Select the dimensional formula of $\frac{B^2}{2\mu_0}$.

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 beam is moving from north to south and an electron beam is moving from south to north. Neglecting the Earth's magnetic field,the electron beam will be deflected (assuming zero gravity):

Two circular coils $X$ and $Y$,having an equal number of turns,carry equal currents in the same sense and subtend the same solid angle at point $O$. If the smaller coil $X$ is midway between $O$ and $Y$,then if we represent the magnetic induction due to the bigger coil $Y$ at $O$ as $B_Y$ and that due to the smaller coil $X$ at $O$ as $B_X$,then:

Two protons move parallel to each other,keeping a distance $r$ between them,both moving with the same velocity $\vec{v}$. Then the ratio of the electric force to the magnetic force of interaction between them is:

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.