Problem 1:

A sphere of radius R has a spherically symmetric charge density given by ρ(r) = ρ₀( 1 - r/R).
(a)  Find the total charge Q in the sphere in terms of ρ₀ and R.
(b)  Find the electric field E both inside and outside the charge distribution using Gauss's law.
(c)  Compute the electrostatic energy stored in the configuration.

Problem 2:

A sphere of radius R has a uniform volume charge density ρ in addition to a surface chare density σ = σ₀sin²θ.  Find the potential Φ(r) everywhere outside of the sphere.

Hint:
P₀(x) = 1
P₁(x) = x
P₂(x) = (3x² - 1)/2
P₃(x) = (5x³ - 3x)/2

Problem 3:

Submit this problem on Canvas as Assignment 2.  If you used an AI as a Socratic tutor, submit a copy of your session leading to your solution and reflect on your session.  If you did not need any help or worked with another student, explain your reasoning, do not just write down formulas. 

An infinitely long charged wire of radius r₀ is parallel to an infinite conducting plane, at a distance h (h >> r₀) from the surface.  The potential difference between the wire and the surface is ∆V.  Derive a formula for the magnitude of the electric field just above the surface below the wire.

Problem 4:

A uniform dielectric round plate has radius R and thickness d (R >> d).   It is uniformly polarized with the polarization P parallel to the plate.  Find the electric field generated by the polarization at the center position of the plate.