Chapter 17.6, Problem 64E

Flux across a cylinder Let S be the cylinder x² + y² = a², for –L ≤ z ≤ L.

a.    Find the outward flux of the field F = 〈x, y, 0〉 across S.

b.    Find the outward flux of the field $F=\frac{\langle x,y,0\rangle }{{\left(x^2+y^2\right)}^{p/2}}=\frac{r}{{\left|r\right|}^p}$ across S. where |r| is the distance from the z-axis and p is a real number.

c.    In part (b), for what values of p is the outward flux finite as a → ∞ (with L fixed)?

d.    In part (b), for what values of p is the outward flux finite as L → ∞ (with a fixed)?