Two points $A$ and $B$ are situated at a distance $x$ and $2x$ respectively from the nearer pole of a magnet $2\,cm$ long. The ratio of magnetic field at $A$ and $B$ is

$(a)$ Magnetic field lines show the direction (at every point) along which a small magnetised needle aligns (at the point). Do the magnetic field lines also represent the lines of force on a moving charged particle at every point?

$(b)$ Magnetic field lines can be entirely confined within the core of a toroid, but not within a straight solenoid. Why?

$(c)$ If magnetic monopoles existed, how would the Gauss’s law of magnetism be modified?

$(d)$ Does a bar magnet exert a torque on itself due to its own field? Does one element of a current-carrying wire exert a force on another element of the same wire?

$(e)$ Magnetic field arises due to charges in motion. Can a system have magnetic moments even though its net charge is zero?

Two like magnetic poles of strength $ 10$  and $40$ $ SI$ units are separated by a distance $30 \,cm$. The intensity of magnetic field is zero on the line joining them

When the $N-$pole of a bar magnet points towards the south and $S-$pole towards the north, the null points are at the

A small current element of length $d \ell$ and carrying current is placed at $(1,1,0)$ and is carrying current in ' $+ z$ ' direction. If magnetic field at origin be $\overrightarrow{ B }_1$ and at point $(2,2,0)$ be $\overrightarrow{ B }_2$ then