Question

The dimensional formula of capacitance is ____. Take $Q$ as the dimension formula of charge
A.${{M}^{1}}{{L}^{-2}}{{T}^{-2}}{{Q}^{-2}}$
B. ${{M}^{1}}{{L}^{2}}{{T}^{-2}}{{Q}^{-2}}$
C. ${{M}^{1}}{{L}^{-2}}{{T}^{2}}{{Q}^{2}}$
D. ${{M}^{-1}}{{L}^{-2}}{{T}^{2}}{{Q}^{2}}$

Answer 1

Hint:
For the dimensional formula of capacitance, we need the formula which relates charge and capacitance and other physical quantity which is voltage.
Then expressing the dimensional formula for voltage and charge (which is given in the question), we will derive the dimensional formula for capacitance.
Voltage is defined as work done per unit charge.

Complete step by step answer:
The formula of capacitance $C$ in terms of charge $Q$ and voltage $V$ is,
$C=\dfrac{Q}{V}$
Dimensional formula for charge =$\left[ {{Q}^{1}} \right]$
Now for finding capacitance dimensional formula we need dimensional formula for voltage
As we know, voltage is work done per unit charge,$\dfrac{W}{Q}$
Dimensional formula for work done is same as energy which is $\left[ M{{L}^{2}}{{T}^{-2}} \right]$
Putting in the formula for voltage
The dimensional formula for voltage is $M{{L}^{2}}{{T}^{-2}}{{Q}^{-1}}$
So now putting the above formula in the main formula for capacitance,
$\Rightarrow C=\dfrac{Q}{V}$
$\Rightarrow C=\dfrac{\left[ Q \right]}{\left[ M{{L}^{2}}{{T}^{-2}}{{Q}^{-1}} \right]}$
$\Rightarrow C=\left[ {{M}^{-1}}{{L}^{-2}}{{T}^{2}}{{Q}^{2}} \right]$

So, we have derived the dimensional formula for capacitance as above and according to it the correct formula is D.

Note:
There are many applications of dimensional analysis in physics like
Dimensional analysis is used to convert the value of a physical quantity from one system of units to another system of units.
To reduce the number of variables required in an experimental program.
 To check the dimensional homogeneity of an equation.
 Base quantities are the Physical Quantities which are represented by a number followed by a unit. The number along with the unit is called the magnitude of the physical quantity. Are expressed in terms of standard units. There are seven basic physical quantities. Examples- mass, length, time, electric current, temperature, luminous intensity and amount of substance.