How Does the Feynman Paradox Apply to Solenoid Flux Calculations?

[tzzzsh]

I am currently going through a friend's Mechanics II notes and homework before I take the course at a different university next semester. I have a few problems that I am having trouble understanding and am posting here for help.

1. Homework Statement
1. A solenoid is on, Flux is Φ.
2. The solenoid is turned off. A charge of Δmv = eΦ/2πr is observed.
3. Find A(initial) assuming A(final) = 0 from this data.
4. Show that A(initial) is consistent with ∫A*dx = ∫d²x∇xA = ∫d²x*B = Φ

Homework Equations

Φ = ∫d²x*B
∇xE = =d/dt∇xA = -dB/dt
dmv/dt-eE = e dA/dt

The Attempt at a Solution

3. dmv/dt-eE = e*dA/dt => mv-e∫E*dx = eA => mv/e-A = ∫E*dx = -dΦ/dt => A(initial) - mv/e = A final = 0 => A(inital) = mv/e
A(initial) = mΦ/2πr
4. I'm not sure where to start here.

 

[MathematicalPhysicist]

First thing the Lorentz force is $F= q(E+v\times B)$;
and in your three you are missing the curl of A, and btw it should be $\nabla \times A = B$, without the minus sign as you wrote over there.

 

[MathematicalPhysicist]

Another thing, your integral should be with respect to $dt$, i.e. $\int curl A dt$.