Match List-$I$ with List-$II$ and choose the correct answer from the options given below:
| List-$I$ ($Y$ vs $X$) | List-$II$ (Shape of Graph) |
| :--- | :--- |
| $(A)$ $Y$ = magnetic susceptibility, $X$ = magnetising field | $(I)$ Linear graph passing through origin |
| $(B)$ $Y$ = magnetic field, $X$ = distance from centre of a current carrying wire for $x < a$ (where $a$ = radius of wire) | $(II)$ Graph with a curve decreasing towards the axis |
| $(C)$ $Y$ = magnetic field, $X$ = distance from centre of a current carrying wire for $x > a$ (where $a$ = radius of wire) | $(III)$ Horizontal straight line graph |
| $(D)$ $Y$ = magnetic field inside solenoid, $X$ = distance from center | $(IV)$ Linear graph starting from origin |

$A$ proton moving with a constant velocity passes through a region of space without any change in its velocity. If $\overrightarrow{E}$ and $\overrightarrow{B}$ represent the electric and magnetic fields respectively,then this region of space may have

In the following hexagons,made up of two different materials $P$ and $Q$,current enters and leaves from points $X$ and $Y$ respectively. In which case is the magnetic field at its centre not zero?

An electron (mass $9 \times 10^{-31} \ kg$ and charge $1.6 \times 10^{-19} \ C$) moving with speed $v = c/100$ $(c = 3 \times 10^8 \ ms^{-1})$ is injected into a magnetic field $\vec{B}$ of magnitude $9 \times 10^{-4} \ T$ perpendicular to its direction of motion. We wish to apply a uniform electric field $\vec{E}$ together with the magnetic field so that the electron does not deflect from its path. Then:

The figure shows a circular loop of radius $a$ with two long parallel wires (numbered $1$ and $2$) all in the plane of the paper. The distance of each wire from the centre of the loop is $d$. The loop and the wires are carrying the same current $I$. The current in the loop is in the counterclockwise direction if seen from above.
$1.$ When $d \approx a$ but wires are not touching the loop,it is found that the net magnetic field on the axis of the loop is zero at a height $h$ above the loop. In that case
$(A)$ current in wire $1$ and wire $2$ is in the direction $PQ$ and $RS$,respectively and $h \approx a$
$(B)$ current in wire $1$ and wire $2$ is in the direction $PQ$ and $SR$,respectively and $h \approx a$
$(C)$ current in wire $1$ and wire $2$ is in the direction $PQ$ and $SR$,respectively and $h \approx 1.2 a$
$(D)$ current in wire $1$ and wire $2$ is in the direction $PQ$ and $RS$,respectively and $h \approx 1.2 a$
$2.$ Consider $d \gg a$,and the loop is rotated about its diameter parallel to the wires by $30^{\circ}$ from the position shown in the figure. If the currents in the wires are in the opposite directions,the torque on the loop at its new position will be (assume that the net field due to the wires is constant over the loop)
$(A)$ $\frac{\mu_0 I^2 a^2}{d}$ $(B)$ $\frac{\mu_0 I^2 a^2}{2 d}$ $(C)$ $\frac{\sqrt{3} \mu_0 I^2 a^2}{d}$ $(D)$ $\frac{\sqrt{3} \mu_0 I^2 a^2}{2 d}$
Give the answer for question $1$ and $2$.

Two very long straight parallel wires,parallel to the $y-$axis,carry currents $4I$ and $I$ along the $+y$ direction and $-y$ direction,respectively. The wires pass through the $x-$axis at the points $(d, 0, 0)$ and $(-d, 0, 0)$ respectively. The graph of the magnetic field $z-$component as one moves along the $x-$axis from $x=-d$ to $x=+d$ is best given by: