JEE MainPhysicsUnits and Dimensions
The magnetic potential energy U of a system is given by the relation U = (1 + x ₀ I^2 ) , where and are constants, x is a characteristic length, I is electric current, and ₀ is the permeability of free space. The dimensions of will be
Options
- A[ M ^0 L ⁻¹ T ^0 ]
- B[ M ^0 L ⁻² T ^0 ]
- C[ M ^0 L ^1 T ^0 ]
- D[ M ^0 L ^2 T ^0 ]
Correct answer
B. [ M ^0 L ⁻² T ^0 ]
Step-by-step solution
The term ₀ I^2 has the dimensions of force, since the force per unit length between two parallel current-carrying wires is proportional to ₀ I^2 / d . [ ₀ I^2] = [ MLT ⁻²] The argument of the logarithm must be dimensionless, so: [ x ₀ I^2 ] = [ M ^0 L ^0 T ^0] [ ] = [ ₀ I^2] [x] = [ MLT ⁻²] [ L ] = [ MT ⁻²] The dimensions of potential energy are [U] = [ ML ^2 T ⁻²] . Equating the dimensions of the LHS and RHS of the given equation: [U] = [ ] [ ] [ ] = [ ] [U] = [ MT ⁻²] [ ML ^2 T ⁻²] = [ M ^0 L ⁻² T ^0] Answer: [ M