Question

Calculate the divergence of the following radial field. Express the result in terms of the position vector r and its length StartAbsoluteValue Bold r EndAbsoluteValue. FequalsStartFraction left angle x comma y comma z right angle Over x squared plus y squared plus z squared EndFraction equalsStartFraction Bold r Over StartAbsoluteValue Bold r EndAbsoluteValue squared EndFraction Choose the correct answer below. A. The divergence of F is 0. B. The divergence of F is StartFraction negative 2 Over StartAbsoluteValue Bold r EndAbsoluteValue Superscript 4 EndFraction . C. The divergence of F is StartFraction 1 Over StartAbsoluteValue Bold r EndAbsoluteValue squared EndFraction . D. The divergence of F is StartFraction negative 1 Over StartAbsoluteValue Bold r EndAbsoluteValue squared EndFraction

Answer 1

C. The divergence of F is StartFraction 1 Over StartAbsoluteValue Bold r EndAbsoluteValue squared EndFraction

∇•F = 1/|r|²

Explanation:

The position vector r = (x, y, z)

r = xi+yj+zk

|r| = √x²+y²+z²

|r|² = x²+y²+z²

Given the radial field F = r/|r|²

Divergence of the radial field is expressed as:

∇•F = {δ/δx i+ δ/δy j + δ/δy k} • {(r/|r|²)

∇•F = {δ/δx i+ δ/δy j + δ/δy k} • ² + yj/

∇•F = δ/δx(x/|r|²) + δ/δy(y/|r|²)+δ/δz(z/|r|²)

Check the attachment for the complete solution.