In the product $\overrightarrow{F} = q(\vec{v} \times \overrightarrow{B})$ where $\overrightarrow{B} = B \hat{i} + B \hat{j} + B_{0} \hat{k}$. Given $q = 1$,$\vec{v} = 2 \hat{i} + 4 \hat{j} + 6 \hat{k}$,and $\overrightarrow{F} = 4 \hat{i} - 20 \hat{j} + 12 \hat{k}$,what is the complete expression for $\overrightarrow{B}$?

An electric charge $+q$ moves with velocity $\overrightarrow{V} = 3\hat{i} + 4\hat{j} + \hat{k}$ in an electromagnetic field given by $\overrightarrow{E} = 3\hat{i} + \hat{j} + 2\hat{k}$ and $\overrightarrow{B} = \hat{i} + \hat{j} - 3\hat{k}$. The $y$-component of the force experienced by $+q$ is: (in $q$)

$A$ 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 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 $(2)$.

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

$A$ charge $q$ is released in the presence of an electric field $(E)$ and a magnetic field $(B)$. After some time,its velocity is $v$. Then: