A 3-phase star connected slip ring induction motor has the following parameters referred to the stator:

$R_s = 3\, \Omega,\; X_s = 2\, \Omega,\; X_r' = 2\, \Omega,\; R_r' = 2.5\, \Omega$

The per phase stator to rotor effective turns ratio is 3:1. The rotor winding is also star connected. The magnetizing reactance and core loss of the motor can be neglected. To have maximum torque at starting, the value of the extra resistance in ohms (referred to the rotor side) to be connected in series with each phase of the rotor winding is _______ (rounded off to 2 decimal places).

A 5 kW, 220 V DC shunt motor has 0.5 $\Omega$ armature resistance including brushes. The motor draws a no-load current of 3 A. The field current is constant at 1 A. Assuming that the core and rotational losses are constant and independent of the load, the current (in amperes) drawn by the motor while delivering the rated load, for the best possible efficiency, is _______ (rounded off to 2 decimal places).

In the $(x, y, z)$ coordinate system, three point-charges $Q$, $Q$, and $\alpha Q$ are located in free space at $(-1, 0, 0)$, $(1, 0, 0)$, and $(0, -1, 0)$, respectively. The value of $\alpha$ for the electric field to be zero at $(0, 0.5, 0)$ is _________________ (rounded off to 1 decimal place).

The given equation represents a magnetic field strength $\bar{H}(r, \theta, \phi)$ in the spherical coordinate system, in free space. Here, $\hat{r}$ and $\hat{\theta}$ represent the unit vectors along $r$ and $\theta$, respectively. The value of $P$ in the equation should be _________________ (rounded off to the nearest integer).

$$\bar{H}(r, \theta, \phi) = \frac{1}{r^3} ( \hat{r} P \cos \theta + \hat{\theta} P \sin \theta)$$