A balanced three-phase supply is given to a $30 \mathrm{~kW}, 4$-pole, $400 \mathrm{~V}, 50 \mathrm{~Hz}$, wound rotor induction motor with Y-connected stator and rotor windings. The motor is driving a constant torque load. With shorted slip rings, the machine runs at 1476 rpm .

When an external non-inductive resistance of $0.27 \Omega$ per phase is connected in series in the rotor circuit, the steady-state speed drops to 1404 rpm .

Neglecting rotational losses, the actual per phase rotor winding resistance is $\_\_\_\_$ $\Omega$. (Round off to two decimal places)

A uniform ring charge of radius $R$ carries a total charge $Q$. Which one of the following options correctly quantifies the magnitude of the force on a point charge of strength kept at the center of the ring? ( $\in$ is the permittivity of the medium))

A positive point charge with velocity $\vec{v}=5 \hat{x}$ enters a region having electric field $\vec{E}=4 \hat{y}$ and magnetic field $\vec{B}=-6 \hat{z}$. Which one of the following statements is correct for the force on the charge as it enters the region?

The figure shows an arbitrarily shaped planar conducting loop $A$ in the $X Y$ plane. Two nonintersecting regions with areas $a_1$ and $a_2$ within the loop are subjected to magnetic fields $\vec{B}_1=\frac{m}{\sqrt{2}} \sin (\omega t)(1 \hat{x}+0 \hat{y}+1 \hat{z})$, and $\vec{B}_2=-\frac{n}{\sqrt{2}} \cos \left(2 \omega t+\frac{\pi}{4}\right)(0 \hat{x}+1 \hat{y}+1 \hat{z})$, respectively.

What is the expression for the induced rms voltage in loop $A$ ?