A physical quantity X is defined by the relation X = a b^2 c + d^2 . It is given that X has the dimensions of pressure, a has the dimensions of force, and c has the dimensions of length. The dimensional formula of the ratio b d is:

Step-by-step solution

According to the principle of homogeneity of dimensions, quantities added together must have the same dimensions. From the denominator (c + d^2) : [d^2] = [c] = [ L ] [d] = [ L ^ 1/2 ] The dimensional formula of the entire equation is: [X] = [a][b^2] [c] We are given that X is pressure and a is force. Substituting their dimensional formulas: [ M L ⁻¹ T ⁻²] = [ M L T ⁻²][b^2] [ L ] Simplifying the right side: [ M L ⁻¹ T ⁻²] = [ M T ⁻²][b^2] Solving for [b^2] : [b^2] = [ M L ⁻¹ T ⁻²] [ M T ⁻²] = [ L ⁻¹] [b] = [ L ^ -