A bar magnet of length l and magnetic dipole moment M is bent to form an arc which subtends an angle of $120°$ at centre. The new magnetic dipole moment will be

1.  $\frac{3\mathrm{M}}{2\mathrm{\pi }}$

2.  $\frac{3\sqrt{3}\mathrm{M}}{2\mathrm{\pi }}$

3.  $\frac{3\mathrm{M}}{\mathrm{\pi }}$

4.  $\frac{2\mathrm{M}}{\mathrm{\pi }}$

$$\begin{gathered} \mathrm{A} \mathrm{small} \mathrm{bar} \mathrm{magnet} \mathrm{is} \mathrm{placed} \mathrm{with} \mathrm{its} \mathrm{north} \mathrm{pole} \mathrm{facing} \mathrm{magnetic} \mathrm{north} \mathrm{pole}. \\ \mathrm{The} \mathrm{neutral} \mathrm{points} \mathrm{are} \mathrm{located} \mathrm{at} \mathrm{distance} \mathrm{r} \mathrm{from} \mathrm{its} \mathrm{centre}. \mathrm{If} \mathrm{magnet} \mathrm{is} \\ \mathrm{rotated} \mathrm{by} {180}^0, \mathrm{neutral} \mathrm{point} \mathrm{shall} \mathrm{be} \mathrm{obtained} \mathrm{at} \mathrm{distance}: \\ 1. 2\mathrm{r} 2. \sqrt{2}\mathrm{r} \\ 3. 2^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\mathrm{r} 4. \frac{\mathrm{r}}{2\sqrt{2}} \end{gathered}$$

$$\begin{gathered} \mathrm{If} \mathrm{a} \mathrm{magnet} \mathrm{is} \mathrm{suspended} \mathrm{at} \mathrm{angle} 30° \mathrm{to} \mathrm{the} \mathrm{magnetic} \mathrm{meridian}, \mathrm{the} \mathrm{dip} \mathrm{needle} \mathrm{makes} \mathrm{an} \mathrm{angle} \mathrm{of} 45° \mathrm{with} \mathrm{the} \mathrm{horizontal}. \\ \mathrm{The} \mathrm{value} \mathrm{of} \mathrm{re}al \mathrm{dip} \mathrm{angle} \mathrm{at} \mathrm{the} \mathrm{place} \mathrm{is} \end{gathered}$$

1.  ${\tan }^{-1} \frac{\sqrt{3}}{2}$

2.  ${\tan }^{-1} 1$

3.  ${\tan }^{-1 }\frac{1}{2}$

4.  ${\tan }^{-1}\left(\frac{2}{\sqrt{3}}\right)$

The magnetic field at a point x on the axis of a small bar magnet is equal to the field at a point y on the equator of the same magnet. The ratio of the distances of x and y from the centre of the magnet is
(a) $2^{-3}$

(b) $2^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

(c) $2^3$

(d) $2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$