Question

A vector field is defined as f⃗(x,y,z) = x/[x²+y²+z²]^(3/2) î + y/[x²+y²+z²]^(3/2) ĵ + z/[x²+y²+z²]^(3/2) k̂. The surface integral ∬f⃗·dS⃗ evaluated over the inner and outer surfaces of a spherical shell formed by two concentric spheres with origin as centre, and internal and external radii of 1 and 2 respectively, is

Accepted Answer

The correct answer is (A) 0. f⃗ = r̂/r² is a radial inverse-square field whose divergence is zero everywhere except at the origin. Since the origin is excluded from the annular shell region (1≤r≤2), div(f⃗)=0 throughout that region, so by the divergence theorem the net flux out of the shell's total boundary (outer surface flux minus inner surface flux) equals ∫div(f⃗)dV = 0.

Explanation