For a class demonstration, a professor needs to create a solenoid using copper wire with diameter $d=0\mathrm{.}500mm.$ The resistance of the solenoid needs to be $3\mathrm{.}30$ and the magnetic field of the solenoid with a current of $3\mathrm{.}70$ A should be $5\mathrm{.}50{10}^{-2}T.$ Assume the resistance does not change significantly with temperature, and the radius of the solenoid is $1\mathrm{.}00$ cm $.($Use $=1\mathrm{.}70{10}^{-8}*m$ for the resistivity of copper.$)$

$($a$)$ How many turns of wire does the professor need?

turns

$($b$)$ Determine the required length $($in cm $)$ of the solenoid.

$$ cm