Question

If $C,R,L$ and $I$ denote capacitance, resistance, inductance and electric current respectively, then the quantities having the same dimensions of time are

$$\begin{gathered} \left(i\right)CR \\ \left(ii\right)\frac{L}{R} \\ \left(iii\right)\sqrt{LC} \\ \left(iv\right)L{I}^2 \end{gathered}$$

• A. $\left(i\right)$ and $\left(ii\right)$
• B. $\left(i\right)$ and $\left(iii\right)$
• C. $\left(i\right)$ and $\left(iv\right)$
• D. $\left(i\right)$,$\left(ii\right)$ and $\left(iii\right)$

Answer 1

The correct option is D

$\left(i\right)$,$\left(ii\right)$ and $\left(iii\right)$

Step 1: Dimensions of $C,R,L$ and $I$

$$\begin{gathered} C=\left[M^{-1}{L}^{-2}{T}^4{A}^2\right] \\ R=\left[M{L}^2{T}^{-3}{A}^{-2}\right] \\ L=\left[M{L}^2{T}^{-2}{A}^{-2}\right] \\ I=\left[A\right] \end{gathered}$$

Step 2: Dimensions of $\left(i\right),\left(ii\right),\left(iii\right)$ and $\left(iv\right)$

$$\begin{gathered} \left(i\right)CR=\left[M^{-1}{L}^{-2}{T}^4{A}^2\right]\left[M{L}^2{T}^{-3}{A}^{-2}\right] \\ \Rightarrow CR=\left[T\right] \\ \left(ii\right)\frac{L}{R}=\frac{\left[M{L}^2{T}^{-2}{A}^{-2}\right]}{\left[M{L}^2{T}^{-3}{A}^{-2}\right]} \\ \Rightarrow \frac{L}{R}=\left[T\right] \\ \left(iii\right)\sqrt{LC}={\left(\left[M{L}^2{T}^{-2}{A}^{-2}\right]\left[M^{-1}{L}^{-2}{T}^4{A}^2\right]\right)}^{\frac{1}{2}} \\ \Rightarrow \sqrt{LC}={\left[T^2\right]}^{\frac{1}{2}} \\ \Rightarrow \sqrt{LC}=\left[T\right] \\ \left(iv\right)L{I}^2=\left[M{L}^2{T}^{-2}{A}^{-2}\right]{\left[A\right]}^2 \\ \Rightarrow L{I}^2=\left[M{L}^2{T}^{-2}\right] \end{gathered}$$

Dimension of time is $\left[T\right]$

Here we can see $\left(i\right)$,$\left(ii\right)$ and $\left(iii\right)$ has dimension of time.

Therefore, Option D is correct