The power P delivered by a system is given by the equation P = x + ( B^2 ₀ ) v where x is distance, B is magnetic field, ₀ is the permeability of free space, and v is velocity. The dimensional formula for the ratio is

Step-by-step solution

By the principle of dimensional homogeneity, each term in the equation must have the dimensions of power. The dimensional formula for power is [P] = [ML^2T⁻³] . For the first term: [ x ] = [P] [ ] [L] = [ML^2T⁻³] [ ] = [ML^3T⁻³] For the second term, the expression B^2 ₀ represents magnetic energy density (energy per unit volume). [ B^2 ₀ ] = [ML^2T⁻²] [L^3] = [ML⁻¹T⁻²] Now, equating the dimensions of the second term to power: [ ] [ B^2 ₀ ] [v] = [P] [ ] [ML⁻¹T⁻²] [LT⁻¹] = [ML^2T⁻³] [ ] [MT⁻³] = [ML^2T⁻³] [ ] = [L^2