Question

The dimensional formula for permittivity of free space $\left({\epsilon }_0\right)$ in equation $F=\frac{1}{4\pi {\epsilon }_0}.\frac{q_1{q}_2}{r^2}$ where symbols have their usual meaning is ________.

Answer 1

1. Permittivity: It is the property of every material, which measures the opposition offered against the formation of an electric field.
2. The permittivity of free space: The vacuum characterizes the least possible permittivity value. This is called the Permittivity of Free Space or electric constant.
3. The dimensional formula for permittivity of free space:
   $$\begin{gathered} F=\frac{1}{4\pi {\epsilon }_0}.\frac{q_1{q}_2}{r^2} \\ \Rightarrow {\epsilon }_0=\frac{1}{4\pi F}.\frac{q_1{q}_2}{r^2} \end{gathered}$$
4. Dimensions of F $=\left[ML{T}^{-2}\right]$
5. Dimensions of q $=\left[AT\right]$
6. Dimensions of r $=\left[L\right]$
7. Now put the dimensional formula of all other terms:
   $$\begin{gathered} {\epsilon }_0=\frac{1}{4\pi F}.\frac{q_1{q}_2}{r^2} \\ \Rightarrow \left[{\epsilon }_0\right]=\frac{\left[AT\right]\left[AT\right]}{\left[ML{T}^{-2}\right]\left[L^2\right]} \\ \Rightarrow \left[{\epsilon }_0\right]=\left[M^{-1}{L}^{-3}{T}^4{A}^2\right] \end{gathered}$$

Thus, the dimensional formula for permittivity of free space is $\left[M^{-1}{L}^{-3}{T}^4{A}^2\right]$.