JEE MainPhysicsUnits and Dimensions
The interaction force F between two particles in a specific medium is modeled by the equation F = (- x^2 k T ) , where x is the distance, k is the Boltzmann constant, T is the absolute temperature, and , are constants. The dimensional formula of the constant is:
Options
- A[ M L T ⁻²]
- B[ M ^2 L T ⁻⁴]
- C[ M ⁻² L ⁻¹ T ^4]
- D[ M ^2 L ^0 T ⁻⁴]
Correct answer
B. [ M ^2 L T ⁻⁴]
Step-by-step solution
The argument of an exponential function must be dimensionless. Therefore, [ x^2 k T ] = [ M ^0 L ^0 T ^0] The product kT represents thermal energy, so its dimensional formula is [kT] = [ ML ^2 T ⁻²] . Equating dimensions for the exponent: [ ] = [x^2] [kT] = [ L ^2] [ ML ^2 T ⁻²] = [ M ⁻¹ T ^2] Since the entire exponential term (- x^2 k T ) is dimensionless, the dimensions of the left hand side must equal the dimensions of the pre-factor on the right hand side. [F] = [ ][ ] Substituting the dimensional formula for f