Electromagnetism

Question

Show that for a vector field that has the form E = F(r) r̂, the rotational is zero and the divergent is given by

DIV(E) = 2F(r) ⁄ r + dF⁄dr

Answer 1

Let g(r) be an antiderivative of F(r). Then grad[g(r) r̂] = g'(r) r̂ = F(r) r̂ = E, so E is a gradient, forcing curl E = 0. Now div E = 1/r² d/dr(r² F(r)), which, by the product rule, equals

1/r² (2r F(r) + r² dF/dr) = 2 F(r)/r + dF/dr

as desired.