Problems with electromagnetic fields inside matter

[valleo4]

Homework Statement

see attachment/picture
im having problems with fields inside matter. i get the left side of gaussian surfaces/ampere loops. but for example this first equation's right side i can't understand.

The Attempt at a Solution

i would think the right side should be
$I$
since its the total enclosed current by my ampere loop

but the book says
$I\frac{\pi s^2}{\pi R^2}$
it has something to do with the dependence of the current enclosed and the distance, which you don't have outside the cylinder. but i can't figure it out. can someone explain?

Homework Statement

 

[nickjer]

It says $s\leq R$. The loop that you draw is inside the wire, so you get a fraction of the total current. That fraction being the fraction of area of your loop compared to the total area of the wire. Outside the wire, you would just set $s=R$ and get back the total current you were talking about.