Give the definition of magnetic field and provide its unit.

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 $(2)$.

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

An electron is moving with a velocity $\vec{v} = (2 \hat{i} + 3 \hat{j}) \text{ m/s}$ in an electric field $\vec{E} = (3 \hat{i} + 6 \hat{j} + 2 \hat{k}) \text{ V/m}$ and a magnetic field $\vec{B} = (2 \hat{j} + 3 \hat{k}) \text{ T}$. Calculate the magnitude and direction (with $x$-axis) of the Lorentz force acting on the electron.

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

What is Lorentz force? Write an expression for it.