Question

- The formula for the capacity of condenser is given by $C = \dfrac{A}{d}$ When $A$ is the area of each plate and $d$ is the distance between the plates. Then the dimensions of missing quantity is:
A)${ \in _0} = {M^{ - 1}}{L^{ - 3}}{T^4}{A^2}$
B)${ \in _0} = {M^1}{L^3}{T^{ - 4}}{A^{ - 2}}$
C)${ \in _0} = {M^{ - 1}}{L^3}{T^4}{A^{ - 2}}$
D)${ \in _{_0}} = {M^{ - 1}}{L^{ - 2}}{T^4}{A^2}$

Answer 1

Hint: Condenser capacity defined as the ability of heat transfer from hot gas (vapours) to the surrounding condensing medium Liquid condensing medium (water) is more effective than the gaseous condensing than the gaseous condensing medium (air).

Complete step by step answer:
Whenever an electric charge is deposited on a $4$ conductor, its potential increases the deposited charge spreads over its surface for any conductor, the electric potential $(v)$ is directly proportional to the charge store $(Q)$.
Hence $Q$ $\alpha $ $v$
$Q = cv$
Where $c$is a constant known as capacity of the conductor.
In this given problem we have to find the dimension of missing quantity.
Let missing quantity be ${ \in _0}$
$C = \dfrac{{{ \in _0}A}}{d}$
As per the given details
Now,
Applying the dimensional formula per primitively of free space
\[{M^{ - 1}}{L^{ - 2}}{T^4}{A^2} = { \in ^0}\dfrac{{{L^2}}}{L}\]
$ \Rightarrow { \in _0} = {M^{ - 1}}{L^{ - 3}}{T^4}{A^2}$

So the correct option is (A) ${ \in _0} = {M^{ - 1}}{L^{ - 3}}{T^4}{A^2}$

Additional Information:
The capacity of a conductor $(c)$is defined as the amount of charge required to make the potential $1$ unit $(1volt)$
Capacity of a conductor is given by
$c = \dfrac{Q}{v}$
Let $v = 1v$, then $c = Q$
The capacity of conductor $(c)$is defined as the amount of charge required to make the potential $1$ unit $(1volt)$
SI unit of capacity is farad $(F)$
We have $c = $ $c = \dfrac{Q}{v}$
SI unit of capacity $ = $SI unit of electric charge $/$ SI unit of potential difference.
Therefore, 1 farad = 1 coulomb $/$1 volt
The capacity of conductor is $1$ farad if a charge of $1$ Coulomb raises its potential by $1$ volt.
The farad is a very large unit hence smaller partial units are used.
Smaller units are,
$1$ micro farad $(1\mu F) = {10^{ - 6}}F$
1 nanofarad $(1\eta F) = {10^{ - 9}}F$
$1$ picofarad $(1\rho F) = {10^{ - 12}}F4$

Note:
we can choose a condenser capacity by applying following steps:
Step1: Select a solvent having high volatility
Step2: Calculate the maximum boil-up of the selected solvent.
Step3: Calculate the condenser capacity for that boil-up