Problem 1:

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 2:

Problem 3:

An electron at a distance d = 1 mm is projected parallel to a grounded perfectly conducting sheet with an energy of 100 electron volts.  Let the grounded conducting sheet lie in the xy-plane and the initial velocity of the electron point in the x-direction.  Let z(0) = d. 
(a)  Find the z-component v_z of the electron's velocity as a function of its distance z from the sheet.
(b)  Find the distance that the electron travels until it hits the plate.  Neglect the force of gravity.
(c)  Find the magnitude and direction of a magnetic field parallel to the surface of the plate and perpendicular to the electron velocity that keeps the electron from hitting the plate.
Give numerical answers for parts (b) and (c).
Hint: z = dsin²θ may be a useful change of variable.  ∫sin²θ dθ = θ/2 - sin(2θ)/4.

Problem 4:

Assume in a region of space the electric field E lies in the (x,y) plane and is rotating with frequency ω counterclockwise about the z-axis.  At t = 0, E = E₀ i.  Assume ω is small and induced magnetic fields can be neglected.
(a)  How could you produce such an electric field?
(b)  Find general solution for x(t), y(t), v_x(t) and v_y(t) for a particle with mass m and charge q located in that region of space?
(c)  Is it possible for the particle to stay confined in that region of space?  What initial conditions are required?

Problem 5:

The displacement vector from electric dipole p₁ to dipole p₂ is r.
(a)  Calculate the electric potential energy W.
(b)  Calculate the force F₂₁ that p₁ exerts on p₂.
(c)  Calculate the torque τ₁₂ that p₁ exerts on p₂.
(d)  Let p₁ = (10^-9 Cm)k be located at the origin and p₂ = (10^-9 Cm)i at r = (3 m)i + (4 m)k.
Provide numeric answers for  F₂₁ and τ₁₂.