The physical quantity which has the dimensional formula ${M^1}{T^{ - 3}}$ is

Consider two physical quantities $A$ and $B$ related to each other as $E = \frac{B - x^2}{At}$,where $E, x,$ and $t$ have dimensions of energy,length,and time respectively. The dimension of $AB$ is

$A$ quantity $x$ is given by $(IF v^{2} / WL^{4})$ in terms of moment of inertia $I$,force $F$,velocity $v$,work $W$,and length $L$. The dimensional formula for $x$ is the same as that of:

In the $CGS$ system, the magnitude of the force is $100 \ dynes$. In another system where the fundamental physical quantities are $kilogram$, $meter$, and $minute$, the magnitude of the force is:

What is Dimensional Analysis? State the uses of Dimensional Analysis.

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:
$(A)$ $[E] = [B][L][T]$
$(B)$ $[E] = [B][L]^{-1}[T]$
$(C)$ $[E] = [B][L][T]^{-1}$
$(D)$ $[E] = [B][L]^{-1}[T]^{-1}$
$(2)$ The relation between $[\varepsilon_0]$ and $[\mu_0]$ is:
$(A)$ $[\mu_0] = [\varepsilon_0][L]^2[T]^{-2}$
$(B)$ $[\mu_0] = [\varepsilon_0][L]^{-2}[T]^2$
$(C)$ $[\mu_0] = [\varepsilon_0]^{-1}[L]^2[T]^{-2}$
$(D)$ $[\mu_0] = [\varepsilon_0]^{-1}[L]^{-2}[T]^2$
Give the answers for questions $(1)$ and $(2)$.