
-e) 1.01E+04  
===16===
{<!--c19ElectricPotentialField_SurfaceIntegral_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,  <br> <math>\vec\mathfrak F = (1.96+2.52z)\rho^2\hat\rho +7.11z^2\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the entire surface of the cylinder.}
+a) 5.91E+02  
-b) 7.16E+02  
-c) 8.68E+02  
-d) 1.05E+03  
-e) 1.27E+03  
===17===
{<!--c19ElectricPotentialField_SurfaceIntegral_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,  <br> <math>\vec\mathfrak F = (1.86+2.43z)\rho^2\hat\rho +9.75z^2\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the entire surface of the cylinder.}
-a) 4.63E+02  
+b) 5.61E+02  
-c) 6.80E+02  
-d) 8.23E+02  
-e) 9.98E+02  
===18===
{<!--c19ElectricPotentialField_SurfaceIntegral_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,  <br> <math>\vec\mathfrak F = (2.24+2.08z)\rho^2\hat\rho +8.93z^3\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the entire surface of the cylinder.}
-a) 3.13E+03  
-b) 3.79E+03  
-c) 4.59E+03  
-d) 5.56E+03  
+e) 6.74E+03  
===19===
{<!--c19ElectricPotentialField_SurfaceIntegral_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,  <br> <math>\vec\mathfrak F = (1.89+1.31z)\rho^3\hat\rho +8.35z^2\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the entire surface of the cylinder.}
-a) 9.41E+02  
+b) 1.14E+03  
-c) 1.38E+03  
-d) 1.67E+03  
-e) 2.03E+03  
===20===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=3, 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,  <br> <math>\vec\mathfrak F = (2.37+2.6z)\rho^2\hat\rho +8.84z^3\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the entire surface of the cylinder.}
-a) 4.63E+03  
+b) 5.61E+03  
-c) 6.79E+03  
-d) 8.23E+03  
-e) 9.97E+03  
===21===
{<!--c19ElectricPotentialField_SurfaceIntegral_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,  <br> <math>\vec\mathfrak F = (2.45+2.26z)\rho^2\hat\rho +8.92z^3\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the entire surface of the cylinder.}
-a) 1.29E+03  
-b) 1.56E+03  
-c) 1.89E+03  
+d) 2.29E+03  
-e) 2.77E+03  
===22===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=3, 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,  <br> <math>\vec\mathfrak F = (1.88+1.29z)\rho^2\hat\rho +7.2z^2\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the entire surface of the cylinder.}
-a) 1.08E+03  
-b) 1.30E+03  
-c) 1.58E+03  
+d) 1.91E+03  
-e) 2.32E+03  
===23===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=3, 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,  <br> <math>\vec\mathfrak F = (2.44+2.86z)\rho^2\hat\rho +7.42z^3\hat z </math><br> Let <math>\hat n</math> be the outward unit normal to this cylinder and evaluate ,<br> <math>\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,</math><br>over the entire surface of the cylinder.}
-a) 9.41E+03  
-b) 1.14E+04  
+c) 1.38E+04  
-d) 1.67E+04  
-e) 2.03E+04

Template:Cob

Physics equations/19-Electric Potential and Electric Field/Q:SurfaceIntegralsCalculus/testbank

c19ElectricPotentialField_SurfaceIntegral_v1

c19ElectricPotentialField_SurfaceIntegral_v1

c19ElectricPotentialField_SurfaceIntegral_v1

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