Question

The square root of the product of inductance and capacitance has the dimension of

• A. length
• B. mass
• C. time
• D. No dimension

Answer 1

The correct option is C

time

Step 1: Given Data:

Let the capacitance be $C$ and inductance be $L$

It is given that the square root of the product of inductance and capacitance $i.e.,\sqrt{LC}$

Step 2: Formula Used:

The dimension of capacitance is $\left[C\right]=\left[M^{-1}{L}^{-2}{T}^4{A}^2\right]$

We know that, $C=\frac{Q}{V}\dots \dots \dots \dots \dots \dots \left(1\right)$ where $QandV$are charge and potential respectively.

Dimension of $Q=\left[AT\right]\dots \dots \dots \dots \dots \dots \left(2\right)$

Dimension of potential $V$ is given by-

$V=\frac{W}{Q}\dots \dots \dots \dots \dots \dots \left(3\right)$

Work has the dimension

$\begin{array}{rcl}\left[W\right]& =& F\times d\\ & =& \left[ML{T}^{-2}\right]\times \left[L\right]\\ & =& \left[M{L}^2{T}^{-2}\right]\dots \dots \dots \dots \dots \dots \dots \left(4\right)\end{array}$

then the dimension of potential, using equations (2) and (4)

$\begin{array}{rcl}V& =& \frac{W}{Q}\\ & =& \frac{\left[M{L}^2{T}^{-2}\right]}{\left[AT\right]}\\ & =& \left[M{L}^2{I}^{-1}{T}^{-3}\right]\dots \dots \dots \dots \dots \dots \dots \left(5\right)\end{array}$

Now the dimension of capacitance has been calculated as, using equations (2) and (5)

$\begin{array}{rcl}C& =& \frac{Q}{V}\\ & =& \frac{\left[AT\right]}{\left[M{L}^2{T}^{-3}{A}^{-1}\right]}\\ & =& \left[M^{-1}{L}^{-2}{T}^4{A}^2\right]\dots \dots \dots \dots \dots \dots \dots \dots \left(6\right)\end{array}$

The dimension of inductance $\left[L\right]=\begin{array}{l}\left[M{L}^2{T}^{-2}{A}^{-2}\right]\end{array}$

We know that, $L=\frac{V}{{\displaystyle \frac{dI}{dt}}}$ where $Vand\frac{dI}{dt}$are potential and rate of flow of current respectively.

Using equation (3)

Dimension of $V=\left[M{L}^2{A}^{-1}{T}^{-3}\right]$

Dimension of $\frac{dI}{dt}=\left[A{T}^{-1}\right]$

Now for the dimension of inductance,

$\begin{array}{rcl}L& =& \frac{V}{{\displaystyle \frac{dI}{dt}}}\\ & =& \frac{\left[M{L}^2{T}^{-3}{A}^{-1}\right]}{\left[A{T}^{-1}\right]}\\ & =& \left[M{L}^2{T}^{-2}{A}^{-2}\right]\end{array}$

The dimension of inductance is $\left[L\right]=\begin{array}{rcl}& & \left[M{L}^2{T}^{-2}{A}^{-2}\right]\end{array}\dots \dots \dots \dots \dots \left(7\right)$

Step 3: Calculation of the dimension of $\sqrt{LC}$

According to the question, it is required to find the dimension of the square root of the product of inductance and capacitance $i.e.,\sqrt{LC}$

It can be calculated using the equation (6) and (7)

$$\begin{gathered} =\sqrt{LC} \\ =\sqrt{\left[M{L}^2{T}^{-2}{A}^{-2}\right]\left[M^{-1}{L}^{-2}{T}^4{A}^2\right]} \\ =\sqrt{T^2} \\ =T \end{gathered}$$

Hence option C is the correct answer.