Question

A quantity $X$ is given by ${\varepsilon _0}L\dfrac{{\Delta V}}{{\Delta t}}$ where ${\varepsilon _0}$ is the permittivity of free space, $L$ is the length, $\Delta V$ is a potential difference and $\Delta t$ is the time interval. The dimensional formula for $X$ is the same as that of
A. resistance
B. charge
C. voltage
D. current

Answer 1

Hint: To find the dimensional formula of the quantity ‘$X$’, we need to first know the dimensional formulas for the individual quantities in the given formula. Then, we can substitute the dimensional formulas of these individual quantities in the formula.

Complete step-by-step answer:
First, we’ll find the dimensional formula of the quantity $X$ from the given formula for $X$. In order to do that, we have to know the dimensional quantities of the individual quantities which are given by

Now, that we know the dimensional formula of individual quantities, we can simply substitute the dimensional formulae in the given formula of $X$.
$\eqalign{
  & \left[ X \right] = \left[ {{\varepsilon _0}L\dfrac{{\Delta V}}{{\Delta t}}} \right] \cr
  & \Rightarrow \left[ X \right] = \left[ {{\varepsilon _0}} \right]\left[ L \right]\left[ {\Delta V} \right]\dfrac{1}{{\left[ {\Delta t} \right]}} \cr
  & \Rightarrow \left[ X \right] = \dfrac{{\left[ {{M^{ - 1}}{L^{ - 3}}{T^4}{I^2}} \right]\left[ {{M^0}{L^1}{T^0}} \right]\left[ {{M^1}{L^2}{T^{ - 3}}{I^{ - 1}}} \right]}}{{\left[ {{M^0}{L^0}{T^1}} \right]}} \cr
  & \Rightarrow \left[ X \right] = \dfrac{{\left[ {{M^0}{L^0}{T^1}{I^1}} \right]}}{{\left[ {{M^0}{L^0}{T^1}} \right]}} \cr
  & \Rightarrow \left[ X \right] = \left[ {{M^0}{L^0}{T^0}{I^1}} \right] \cr
  & \therefore \left[ X \right] = \left[ I \right] = current \cr} $
Therefore, the dimensional formula of the given quantity $X$ is equivalent to that of current.

So, the correct answer is “Option D”.

Additional Information: Dimensional analysis is based on the fundamental and derived quantities in physics. That is the derived physical quantities are written in the powers of the fundamental physical quantities. The fundamental physical quantities and their dimensional formulas are:

Fundamental quantity	Units in SI system	Dimensional formula
Mass	kilogram	$\left[ M \right]$
Length	meter	$\left[ L \right]$
Time	second	$\left[ T \right]$
Temperature	Kelvin	$\left[ K \right]$
Current	Ampere	$\left[ I \right]$
Amount of Substance	Moles	$\left[ N \right]$
Luminous Intensity	Candela	$\left[ J \right]$

Note: If you find it hard to remember the dimensional formula for permittivity of free space, . Try to derive it from the formula for electrostatic force, given by
$F = \dfrac{1}{{4\pi {\varepsilon _0}}}\dfrac{{{q_1}{q_2}}}{{{r^2}}}$
Similarly, you can derive the dimensional formula for the potential difference that can be remembered as the work done in bringing a charge q from infinity to a point in a field or simply work done per unit charge. So, its formula will be given by
$\Delta V = \dfrac{W}{q}$
One can also solve the above question if one remembers the units of the given quantities well.