Let C be the portion of the cylinder $x ^ { 2 } + y ^ { 2 } = R ^ { 2 }$ of fixed radius R with $\pi$/3 $\le$ $\theta$ $\le$ 2 $\pi$/3 and -a $\le$ z$\le$ a oriented outward for some positive number a.Let S be the portion of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = R ^ { 2 }$ with $\pi$/3 $\le$$\theta$$\le$2 $\pi$/3 and $\pi$/3 $\le$ $\varphi$ $\le$ 2 $\pi$/3 oriented outward.Determine the value of a for which the flux of $\vec { r }$ through each of these surfaces is equal in magnitude but opposite in sign for any choice of R.

Question

Let C be the portion of the cylinder $x ^ { 2 } + y ^ { 2 } = R ^ { 2 }$ of fixed radius R with  $\pi$/3 $\le$ $\theta$ $\le$ 2 $\pi$/3 and -a $\le$ z$\le$ a oriented outward for some positive number a.Let S be the portion of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = R ^ { 2 }$ with $\pi$/3 $\le$$\theta$$\le$2 $\pi$/3 and  $\pi$/3 $\le$ $\varphi$ $\le$ 2 $\pi$/3 oriented outward.Determine the value of a for which the flux of $\vec { r }$ through each of these surfaces is equal in magnitude but opposite in sign for any choice of R.

Quizplus · Accepted Answer

The Answer of Let C be the portion of the cylinder $x ^ { 2 } + y ^ { 2 } = R ^ { 2 }$ of fixed radius R with  $\pi$/3 $\le$ $\theta$ $\le$ 2 $\pi$/3 and -a $\le$ z$\le$ a oriented outward for some positive number a.Let S be the portion of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = R ^ { 2 }$ with $\pi$/3 $\le$$\theta$$\le$2 $\pi$/3 and  $\pi$/3 $\le$ $\varphi$ $\le$ 2 $\pi$/3 oriented outward.Determine the value of a for which the flux of $\vec { r }$ through each of these surfaces is equal in magnitude but opposite in sign for any choice of R.

Let C Be the Portion of the Cylinder x2+y2=R2x ^ { 2 } + y ^ { 2 } = R ^ { 2 }x2+y2=R2

Let C Be the Portion of the Cylinder $x ^ { 2 } + y ^ { 2 } = R ^ { 2 }$