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

question 24

Short Answer

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.

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