Mark the correct statement.

$A$ proton moving with a constant velocity passes through a region of space without any change in its velocity. If $\vec{E}$ and $\vec{B}$ represent the electric and magnetic fields respectively,then the region of space may have :
$(A)$ $E=0, B=0$
$(B)$ $E=0, B \neq 0$
$(C)$ $E \neq 0, B=0$
$(D)$ $E \neq 0, B \neq 0$
Choose the most appropriate answer from the options given below :

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 |

Answer the following questions:
$(a)$ $A$ magnetic field that varies in magnitude from point to point but has a constant direction (east to west) is set up in a chamber. $A$ charged particle enters the chamber and travels undeflected along a straight path with constant speed. What can you say about the initial velocity of the particle?
$(b)$ $A$ charged particle enters an environment of a strong and non-uniform magnetic field varying from point to point both in magnitude and direction,and comes out of it following a complicated trajectory. Would its final speed equal the initial speed if it suffered no collisions with the environment?
$(c)$ An electron travelling west to east enters a chamber having a uniform electrostatic field in north to south direction. Specify the direction in which a uniform magnetic field should be set up to prevent the electron from deflecting from its straight line path.

The dimensional formula of $\frac{1}{2} \mu_0 H^2$ (where $\mu_0$ is the permeability of free space and $H$ is the magnetic field intensity) is:

$A$ uniform beam of positively charged particles is moving with a constant velocity parallel to another beam of negatively charged particles moving with the same velocity in the opposite direction,separated by a distance $d$. The variation of the magnetic field $B$ along a perpendicular line drawn between the two beams is best represented by: