Question

With usual notation, amongst the following the one, which does not represent the dimension of time ?

• A. $\frac{L}{R}$
• B. $RC$
• C. $\sqrt{LC}$
• D. $\frac{1}{\sqrt{LC}}$

Answer 1

The correct option is D $\frac{1}{\sqrt{LC}}$

From ohm's law
$\left[V\right]=\left[IR\right]$
$\left[R\right]=\frac{\left[V\right]}{\left[I\right]}$
Dimension of $V=\frac{W}{q}=\frac{\left[M{L}^2{T}^{-2}\right]}{\left[AT\right]}$
$\left[R\right]=\frac{\left[M{L}^2{T}^{-2}\right]{\left[AT\right]}^{-1}}{A}$
$\left[R\right]=\left[M{L}^2{T}^{-3}{A}^{-2}\right]$
From induced emf in conductor,
$e=L\left(\frac{dI}{dt}\right)$
$\left[L\right]=\frac{\left[e\right]\left[dt\right]}{\left[dI\right]}=\frac{\left[\frac{W}{q}\right]\left[dt\right]}{\left[dI\right]}=\frac{\left[W\right]\left[dt\right]}{\left[q\right]\left[dI\right]}=\frac{\left[W\right]}{[dI{]}^2}=\frac{\left[M{L}^2{T}^{-2}\right]}{[A{]}^2}=\left[M{L}^2{T}^{-2}{A}^{-2}\right]$
$\left[C\right]=\frac{\left[q\right]}{\left[V\right]}=\frac{[q{]}^2}{\left[W\right]}=\frac{\left[A^2{T}^2\right]}{\left[M{L}^2{T}^{-2}\right]}=\left[M^{-1}{L}^{-2}{T}^4{A}^2\right]$
Option (a)
$\frac{\left[L\right]}{\left[R\right]}=\frac{\left[M{L}^2{T}^{-2}{A}^{-2}\right]}{\left[M{L}^2{T}^{-3}{A}^{-2}\right]}=\left[T\right]$
Option (b)
$\left[RC\right]=\left[M{L}^2{T}^{-3}{A}^{-2}\right]\left[M^{-1}{L}^{-2}{T}^4{A}^2\right]=\left[T\right]$
Option (c)
$\left[\sqrt{LC}\right]=\left[M{L}^2{T}^{-2}{A}^{-2}\right]\times \left[M^{-1}{L}^{-2}{T}^4{A}^2{]}^{\frac{1}{2}}\right[T^2{]}^{\frac{1}{2}}=\left[T\right]$
Option (d)
$\left[\frac{1}{\sqrt{LC}}\right]=\left[T^{-1}\right]$