In a certain region, a charge distribution exists that is spherically symmetric but
non-uniform. When a positive point charge q is located at (r,θ,φ) near this charge
distribution, there is a resulting electric potential energy for the system given by:
U(r,θ,φ) = ρ(naught)a^2q/18ε(naught)(1-3((r/a)^2) + 2((r/a)^3) for r ≤ a
and 0 for r ≥ a
where ρo is a constant having units of C/m^3 (volume charge density) and a is a
constant having units of m. Note that there is no θ or φ dependence here since the
charge distribution is spherically symmetric.
Determine the electric force F
exerted on charge q as a function of its location
a) r ≤ a
b) r > a
Check that the units of your answer make sense. (Show your work.)
The Attempt at a Solution'
I derived ∇U=dU/dr r from the spherical coordinate gradients and since there is no dependence on phi and theta we will just be using the r vector.
d/drρ(naught)a^2q/18ε(naught)(1-3((r/a)^2) + 2((r/a)^3) r
= 6r(r-a)/a^3 r
This is where I don't know what to do, how would I express my answer?