What is the surface current density at a general point on a conducting sphere?

[RobTwox]

Homework Statement

Electric current of I amperes flows along the z-axis from (0, 0,-∞) to (0, 0, -a) and from there it spreads over a conducting sphere r = a in the -a_θ direction, comes to the point (0; 0; a) and goes to (0, 0, ∞) again along the z-axis. What is the surface current density at a general point on the sphere where z = h?

Homework Equations

surface current density = J_s = Δi / Δl ; l = length , i = current

The Attempt at a Solution

I am pretty lost with this one. If I use the above equation for surface current density it seems like I need to do this

∫ i / L dl = ∫ i / z dz from z = -a to a

I know this isn't the answer. Its to simple and my professor would never assign something as simple as this. I think I have some conceptual issues and I fear that my relevant equation is incorrect. I had to miss lecture due to a trip to the ER and I am relying on a friends notes, nice guy but less than exemplary student.

 

[RobTwox]

Homework Statement

Electric current of I amperes flows along the z-axis from (0, 0,-∞) to (0, 0, -a) and from there it spreads over a conducting sphere r = a in the -a_θ direction, comes to the point (0; 0; a) and goes to (0, 0, ∞) again along the z-axis. What is the surface current density at a general point on the sphere where z = h?

Homework Equations

surface current density = J_s = Δi / Δl ; l = length , i = current

The Attempt at a Solution

I am pretty lost with this one. If I use the above equation for surface current density it seems like I need to do this

∫ i / L dl = ∫ i / z dz from z = -a to a

I know this isn't the answer. Its to simple and my professor would never assign something as simple as this. I think I have some conceptual issues and I fear that my relevant equation is incorrect. I had to miss lecture due to a trip to the ER and I am relying on a friends notes, nice guy but less than exemplary student.

I have a new thought. I looked up surface current density and found that the units are A / m^2. if this is true then this boils down to current per area.

So, could I say (and be correct) that the current I will be uniformly distributed and constant. Then to write a general expression for current density is it just
J_s = i / (4 pi h^2).

This still seems too easy.