A synchronous motor operates at $$0.8$$ $$p.f.$$ lagging. If the field current of the motor is continuously increased

$$(1)$$ the power factor decreases upto a certain value of field current and thereafter it increases
$$(2)$$ the armature current increases upto a certain value of field current and thereafter it decreases
$$(3)$$ the power factor increases upto a certain value of field current and thereafter it decreases
$$(4)$$ the armature current decreases upto a certain value of field current and thereafter it increases. From these the correct one is

A three phase alternator has negligible stator resistance. A short circuit test is conducted on this alternator. At a particular speed a field current of $${{\rm I}_f}$$ is required to drive the rated armature current. If the speed of the alternator is reduced to half, the field current required to maintain rated armature current

A three phase slip ring induction motor is fed from the rotor side with stator winding short circuited. The frequency of the currents flowing in the short circuited stator is

The line integral of the vector potential A around the boundary of a surface S represents