Question

Write the dimensional formula of capacitance.
A. $[{{M}^{-1}}{{L}^{-2}}{{T}^{4}}{{A}^{2}}]$
B. $[{{M}^{-2}}{{L}^{-4}}{{T}^{5}}{{A}^{3}}]$
C. $[M{{L}^{2}}{{T}^{-2}}{{A}^{0}}]$
D. None of these

Answer 1

Hint: We can find the dimensional formula with the following equation. $\text{Capacitance = }\dfrac{\text{Charge}}{\text{Voltage}}$. You can use the voltage equation either $V=\dfrac{W}{q}$ or $V=E\times d$.

Complete step by step answer:
Capacitance can be written as,
$\text{Capacitance = }\dfrac{\text{Charge}}{\text{Voltage}}$………………..(1)
Charge can be written as the product of current and time. So it can be written as,
$\text{Charge = Current }\times \text{ Time}$……………………(2)
So, the dimensional formula of charge is,
$\Rightarrow \text{ }\\!\\![\\!\\!\text{ A }\\!\\!]\\!\\!\text{ }\\!\\!\times\\!\\!\text{ }\\!\\![\\!\\!\text{ T }\\!\\!]\\!\\!\text{ = }\\!\\![\\!\\!\text{ AT }\\!\\!]\\!\\!\text{ }$………………….(3)
To find the dimensional formula of current, we have to reduce it in the basic form.
$\text{Voltage = Electric field }\times \text{ Distance}$…………(4)
But $\text{Electric field = }\dfrac{\text{Force}}{Ch\arg e}$
So, the dimensional formula of electric field will be,
$\text{ }\\!\\![\\!\\!\text{ Electric field }\\!\\!]\\!\\!\text{ = }\dfrac{\text{ }\\!\\![\\!\\!\text{ ML}{{\text{T}}^{\text{-2}}}\text{ }\\!\\!]\\!\\!\text{ }}{\text{ }\\!\\![\\!\\!\text{ AT }\\!\\!]\\!\\!\text{ }}$
$\Rightarrow \text{ }\\!\\![\\!\\!\text{ ML}{{\text{A}}^{-1}}{{\text{T}}^{\text{-3}}}\text{ }\\!\\!]\\!\\!\text{ }$……………..(5)
We can substitute the dimensional formula of electric field in equation 4.
So, the dimensional formula of voltage is,
$[\text{Voltage}]=\text{ }\\!\\![\\!\\!\text{ ML}{{\text{A}}^{-1}}{{\text{T}}^{\text{-3}}}\text{ }\\!\\!]\\!\\!\text{ }\times \text{ }\\!\\![\\!\\!\text{ L }\\!\\!]\\!\\!\text{ = }\\!\\![\\!\\!\text{ M}{{\text{L}}^{2}}{{\text{A}}^{-1}}{{\text{T}}^{\text{-3}}}\text{ }\\!\\!]\\!\\!\text{ }$………….(6)
We can assign these dimensional formulas in equation 1 to get the dimensional formula of capacitance.
$\text{ }\\!\\![\\!\\!\text{ Capacitance }\\!\\!]\\!\\!\text{ = }\dfrac{\text{ }\\!\\![\\!\\!\text{ AT }\\!\\!]\\!\\!\text{ }}{\text{ }\\!\\![\\!\\!\text{ M}{{\text{L}}^{\text{2}}}{{\text{A}}^{\text{-1}}}{{\text{T}}^{\text{-3}}}\text{ }\\!\\!]\\!\\!\text{ }}$
$\text{ }\\!\\![\\!\\!\text{ Capacitance }\\!\\!]\\!\\!\text{ = }\\!\\![\\!\\!\text{ }{{\text{M}}^{-1}}{{\text{L}}^{\text{-2}}}{{\text{A}}^{2}}{{\text{T}}^{4}}\text{ }\\!\\!]\\!\\!\text{ }$
So, the correct option is A.

Additional information:
Capacitor is a device that can be used for the storage of electric charge. When a capacitor is connected to a battery, electrons from one plate transfer to another plate and become negatively charged. Electron received plates will be positively charged. So, a potential difference is developed between these plates. It will be the same as the battery's terminal voltage.
Generally, we can say that the charge stored in the capacitor is proportional to the potential difference between the plates. So that,
$Q=CV$, where $C$ is the proportionality constant known as capacitance. It is the ratio of magnitude of charge on conductor plates to the potential difference existing between the conductors.

Note: It is better to make complex equations into basic forms to find the dimensional formulas. So, we can find the dimensional formula of capacitance by using the formulas of force, current and time. It is better to remember that the capacitor needs current to be active. So, we can avoid wrong options logically also.