Line charge creating induced emf and displacement current

[gsmtiger18]

Homework Statement

A long, straight wire has a line charge, λ, that varies in time according to: λ = λ₀ exp(-βt). A square wire loop of dimension a is located adjacent to the wire at a distance of a from the wire. Calculate expressions for the displacement current at the center of the wire loop and the magnetic flux through the loop.

a = the side length of the square wire loop.
λ₀ = the initial charge of the line charge at time t = 0
β = I assume some kind of constant. I haven't encountered this variable in a displacement current problem before.

Homework Equations

i_disp = ε₀ * dΦ_e/dt
ε = -dΦ_m/dt
Φ_m = ∫B ⋅ dA
Φ_e = ∫E ⋅ dA = q_enc / ∈₀
∫ E ⋅ dl = -dΦ_m / dt

The Attempt at a Solution

I'm pretty sure I can find an expression for the electric field first by integrating the equation for the time-varying charge, but I'm not sure how to set that up. I think I should integrate with respect to time and treat radius as a constant. From the electric field I can use Faraday's Law to find the emf, but again I'm unsure as to how exactly to set up the math.

 

[rude man]

What is the juxtaposition between the wire direction vector and the normal to the loop? I.e if the wire vector is in the x direction, is the loop normal in the y or the z direction?

Is "a" the distance from the loop's center to the perpendicular distance to the wire, or is "a" the perpendicular distance from the closer collinear loop segment to the wire?

A picture would help immensely.