Answer to Question #86238 in Field Theory for Ajay

Question

Using Gauss’ Theorem calculate the flux of the vector field F ˆ
i ˆ
j kˆ = x + y + z r through
the surface of a cylinder of radius A and height H, which has its axis along the z-axis
and the base of the cylinder is on the xy-plane.

Answer 1

Gauss' theorem states

$$\rm{Flux}=\int \vec F d\vec A=\intop \rm{div}\vec F dV$$

Since

$$\rm{div}\vec F=\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}=\frac{\partial x}{\partial x}+\frac{\partial y}{\partial y}+\frac{\partial z}{\partial z}=3$$

we obtain

$$\rm{Flux}=\intop \rm{div}\vec F dV=\intop 3 dV=3V=3 \pi A^2 H$$