$A$ circular loop of radius $R$ carrying current $I$ lies in the $x-y$ plane with its centre at the origin. The total magnetic flux through the $x-y$ plane 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?

$A$ circular disc of radius $0.2 \; m$ is placed in a uniform magnetic field of induction $\frac{1}{\pi} \; Wb/m^2$ in such a way that its axis makes an angle of $60^{\circ}$ with $\vec{B}$. The magnetic flux linked with the disc is..... $Wb$.

$A$ uniform magnetic field of strength $B=2 \text{ mT}$ exists vertically downwards. These magnetic field lines pass through a closed surface as shown in the figure. The closed surface consists of a hemisphere $S_1$,a right circular cone $S_2$,and a circular surface $S_3$. The magnetic flux through $S_1$ and $S_2$ are respectively:

Write the dimensional formula for magnetic flux.

Given below are two statements $:$ one is labelled as Assertion $(A)$ and the other is labelled as Reason $(R)$.
Assertion $(A) :$ Magnetic monopoles do not exist.
Reason $(R) :$ Magnetic field lines are continuous and form closed loops.
In the light of the above statements, choose the most appropriate answer from the options given below.