Question

The energy stored in an electric device known as a capacitor is given by $u=\dfrac{{{q}^{2}}}{2C}$
Where, $u$ = energy stored in capacitor
$C$ = capacity of the capacitor
$q$ = charge on the capacitor
The dimensions of capacity of the capacitor is-
(A). $[{{M}^{-1}}{{L}^{-2}}{{T}^{4}}{{A}^{2}}]$
(B). $[{{M}^{-1}}{{L}^{-2}}{{T}^{4}}A]$
(C). $[{{M}^{-2}}{{L}^{-2}}{{T}^{4}}{{A}^{2}}]$
(D). $[{{M}^{0}}{{L}^{-2}}{{T}^{4}}{{A}^{0}}]$

Answer 1

Capacitance is the capacity of a conductor to store charge on it. For a capacitor connected in a circuit, it is related to potential difference and the charge. We can use this relation to calculate the dimensional formula of capacitance. The representation of a quantity in terms of fundamental units is called dimensional formula

Formula used:
$C=\dfrac{q}{V}$
$u=\dfrac{{{q}^{2}}}{2C}$

Complete step-by-step solution:
The dimensional formula of a physical quantity is an equation which represents the unit for that physical quantity in terms of fundamental units. There are seven fundamental units; length (length (metres $[L]$), mass (kilogram $[M]$), time (seconds $[T]$), current (ampere $[A]$), temperature (kelvin $[T]$), amount of substance (mole $[mol]$), luminous intensity (candela $[cd]$).

The capacity of a conductor to store charge is known as its capacitance. Its SI unit is Farad ($F$). It is given as
$C=\dfrac{q}{V}$
Here, $C$ is the capacitance
$q$ is the charge on the conductor
$V$ is the potential difference across the conductor

A capacitor is a device which can store charge on it and its capacity to store charge is known as its capacitance. Give, the energy stored on a capacitor is-
$u=\dfrac{{{q}^{2}}}{2C}$
$\Rightarrow C=\dfrac{{{q}^{2}}}{2u}$ - (1)
From the above equation, we can determine the dimensional formula of capacitance.

The flow of charge from one point to the other per unit time is called current. It is given by-
$\begin{aligned} & I=\dfrac{q}{t} \\\ & \Rightarrow q=I\times t \\\ \end{aligned}$

From the above equation, the dimensional formula of charge will be-
$[q]=[It]$
$\Rightarrow [q]=[AT]$ - (2)

The dimensional formula of energy is $[{{M}^{1}}{{L}^{2}}{{T}^{-2}}]$. Using dimensional formula for energy and charge to calculate the dimensional formula for capacitance from eq (1), we get,
$\begin{aligned} & C=\dfrac{{{q}^{2}}}{2u} \\\ & \Rightarrow [C]={{\dfrac{[q]}{[u]}}^{2}} \\\ & \Rightarrow [C]=\dfrac{{{[AT]}^{2}}}{[M{{L}^{2}}{{T}^{-2}}]} \\\ & \Rightarrow [C]=\dfrac{{{A}^{2}}{{T}^{2}}}{[M{{L}^{2}}{{T}^{-2}}]} \\\ & \therefore [C]=[{{M}^{-1}}{{L}^{-2}}{{A}^{2}}{{T}^{4}}] \\\ \end{aligned}$
Therefore, the dimensions of capacitance are $[{{M}^{-1}}{{L}^{-2}}{{A}^{2}}{{T}^{4}}]$.

Hence, the correct option is (A).

Note:
The capacitance depends on the dimensions of the conductor as well as the medium. The work done by the battery is stored as energy in the capacitor. The combinations of capacitors are analogous to resistors. Capacitance is a derived quantity, derived from the fundamental units; force, displacement between plates of capacitor, current and time taken.