Question

The quantities $x=\frac{l}{\sqrt{{\mu }_0{ϵ}_0}},y=\frac{E}{B}$ and $z=\frac{l}{CR}$ are defined where C - capacitance R - Resistance, l - length, E - Electric field, B - magnetic field and ${ϵ}_0,{\mu }_0,$ - free space permittivity and permeability respectively. Then :

• A. Only $x$ and $y$ have the same dimension
• B. $x,y$ and $z$ have the same dimension
• C. Only $x$ and $z$ have the same dimension
• D. Only $y$ and $z$ have the same dimension

Answer 1

Answer: (B)

$x=\frac{1}{\sqrt{{\mu }_0{\epsilon }_0}}=$ speed
$\Rightarrow [x]=\left[L^1{T}^{-1}\right]$
$y=\frac{E}{B}=$ speed
$\Rightarrow [y]=\left[L^1{T}^{-1}\right]$
$z=\frac{\ell }{RC}=\frac{\ell }{\tau }$
$\Rightarrow [z]=\left[L^1{T}^{-1}\right]$
So, $x,y,z$ all have the same dimensions.