Question

The quantities $x=\frac{1}{\sqrt{\mu_{0} \epsilon_{0}}}, y=\frac{E}{B}$ and $z =\frac{1}{ CR }$ are defined where C - capacitance R - Resistance, l - length, E - Electric field, B - magnetic field and $\epsilon_{0}, \mu_{0},$ - 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)

$x, y$ and $z$ have the same dimension

Answer 2

$x =\frac{1}{\sqrt{\mu_{0} \varepsilon_{0}}}=$ 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.