A physical quantity is defined as X = L^x C^y R^z , where L , C , and R are self-inductance, capacitance, and resistance respectively, and x, y, z are non-zero real numbers. If X is found to be entirely independent of the fundamental dimension of electric current [A] , what must be true about its dependence on the dimensions of mass [M] and length [L] ?

Step-by-step solution

The dimensional formulas for the given quantities are: [L] = [M L^2 T⁻² A⁻²] [C] = [M⁻¹ L⁻² T^4 A^2] [R] = [M L^2 T⁻³ A⁻²] For the quantity X = L^x C^y R^z , the combined dimensional formula is: [X] = [M L^2 T⁻² A⁻²]^x [M⁻¹ L⁻² T^4 A^2]^y [M L^2 T⁻³ A⁻²]^z Grouping the fundamental dimensions: [X] = [M^ x - y + z L^ 2x - 2y + 2z T^ -2x + 4y - 3z A^ -2x + 2y - 2z ] It is given that X is independent of the dimension of electric current [A] . Therefore, the exponent of A must be zero: -2x + 2y - 2z = 0 y = x + z Now, s