Question

Multiple Correct Answers Type
Which of the following pairs have the same dimensions?
(L = inductance, C = capacitance, R = Resistance)

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

Answer 1

Answer: (A), (C)

The correct options are
A $\frac{L}{R}$ and CR
C $\frac{L}{R}$ and $\sqrt{LC}$

$\left[L\right]=\left[\frac{\varphi }{I}\right]$

Where $\varphi$ is the magnetic flux and $I$ is the current.

$\left[\varphi \right]=\left[B\right]\left[L^2\right]=\left[{MT}^2{A}^{-1}\right]L^2$

$\Rightarrow \left[\varphi \right]=\left[{ML}^2{T}^{-2}{A}^{-1}\right]$

So,

$\left[L\right]=\left[{MT}^{-2}{L}^2{A}^{-2}\right]$

Now for $R$

$\left[R\right]=\left[\frac{V}{I}\right]=\left[\frac{{ML}^2{T}^{-3}{A}^{-1}}{A^{-1}}\right]$

$\left[R\right]=\left[{ML}^2{T}^{-3}{A}^{-2}\right]$

Now, $\left[\frac{L}{R}\right]=\left[\frac{{MT}^2{L}^2{A}^{-2}}{{ML}^2{T}^{-3}{A}^{-2}}\right]$

$\left[\frac{L}{R}\right]=\left[T\right]$

$\left[C\right]=\left[\frac{AT}{V}\right]=\left[\frac{AT}{{ML}^2{T}^{-3}{A}^{-1}}\right]=\left[M^{-1}{L}^{-2}{T}^4{A}^2\right]$

So $\left[CR\right]=\left[T\right]$

$\left[LC\right]=\left[T^2\right]$

${\left[LC\right]}^{1/2}=\left[T\right]$

$\left[RC\right]=T$