Answer
$B(r) = \frac{\mu_0~i~r}{2\pi~c^2}$
Work Step by Step
We can find an expression for the current enclosed in a loop of radius $r$ where $r \lt c$:
$i_{enc} = \frac{\pi~r^2}{\pi~c^2}~i = \frac{i~r^2}{c^2}$
We can find an expression for $B(r)$:
$\int~B\cdot ds = \mu_0~i_{enc}$
$B\cdot 2\pi~r = \frac{\mu_0~i~r^2}{c^2}$
$B\cdot 2\pi = \frac{\mu_0~i~r}{c^2}$
$B(r) = \frac{\mu_0~i~r}{2\pi~c^2}$
