A long solenoid with 1000 turns m$$^{-1}$$ 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 $$\frac{x}{499}$$. Find the value of $$x$$.

We have a long solenoid that originally contains a core material whose relative permeability is given as $$\mu_{r1}=500$$. The number of turns per unit length of the solenoid is $$n=1000\;\text{turns m}^{-1}$$ and the exciting current is $$I=0.75\;\text{A}$$. Later, the core is replaced by another material of the same volume but with a different relative permeability $$\mu_{r2}=750$$. We have to find the fractional change in the magnetic moment of the core.

First, recall the magnetic field strength inside a long solenoid. The standard result is

$$H = nI,$$

where $$H$$ is the magnetic field intensity, $$n$$ is the number of turns per unit length and $$I$$ is the current. Because both $$n$$ and $$I$$ remain the same before and after the replacement of the core, the quantity $$H$$ remains unchanged throughout this problem.

Next, recall the relationship between magnetisation $$M$$, magnetic susceptibility $$\chi_m$$ and the field strength $$H$$. The definition is

$$M = \chi_m\,H.$$

Magnetic susceptibility and relative permeability are connected through the familiar formula

$$\chi_m = \mu_r - 1.$$

Hence, for the first core material we have

$$\chi_{m1} = \mu_{r1} - 1 = 500 - 1 = 499,$$

and for the second core material we have

$$\chi_{m2} = \mu_{r2} - 1 = 750 - 1 = 749.$$

The magnetisations therefore become

$$M_1 = \chi_{m1}H = 499\,H,$$

$$M_2 = \chi_{m2}H = 749\,H.$$

The magnetic moment $$\mu$$ of a uniformly magnetised body is given by the product of its magnetisation $$M$$ and its volume $$V$$:

$$\mu = M\,V.$$

Because the volume of the core is stated to remain the same (namely $$V = 10^3\;\text{cm}^3$$, which we need not convert because it will cancel out), we may write

$$\mu_1 = M_1V = 499\,HV,$$

$$\mu_2 = M_2V = 749\,HV.$$

Now we compute the fractional change in the magnetic moment. The fractional change is the ratio of the change to the original value:

$$\text{Fractional change} = \frac{\mu_2 - \mu_1}{\mu_1}.$$ Substituting the expressions for $$\mu_2$$ and $$\mu_1$$ we get

$$\frac{749\,HV - 499\,HV}{499\,HV} = \frac{(749 - 499)\,HV}{499\,HV} = \frac{250}{499}.$$ Hence the fractional change can be written as $$\dfrac{x}{499}$$ with

$$x = 250.$$ Therefore, the required value is $$x = 250.$$

Hence, the correct answer is Option 250.