Chapter 14.8, Problem 46E

Inverse square fields are special Let F be a radial field F = r/|r|^p, where p is a real number and r = 〈x, y, z〉. With p = 3, F is an inverse square field.

a.    Show that the net flux across a sphere centered at the origin is independent of the radius of the sphere only for p = 3.

b.    Explain the observation in part (a) by finding the flux of F = r/|r|^p across the boundaries of a spherical box {(ρ, ϕ, θ): a ≤ ρ ≤ b, ϕ₁ ≤ ϕ ≤ ϕ₂, θ₁ ≤ θ ≤θ₂} for various values of p.