The magnitude of magnetic field of a dipole $m$, at a point on its axis at distance $r$, is $\frac{\mu_0}{2 \pi} \frac{m}{r^3}$, where $\mu_0$ is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments, $m_1$ and $m_2$, separated by a distance $r$ on the common axis, with their north poles facing each other, is $\frac{k m_1 m_2}{r^4}$, where $k$ is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.

($1$) When the dipole $m$ is placed at a distance $r$ from the center of the loop (as shown in the figure), the current induced in the loop will be proportional to

$(A)$ $\frac{m}{r^3}$ $(B)$ $\frac{m^2}{r^2}$ $(C)$ $\frac{m}{r^2}$ $(D)$ $\frac{m^2}{r}$

($2$) The work done in bringing the dipole from infinity to a distance $r$ from the center of the loop by the given process is proportional to

$(A)$ $\frac{m}{r^5}$ $(B)$ $\frac{m^2}{r^5}$ $(C)$ $\frac{m^2}{r^6}$ $(D)$ $\frac{m^2}{r^7}$

Give the answer or qution ($1$) and ($2$)

[Given: Mass of neutron/proton $=(5 / 3) \times 10^{-27} kg$, charge of the electron $=1.6 \times 10^{-19} C$.]

Which of the following option($s$) is(are) correct?

$(A)$ The value of $x$ for $H^{+}$ion is $4 cm$.

$(B)$ The value of $x$ for an ion with $A_M=144$ is $48 cm$.

$(C)$ For detecting ions with $1 \leq A_M \leq 196$, the minimum height $\left(x_1-x_0\right)$ of the detector is $55 cm$.

$(D)$ The minimum width $w$ of the region of the magnetic field for detecting ions with $A_M=196$ is $56 cm$.