Question

The dimensional formula for permittivity of free space (${ϵ}_0$) in the equation $F={\displaystyle \frac{1}{4\pi {ϵ}_0}}.{\displaystyle \frac{q_1{q}_2}{r^2}}$ where, symbols have their usual meanings is
$A.[M^1{L}^3{A}^{-2}{T}^{-4}]$
$B.[M^{-1}{L}^{-3}{T}^4{A}^2]$
$C.[M^{-1}{L}^{-3}{A}^{-2}{T}^{-4}]$
$D.[M^1{L}^3{A}^2{T}^{-4}]$

Answer 1

Hint: We will know the fundamental dimensional formulas first. Then we are to find the formulas for different quantities present in the expression. Therefore, 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(F), charges ($q_2,q_2$) separately.
Force is given as the multiplication of mass and acceleration. Also, acceleration is the change in velocity per unit time. So, the dimension of acceleration is $L{T}^{-2}$ . Hence, the dimension of force is given by,
$[F]=[\text{mass}].[\text{acceleration}]=ML{T}^{-2}$
Again, electric charge is given by the product of current and time. So, the dimension of electric charge is, $[q]=AT$ .
Now, the given formula is
$F={\displaystyle \frac{1}{4\pi {ϵ}_0}}.{\displaystyle \frac{q_1{q}_2}{r^2}}$ .
From this we obtain,
${ϵ}_0={\displaystyle \frac{1}{4\pi F}}.{\displaystyle \frac{q_1{q}_2}{r^2}}$
Hence, the dimension of permittivity in free space is given as,
$[{ϵ}_0]={\displaystyle \frac{[q_1].[q_2]}{[F].[r^2]}}$
The factor $4\pi$ is a 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,
${ϵ}_0={\displaystyle \frac{1}{4\pi F}}.{\displaystyle \frac{q_1{q}_2}{r^2}}$
Hence, option B is the correct answer.
Additional information: There are some quantities that 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 Angle. Dielectric constant is another example of dimensionless quantity. It doesn’t have any dimension since it is the ratio of two permittivity.
There are some more fundamental dimensional formulas like,
[Temperature]=$\mathrm{\Theta }$, [Amount of matter] =N, [Luminous intensity] =J.
The dimensional formulas for any other physical quantity can be obtained by the seven fundamental dimensions.

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