Final answer:
The flux of a vector field V = α/(x²+y²) through a rectangle in the xy-plane or yz-plane is zero because the field lines do not pass through these planes, making the angle between the vector field and the surface 90°.
Step-by-step explanation:
To calculate the flux of a vector field through a surface, we use the formula Φ = ∨ Φ ⋅ dA, where Φ is the vector field, and dA is the differential area vector, which is perpendicular to the surface and has a magnitude equal to the area of the differential element. In the case of your question, the vector field is given by V = α/(x²+y²) and points along the z-axis.
Flux Through the xy-Plane Rectangle: The differential area vector for a rectangle in the xy-plane will be perpendicular to the xy-plane, which means it will be in the z-direction. Since the vector field V also points in the z-direction, the flux would be the integral of V over the rectangle. However, when calculating the flux through a plane parallel to the vector field (like in the xy-plane for a field in the z direction), the flux will be zero since the angle between the vector field and area vector is 90°. The integral will thus evaluate to zero as well.
Flux Through the yz-Plane Rectangle: For a rectangle in the yz-plane, the differential area vector will be perpendicular to the yz-plane and will lie in the xy-plane. Since the vector field is parallel to the z-axis, there is also no component of the vector field through this rectangle, so the flux through this rectangle is again zero since the field lines do not pass through this plane.