JEE MainPhysicsUnits and Dimensions
The energy E of a particle as a function of its position x and time t is given by the empirical relation E = a - x^2 b t , where a and b are unknown constants. The dimensional formula of the product (a b) is:
Options
- A[ M ⁻¹ L ^2 T ]
- B[ M L ^2 T ⁻¹]
- C[ M ⁻¹ L ^3 T ]
- D[ M ⁻¹ L ⁻² T ]
Correct answer
A. [ M ⁻¹ L ^2 T ]
Step-by-step solution
According to the principle of dimensional homogeneity, quantities added or subtracted must have the same dimensions. Therefore, the dimension of a must be equal to the dimension of x^2 . [a] = [x^2] = [ L ^2] Now, equating the dimensions of both sides of the given equation: [E] = [a] [b][t] Substituting the dimensional formulas for energy [E] = [ ML ^2 T ⁻²] and time [t] = [ T ] : [ ML ^2 T ⁻²] = [ L ^2] [b][ T ] [b] = [ L ^2] [ ML ^2 T ⁻²][ T ] = [ L ^2] [ ML ^2 T ⁻¹] = [ M ⁻¹ T ] The dimensional formula of the pr