Question

In terms of potential difference $V$, electric current $I$, permittivity ${\epsilon }_0$, permeability ${\mu }_0$ and speed of light $c$, the dimensionally correct equation(s) is/are

• A. ${\mu }_0{I}^2={\epsilon }_0{V}^2$
  
• B. ${\epsilon }_{0}I={\mu }_{0}V$
  
  
• C. $I={\epsilon }_{0}cV$
  
• D. ${\mu }_{0}cI={\epsilon }_{0}V$

Answer 1

Answer: (A), (C)

The correct options are
A ${\mu }_0{I}^2={\epsilon }_0{V}^2$

C $I={\epsilon }_{0}cV$


Dimensions of different variables and constant
$\left[V\right]=\left[M^1{L}^2{T}^{-3}{A}^{-1}\right]$
$\left[I\right]=\left[A\right]$
$\left[{ϵ}_o\right]=\left[M^{-1}{L}^{-3}{T}^4{A}^2\right]$
$\left[{\mu }_o\right]=\left[M^1{L}^1{T}^{-2}{A}^{-2}\right]$
$\left[c\right]=\left[L^1{T}^{-1}\right]$

(A) ${\mu }_0{I}^2={\epsilon }_0{V}^2$
$\left[M^1{L}^1{T}^{-2}\right]=\left[M^1{L}^1{T}^{-2}\right]$
Dimensionally correct

(B) ${\epsilon }_{0}I={\mu }_{0}V$
$\left[M^1{L}^{-3}{T}^4{A}^3\right]\ne \left[M^2{L}^3{T}^{-5}{A}^{-3}\right]$
Dimensionally Incorrect

(C) $I={\epsilon }_{0}cV$
$\left[M^0{L}^0{T}^0{A}^1\right]=\left[M^0{L}^0{T}^0{A}^1\right]$
Dimensionally correct

(D) ${\mu }_{0}cI={\epsilon }_{0}V$
$\left[M^1{L}^2{T}^{-3}{A}^{-1}\right]\ne \left[L^{-1}{T}^1{A}^1\right]$
Dimensionally Incorrect