$A$ charge $q$ moves with velocity $\vec{V}$ through an electric field $\vec{E}$ as well as a magnetic field $\vec{B}$. Then the force acting on it is:

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 $E$ and $B$ are the magnitudes of electric and magnetic fields respectively in some region of space, then the possibilities for which a charged particle may move in that space with a uniform velocity of magnitude $v$ are

$A$ charge $q$ moves in a region where both an electric field and a magnetic field exist. The total force on it is:

$A$ charge moves with velocity $\overrightarrow{V}$ through an electric field $\overrightarrow{E}$ as well as a magnetic field $\overrightarrow{B}$. Then the force acting on it is:

In a certain region,static electric and magnetic fields exist. The magnetic field is given by $\vec B = B_0(\hat i + 2\hat j - 4\hat k)$. If a test charge moving with a velocity $\vec v = v_0(3\hat i - \hat j + 2\hat k)$ experiences no force in that region,then the electric field in the region,in $SI$ units,is