If ₀ is the permeability of free space and ₀ is the permittivity of free space, then which of the following expressions has the dimensional formula [ M L ^2 T ⁻³ A ⁻²] ?

Step-by-step solution

The given dimensional formula [ M L ^2 T ⁻³ A ⁻²] corresponds to electrical resistance ( R ). We know the dimensions of permeability and permittivity of free space are: [ ₀] = [ M L T ⁻² A ⁻²] [ ₀] = [ M ⁻¹ L ⁻³ T ^4 A ^2] Let us evaluate the ratio ₀ ₀ : [ ₀] [ ₀] = [ M L T ⁻² A ⁻²] [ M ⁻¹ L ⁻³ T ^4 A ^2] = [ M ^2 L ^4 T ⁻⁶ A ⁻⁴] Taking the square root of this ratio gives: ₀ ₀ = ( [ M ^2 L ^4 T ⁻⁶ A ⁻⁴] )^ 1/2 = [ M L ^2 T ⁻³ A ⁻²] Alternatively, one can recognize that the characteristic impedance of free space is