Question

Using Gauss’s theorem, calculate the flux of the vector field F =xˆi + y ˆj + zkˆ through the surface of a cylinder of radius A and height H, which has its axis along the z-axis. The base of the cylinder is on the xy plane.

Answer 1

The Gauss’s theorem says

$${\rm flux}=\oiint {\bf F\cdot dA}=\intop \rm div{\bf F}dV$$ $$\rm div{\bf F}=\rm div(x{\bf i}+y{\bf j}+z{\bf k})=1+1+1=3.$$

Hence

$$\rm flux=3\intop dV=3V=3\pi a^2H.$$