Question

If is the unit sphere centered at the origin, oriented outward and the flux integral ∫∫ₛ F . dA is zero, then F(x,y,z) is perpedicular to r = (x,y,z) at every point of S.

a. true
b. false

Answer 1

It is false that F(x,y,z) must be perpendicular to r at every point on the unit sphere S merely because the flux integral over S is zero. Zero flux can occur under various conditions, not just perpendicularity between F(x,y,z) and r.

Step-by-step explanation:

The question is about whether the vector field F(x,y,z) being perpendicular to the position vector r = (x,y,z) at every point on the unit sphere S, assuming that the flux integral ∫∫ₛs F . dA is zero. It is false that F(x,y,z) must be perpendicular to r at every point on S just because the flux integral is zero.

The flux integral being zero indicates that the net amount of the field passing through the surface is zero, which can happen in several scenarios, not just when F is perpendicular to r. For instance, inside a magnetic field like that of a toroid, the magnetic field lines are tangent to any circle within the same plane, but the net flux through a surface within this field could still be zero without the field being perpendicular to the position vector at every point.