Question

The dimensional formula for electric flux is
$\text{A}\text{. }\left[ M{{L}^{3}}{{I}^{-1}}{{T}^{-3}} \right]$
$\text{B}\text{. }\left[ {{M}^{2}}{{L}^{2}}{{I}^{-1}}{{T}^{-2}} \right]$
$\text{C}\text{. }\left[ M{{L}^{3}}{{I}^{1}}{{T}^{-3}} \right]$
$\text{D}\text{. }\left[ M{{L}^{-3}}{{I}^{-1}}{{T}^{-3}} \right]$

Answer 1

Hint: When an electric field is passing to an area, we define something that is called electric flux through this area given as $\phi =\overrightarrow{E}.\overrightarrow{A}$. The dimensional formula of flux will be $\left[ \phi \right]=\left[ EA\cos \theta \right]=\left[ E \right]\left[ A \right]$. Calculate the dimensional formula of E and A . Use the definition of electric field to find its dimensional formula.

Formula used:
$\phi =\overrightarrow{E}.\overrightarrow{A}$
$E=\dfrac{F}{q}$

Complete step by step answer:
When an electric field is passing to an area, we define something that is called electric flux through this area. Suppose an uniform electric field of magnitude E is passing through an area A. Then the electric field is given as $\phi =\overrightarrow{E}.\overrightarrow{A}$.
On solve the dot product we get,
$\phi =EA\cos \theta $ ……….(i).
where $\theta $ is the angle between the electric field vector and the area vector.
With the help of equation (i) let us calculate the dimensional formula of electric flux.
 The dimensional formula of flux will be $\left[ \phi \right]=\left[ EA\cos \theta \right]=\left[ E \right]\left[ A \right]$ …… (ii).
The term $\cos \theta $ is just a number and has no dimension. Hence, it is not included in the dimensional formula.
For the dimensional formula of E, let us use the definition of electric field.
Electric field (E) is the force (F) that a charge Q exerts on a unit charge that is at distance r from it.
Therefore, the electric field is forced upon charge.
i.e. $E=\dfrac{F}{q}$,
Therefore dimensional formula of electric field will be $\left[ E \right]=\left[ \dfrac{F}{q} \right]=\dfrac{\left[ F \right]}{\left[ q \right]}$
We know that the dimensional formula of force is $\left[ F \right]=\left[ ML{{T}^{-2}} \right]$
The dimensional formula of charge is $\left[ q \right]=\left[ IT \right]$.
[I] is the dimensional formula of current.
Therefore,
$\left[ E \right]=\left[ \dfrac{F}{q} \right]=\dfrac{\left[ F \right]}{\left[ q \right]}=\dfrac{\left[ ML{{T}^{-2}} \right]}{\left[ IT \right]}=\left[ ML{{I}^{-1}}{{T}^{-3}} \right]$.
Hence, the dimensional formula of electric field E is $\left[ ML{{I}^{-1}}{{T}^{-3}} \right]$.
We know that the dimensional formula of area is $\left[ A \right]=\left[ {{L}^{2}} \right]$.
Substitute the found dimensional of electric field and area in equation (ii).
Therefore, we get
$\left[ \phi \right]=\left[ E \right]\left[ A \right]=\left[ MLI{{T}^{-3}} \right]\left[ {{L}^{2}} \right]=\left[ M{{L}^{3}}{{I}^{-1}}{{T}^{-3}} \right]$
This means that the dimensional formula of electric flux is $\left[ M{{L}^{3}}{{I}^{-1}}{{T}^{-3}} \right]$.
Hence, the correct option C.

Note: To find the dimensional formula of electric flux, we may also use Gauss’s law that gives the electric flux through a closed surface as $\phi =\dfrac{{{q}_{enclosed}}}{{{\varepsilon }_{0}}}$.
Hence, the dimensional formula of electric flux is $\left[ \phi \right]=\dfrac{\left[ {{q}_{enclosed}} \right]}{\left[ {{\varepsilon }_{0}} \right]}$.
We already know the dimensional formula of q i.e. $\left[ q \right]=\left[ IT \right]$.
The dimensional formula of ${{\varepsilon }_{0}}$ is $\left[ {{M}^{-1}}{{L}^{-3}}{{I}^{2}}{{T}^{4}} \right]$.
Therefore, $\left[ \phi \right]=\dfrac{\left[ {{q}_{enclosed}} \right]}{\left[ {{\varepsilon }_{0}} \right]}=\dfrac{\left[ IT \right]}{\left[ {{M}^{-1}}{{L}^{-3}}{{I}^{2}}{{T}^{4}} \right]}=\left[ M{{L}^{3}}{{I}^{-1}}{{T}^{-3}} \right]$.
We just have to know an expression for a quantity, in which we know the dimensions of each quantity present in the expression. With this, we can find dimensional formulas and units of the quantity.
Note that charge is not a fundamental physical quantity. Current is considered as one of the fundamental quantities. Since current is charge flowing in one unit of time, charge will be current times the time.