Question

Find the dimensions of electric permittivity

• A. $A^2{M}^{-1}{L}^{-3}{T}^4$
• B. $A^2{M}^{-1}{L}^{-3}{T}^0$
• C. $A{M}^{-1}{L}^{-3}{T}^4$
• D. $A^2{M}^0{L}^{-3}{T}^4$

Answer 1

The correct option is A

$A^2{M}^{-1}{L}^{-3}{T}^4$

Step 1. Known dimensional formulas

We know

Dimensional formula for $q^2$ is $\left[A{T}^2\right]$

Dimensional formula for $F$ is $\left[ML{T}^{-2}\right]$

Dimension formula for $r$ is $L^2$

Step 2. calculating dimensions of electric permittivity

We have,

${\epsilon }_0$ is the permittivity

$q$ is the charge

$F$ is the force

$r$ is the distance between the charges

Now, by substituting these dimensional formula in the formula, we get

, $$\begin{gathered} F=\frac{1}{4\pi {\epsilon }_0}\frac{q^2}{r^2} \\ {\epsilon }_0=\frac{1}{4\pi F}\frac{q^2}{r^2} \\ \left[{\epsilon }_0\right]=\frac{{\left[AT\right]}^2}{\left[ML{T}^{-2}\right]\left[L^2\right]} \\ =\left[A^2{M}^{-1}{L}^{-3}{T}^4\right] \end{gathered}$$

Hence, the correct option is (A).