A $415$ V$,2$ pole, $3$ phase, $50$ Hz$,$ star connected, non$-$salient pole synchronous motor has synchronous reactance of $2$ per phase and negligible stator resistance. At a particular field excitation, it draws $20$ A at unity power factor from a $415$ V$,3$ phase, $50$ Hz supply. The mechanical load on the motor is now increased till the stator current is equal to $50$ A$.$ The field excitation remains unchanged. Determine:

$($a$)$ the per phase open circuit voltage E$0$

$($b$)$ the developed power for the new operating condition and corresponding power factor.