In the relation $P = \frac{\alpha}{\beta} e^{-\frac{\alpha Z}{k\theta}}$,$P$ is pressure,$Z$ is distance,$k$ is Boltzmann constant,and $\theta$ is temperature. The dimensional formula of $\beta$ is:

Applying the principle of homogeneity of dimensions,determine which one is correct. Where $T$ is time period,$G$ is gravitational constant,$M$ is mass,and $r$ is the radius of the orbit.

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 $(2)$.

Stokes' law states that the viscous drag force $F$ experienced by a sphere of radius $a$,moving with a speed $v$ through a fluid with coefficient of viscosity $\eta$,is given by $F=6 \pi \eta a v$. If this fluid is flowing through a cylindrical pipe of radius $r$,length $l$ and pressure difference of $p$ across its two ends,then the volume of water $V$ which flows through the pipe in time $t$ can be written as $\frac{V}{t}=k\left(\frac{p}{l}\right)^a \eta^b r^c$,where $k$ is a dimensionless constant. Correct values of $a, b$ and $c$ are

$L$, $C$, and $R$ represent the physical quantities inductance, capacitance, and resistance, respectively. The dimensional formula $ML^2T^{-4}A^{-2}$ corresponds to . . . . . . .

From the dimensional consideration,which of the following equations is correct?