$A$ proton is projected with velocity $\overrightarrow{V} = 2 \hat{i} \; m/s$ in a region where magnetic field $\overrightarrow{B} = (\hat{i} + 3 \hat{j} + 4 \hat{k}) \; \mu T$ and electric field $\overrightarrow{E} = 10 \hat{i} \; \mu V/m$. Find the net acceleration of the proton (in $m/s^2$).

In electromagnetic theory, electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related. In the questions below, $[E]$ and $[B]$ stand for dimensions of electric and magnetic fields respectively, while $[\varepsilon_0]$ and $[\mu_0]$ stand for dimensions of the permittivity and permeability of free space respectively. $L$ and $T$ are dimensions of length and time respectively. All quantities are in $SI$ units.
$(1)$ The relation between $[E]$ and $[B]$ is:
$(A)$ $[E]=[B][L][T]^{-1}$
$(B)$ $[E]=[B][L][T]$
$(C)$ $[E]=[B][L]^{-1}[T]$
$(D)$ $[E]=[B][L]^{-1}[T]^{-1}$
$(2)$ The relation between $[\varepsilon_0]$ and $[\mu_0]$ is:
$(A)$ $[\mu_0]=[\varepsilon_0][L]^2[T]^{-2}$
$(B)$ $[\mu_0]=[\varepsilon_0]^{-1}[L]^{-2}[T]^2$
$(C)$ $[\mu_0]=[\varepsilon_0][L]^{-2}[T]^2$
$(D)$ $[\mu_0]=[\varepsilon_0]^{-1}[L]^2[T]^{-2}$
Select the correct options for $(1)$ and $(2)$.

If the magnetic field is parallel to the positive $y$-axis and the charged particle is moving along the positive $x$-axis (Figure),which way would the Lorentz force be for
$(a)$ an electron (negative charge),
$(b)$ a proton (positive charge).

What is the behavior of a charged particle moving in a region where the electric field $\vec{E}$ and magnetic field $\vec{B}$ are perpendicular to each other?

$A$ particle of mass $1 \times 10^{-26} \,kg$ and charge $1.6 \times 10^{-19} \,C$ travelling with a velocity $1.28 \times 10^6 \,ms^{-1}$ along the positive $X$-axis enters a region in which a uniform electric field $E$ and a uniform magnetic field of induction $B$ are present. If $E = -102.4 \times 10^3 \hat{k} \,NC^{-1}$ and $B = 8 \times 10^{-2} \hat{j} \,Wbm^{-2}$, the direction of motion of the particle is:

$A$ metal sample carrying a current along the $x-$ axis with current density $J$ is subjected to a magnetic field $B$ (along the $z-$ axis). The electric field $E$ developed along the $y-$ axis is directly proportional to $J$ as well as $B$. The constant of proportionality has the $SI$ unit: