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**Hint:**We will know the fundamental dimensional formulas in the beginning. Then we have to find the formulas for different quantities present in the expression. After putting them in the expression, the required dimensional formula can be found.

**Complete step-by-step solution:**

The dimensional formulas for mass, time, current, and length are $M$ , $T$ , $A$ and $L$ respectively.

Now, we will see the dimensions of force $\left( F \right)$ , charges $\left( {{q_1},{q_2}} \right)$ separately.

Force is known as the multiplication of mass and acceleration. Also, acceleration is the change in velocity per unit of time. So, the dimension is $L{T^{ - 2}}$ . Hence, the dimension of force is given by,

$\left[ F \right] = \left[ {mass} \right].\left[ {acceleration} \right] = ML{T^{ - 2}}$

Again, the electric charge is given by the product of current and time. So, the dimension of the electric charge is, $\left[ q \right] = AT$ .

Now, the given formula is

$F = \dfrac{1}{{4\pi \varepsilon }}.\dfrac{{{q_1}{q_2}}}{{{r^2}}}$

Hence, the dimension of permittivity in free space is known as,

$\left[ \varepsilon \right] = \dfrac{{\left[ {{q_1}} \right].\left[ {{q_2}} \right]}}{{\left[ F \right].\left[ {{r^2}} \right]}}$

The factor $4\pi $ is constant and doesn’t have any dimension. So, after putting all the dimensions in the above equation and adding the different powers appropriately, we obtain,

$\left[ \varepsilon \right] = \dfrac{{\left[ {AT} \right]\left[ {AT} \right]}}{{\left[ {ML{T^{ - 2}}} \right]\left[ {{L^2}} \right]}}$

$ \Rightarrow \left[ \varepsilon \right] = \dfrac{{\left[ {{A^2}{T^2}} \right]}}{{\left[ {M{L^3}{T^{ - 2}}} \right]}}$

On further solving the above equation,

$\left[ \varepsilon \right] = \left[ {{M^{ - 1}}{L^{ - 3}}{T^4}{A^2}} \right]$

**Hence, option (D) is the correct answer.**

**Additional information:**Some quantities don’t have any dimension at all. They are the ratio of two quantities that have equal dimensional formulas. They are called dimensionless quantities. An example of such a quantity is the angle. The dielectric constant is another example of a dimensionless quantity. It doesn’t have any dimension since it is the ratio of two permittivities.

There are some more fundamental dimensional formulas like,

[Temperature] = $\theta $ , [Amount of matter] = $N$ , [Luminous intensity] = $J$ .

The dimensional formulas for any other physical quantities can be obtained by the seven fundamental dimensions.

**Note:**Keep in mind that the dimensions for ${q_1}$ and ${q_2}$ are the same since both are different values of the same physical quantity called electric charge. Be very careful while adding and subtracting the powers of different dimensions. The numerical factors like $3,4,7,....$, etc. don’t have any dimension.

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