Question

If $C$ and $V$ represent capacitance and voltage respectively, then what are the dimensions of $\lambda$ where $\lambda =\frac{C}{V}$ ?

• A. $\left[M^{-2}{L}^{-4}{I}^3{T}^7\right]$
• B. $\left[M^{-2}{L}^{-3}{I}^2{T}^6\right]$
• C. $\left[M^{-1}{L}^{-3}{I}^{-2}{T}^{-7}\right]$
• D. $\left[M^{-3}{L}^{-4}{I}^3{T}^7\right]$

Answer 1

The correct option is A

$\left[M^{-2}{L}^{-4}{I}^3{T}^7\right]$

Step 1: Given data

$\lambda =\frac{C}{V}$

capacitance$=C$

voltage$=V$

We have to find the dimensions of $\lambda$

Step 2: Formula used

We know,

Voltage, $V=\frac{W}{Q}$

Capacitance, $C=\frac{Q}{V}$

Where $W$ represents work done and $Q$ represents charge.

Step 3: Finding the dimension

Dimension of $\lambda$,

$$\begin{gathered} \lambda =\frac{C}{V} \\ \lambda =\frac{{\displaystyle \raisebox{1ex}{$Q$}\!\left/ \!\raisebox{-1ex}{$V$}\right.}}{V}\left(\because C=\frac{Q}{V}\right) \\ \Rightarrow \lambda =\frac{Q}{V^2} \\ \Rightarrow \lambda =\frac{Q}{{\left({\displaystyle \raisebox{1ex}{$W$}\!\left/ \!\raisebox{-1ex}{$Q$}\right.}\right)}^2}\left(\because V=\frac{W}{Q}\right) \\ \Rightarrow \lambda =\frac{Q^3}{W^2} \end{gathered}$$

$$\begin{gathered} \lambda =\frac{Q^3}{W^2} \\ \lambda =\frac{{\left(It\right)}^3}{{(F·s)}^2}\left(\because Q=ItandW=F·s\right) \\ \lambda =\frac{\left[I^3{T}^3\right]}{{\left[M{L}^2{T}^{-2}\right]}^2} \\ \lambda =\left[M^{-2}{L}^{-4}{I}^3{T}^7\right] \end{gathered}$$

Dimensions of $\lambda$ is $\left[M^{-2}{L}^{-4}{I}^3{T}^7\right]$

Option A is correct