A hollow, right circular cone of base radius $R$ and height $h$, with its tip at the origin is rotating about the $Z$-axis with an angular velocity $\omega$, as shown in the figure. The cone carries a total charge $Q$ uniformly distributed on its curved surface. The magnitude of magnetic field at a point $(0,0,z)$, where $z \gg R$ and $z \gg h$, is $$\frac{n\mu_0}{4\pi}\frac{Q R^2 \omega}{z^3}.$$ The value of $n$ is:

A conducting solid sphere of radius $R$ and mass $M$ carries a charge $Q$. The sphere is rotating about an axis passing through its center with a uniform angular speed $\omega$. The ratio of the magnitudes of the magnetic dipole moment to the angular momentum about the same axis is given as $\alpha \frac{Q}{2 M}$. The value of $\alpha$ is ___________.

A rectangular conducting loop of length $4 \mathrm{~cm}$ and width $2 \mathrm{~cm}$ is in the $x y$-plane, as shown in the figure. It is being moved away from a thin and long conducting wire along the direction $\frac{\sqrt{3}}{2} \hat{x}+\frac{1}{2} \hat{y}$ with a constant speed $\mathrm{v}$. The wire is carrying a steady current $I=10 \mathrm{~A}$ in the positive $x$-direction. A current of $10 \mu \mathrm{A}$ flows through the loop when it is at a distance $d=4 \mathrm{~cm}$ from the wire. If the resistance of the loop is $0.1 \Omega$, then the value of $\mathrm{v}$ is ________ $\mathrm{m} \mathrm{s}^{-1}$.

[Given: The permeability of free space $\mu_0=4 \pi \times 10^{-7} \mathrm{~N} \mathrm{~A}^{-2}$ ]

An $$\alpha$$-particle (mass 4 amu) and a singly charged sulphur ion (mass 32 amu) are initially at rest. They are accelerated through a potential V and then allowed to pass into a region of uniform magnetic field which is normal to the velocities of the particles. Within this region, the $$\alpha$$-particle and the sulfur ion move in circular orbits of radii r_$$\alpha$$ and r_s, respectively. The ratio (r_s/r_$$\alpha$$) is __________.