Question

The dimensions of capacitance are:
A. $\left[ {M{L^{ - 2}}{Q^{ - 2}}{T^2}} \right]$
B. $\left[ {{M^{ - 1}}{L^2}{T^{ - 2}}{Q^2}} \right]$
C. $\left[ {{M^{ - 1}}{L^2}{T^{ - 2}}{Q^{ - 2}}} \right]$
D. $\left[ {{M^{ - 1}}{L^{ - 2}}{T^2}{Q^2}} \right]$

Answer 1

- Hint: To solve this question, we need to use the basic theory of unit and measurements chapter. As we know, Capacitance is dependent upon charge and voltage. So, first we need to calculate the Dimensions formula of voltage so that we calculate the Dimensions formula of Capacitance using the below mentioned formula.

Formula used- Capacitance (C) = Charge × ${\left( {Voltage} \right)^{ - 1}}$

Complete step-by-step solution -

Now, as we know-
Capacitance (C) = Charge × ${\left( {Voltage} \right)^{ - 1}}$ ……… (1)
Since, Charge = Current × Time
∴ The dimensional formula of charge =$\left[ {AT} \right]$ ..……. (2)
And, Electric potential = Electric Field × Distance ……… (3)
Electric Field =$\left[ {Force \times Ch\arg {e^{ - 1}}} \right]$
The dimensional formula of force and charge is\[\left[ {ML{T^{ - 2}}} \right]\] and $\left[ {AT} \right]$ respectively.
∴ The dimensional formula of Electric Field =\[\left[ {{M^1}{L^1}{T^{ - 2}}} \right]\]$ \times $${\left[ {AT} \right]^{ - 1}}$
                                                                             $ = \left[ {{M^1}{L^1}{T^{ - 3}}{A^{ - 1}}} \right]$ ….. (4)
On putting equation (4) in equation (3) and we get,
The dimensional formula of Voltage $ = \left[ {{M^1}{L^1}{T^{ - 3}}{A^{ - 1}}} \right]$$ \times \left[ {{L^1}} \right] = \left[ {{M^1}{L^2}{T^{ - 3}}{A^{ - 1}}} \right]$ ……… (5)
On putting equation (5) and (2) in equation (1) and we also get,
Capacitance = Charge × ${\left( {Voltage} \right)^{ - 1}}$
Or C = $\left[ {{A^1}{T^1}} \right] \times {\left[ {{M^1}{L^2}{T^{ - 3}}{A^{ - 1}}} \right]^{ - 1}} = \left[ {{M^{ - 1}}{L^{ - 2}}{T^4}{A^2}} \right]$
And also, we know
Charge = Current × Time
∴ The dimensional formula of charge =$\left[ {AT} \right]$= [Q]
So, Capacitance is dimensionally also represented as $\left[ {{M^{ - 1}}{L^{ - 2}}{T^4}{A^2}} \right]$ or $\left[ {{M^{ - 1}}{L^{ - 2}}{T^2}{Q^2}} \right]$
Therefore, we conclude that Capacitance is dimensionally represented as $\left[ {{M^{ - 1}}{L^{ - 2}}{T^2}{Q^2}} \right]$from given in all options.

Thus, option (D) is the correct answer.

Note: Dimensional Formula showing the powers to which the fundamental units are to be raised to obtain one unit of a derived quantity is called the dimensional formula of that given quantity. For example, If C is the unit of a derived quantity represented by C = ${M^a}{L^b}{T^c}$, then [${M^a}{L^b}{T^c}$] is called dimensional formula and the exponents a, b and, c are called the dimensions.
Where, M represents mass, L represents length and T represents time.