If force ( F ), velocity ( v ), time ( T ), and electric current ( I ) are taken as the fundamental base quantities, what is the dimensional formula for the permittivity of free space ( ₀ )?

Step-by-step solution

The electrostatic force between two point charges is given by Coulomb's law: F = 1 4 ₀ q₁ q₂ r^2 Rearranging for the permittivity of free space ₀ : ₀ = q₁ q₂ 4 F r^2 We need to express the dimensions of charge q and distance r in terms of the new fundamental quantities F , v , T , and I . Electric charge q = current time [q] = I T Distance r = velocity time [r] = v T Substituting these into the dimensional equation for ₀ (ignoring the dimensionless constant 4 ): [ ₀] = [I T]^2 [F] [v T]^2 [ ₀] = I^2 T^2 F v^2 T^2 T