Question

If we use permittivity $\epsilon$, resistance $R$, gravitational constant $G$ and voltage $V$ as fundamental physical quantities, then :

• A. [angular displacement]$={\epsilon }^0{R}^0{G}^0{V}^0$
• B. [velocity]$={\epsilon }^{-1}{R}^{-1}{G}^0{V}^0$
• C. [dipole moment]$={\epsilon }^1{R}^0{G}^0{V}^1$
• D. [force$]={\epsilon }^1{R}^0{G}^0{V}^2$

Answer 1

Answer: (A), (B), (D)

The correct options are
A [angular displacement]$={\epsilon }^0{R}^0{G}^0{V}^0$
B [velocity]$={\epsilon }^{-1}{R}^{-1}{G}^0{V}^0$
D [force$]={\epsilon }^1{R}^0{G}^0{V}^2$

${ϵ}^{-1}=L^{3}M{T}^{-2}{Q}^{-2},R=L^{2}M{T}^{-1}{Q}^{-2},V=M{L}^2{T}^{-2}{Q}^{-1},G=L^3{M}^{-1}{T}^{-2}$
As the angular displacement $\left(\theta \right)$ is dimensionless so
[displacement]$={ϵ}^0{R}^0{G}^0{V}^0$
Velocity, $\left[v\right]=L{T}^{-1}$
here ${ϵ}^{-1}{R}^{-1}{G}^0{V}^0=L^{3}M{T}^{-2}{Q}^{-2}{L}^{-2}{M}^{-1}{T}^1{Q}^2=L{T}^{-1}=\left[v\right]$
dipole moment, $\left[p\right]=QL$
here ${ϵ}^1{R}^0{G}^0{V}^1=\left(L^{-3}{M}^{-1}{T}^2{Q}^2\right)\left(M{L}^2{T}^{-2}{Q}^{-1}\right)=Q{L}^{-1}\ne \left[p\right]$
force, $\left[f\right]={ϵ}^{-1}{L}^{-2}{Q}^2$
here ${ϵ}^1{R}^0{G}^0{V}^2=\left(L^{-3}{M}^{-1}{T}^2{Q}^2\right)\left(M^2{L}^4{T}^{-4}{Q}^{-2}\right)=\left(L^{3}M{T}^{-2}{Q}^{-2}\right)L^{-2}{Q}^2={ϵ}^{-1}{L}^{-2}{Q}^2=\left[f\right]$