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## Homework Statement

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

(r,θ,φ) for:

a) r ≤ a

b) r > a

Check that the units of your answer make sense. (Show your work.)

## Homework Equations

F=-∇U

∇U=dU/dr

**r**

## The Attempt at a Solution

'I derived ∇U=dU/dr

**r**from the spherical coordinate gradients and since there is no dependance on phi and theta we will just be using the

**r**vector.

therefore:

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?