A long solenoid with \(1000\, turns/m\) has a core material with relative permeability \(500\) and volume \(10^{3}\, {cm}^{3} .\) If the core material is replaced by another material having relative permeability of \(750\) with same volume maintaining same current of \(0.75\, {A}\) in the solenoid, the fractional change in the magnetic moment of the core would be approximately \(\left(\frac{{x}}{499}\right) .\) Find the value of \({x}\).

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.