The ratio of speeds of electromagnetic waves in vacuum and a medium, having dielectric constant $k=3$ and permeability of $\mu=2 \mu_0$, is ( $\mu_0=$ permeability of vacuum)

Suppose a long solenoid of 100 cm length, radius 2 cm having 500 turns per unit length, carries a current $I=10 \sin (\omega \mathrm{t}) \mathrm{A}$, where $\omega=1000 \mathrm{rad} . / \mathrm{s}$. A circular conducting loop $(B)$ of radius 1 cm coaxially slided through the solenoid at a speed $v=1 \mathrm{~cm} / \mathrm{s}$. The r.m.s. current through the loop when the coil $B$ is inserted 10 cm inside the solenoid is $${\alpha \over {\sqrt 2 }}\mu A$$. The value of $\alpha$ is $\_\_\_\_$ .

[Resistance of the loop $=10 \Omega$ ]

A body of mass 14 kg initially at rest explodes and breaks into three fragments of masses in the ratio $2: 2: 3$. The two pieces of equal masses fly off perpendicular to each other with a speed of $18 \mathrm{~m} / \mathrm{s}$ each. The velocity of the heavier fragment is

$\_\_\_\_$ $\mathrm{m} / \mathrm{s}$.

A parallel plate capacitor with plate separation 5 mm is charged by a battery. On introducing a mica sheet of 2 mm and maintaining the connections of the plates with the terminals of the battery, it is found that it draws $25 \%$ more charge from the battery. The dielectric constant of mica is $\_\_\_\_$