Lorentz magnetic force is acting on a particle of charge $q$ moving with velocity $\vec{V}$ in magnetic field $\vec{B}$. The work done by this force on the charged particle is

$A$ charge $q$ enters a region having electric field $E$ and magnetic field $B$ with velocity $v$. If it continues to move with the same velocity, then which of the following statements is not true?

Write the Lorentz force equation.

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 $(2)$.

The dimension of magnetic field in $M, L, T$ and $C$ (coulomb) is given as:

$A$ charged particle carrying charge $1\,\mu C$ is moving with velocity $(2 \hat{i} + 3 \hat{j} + 4 \hat{k})\, ms^{-1}$. If an external magnetic field of $(5 \hat{i} + 3 \hat{j} - 6 \hat{k}) \times 10^{-3}\, T$ exists in the region where the particle is moving,then the force on the particle is $\overrightarrow{F} \times 10^{-9}\, N$. The vector $\overrightarrow{F}$ is: