1. Magnetic field inside a long solenoid

For a long solenoid, $H = nI$ where:

• $n = 1000\ \text{turns/m}$
• $I = 0.75\ \text{A}$

So, $H = 1000 \times 0.75 = 750\ \text{A/m}$

2. Magnetization of the core

For a linear magnetic material, $M = \chi_m H$ where magnetic susceptibility $\chi_m = \mu_r - 1$

Hence magnetic moment of the core is $m = MV = \chi_m H V = (\mu_r - 1)HV$

Since $H$ and $V$ remain the same, magnetic moment is proportional to $m \propto (\mu_r - 1)$

3. Initial and final magnetic moments

For the first core: $\mu_{r1} = 500 \Rightarrow \chi_1 = 500 - 1 = 499$

Thus, $m_1 \propto 499$

For the second core: $\mu_{r2} = 750 \Rightarrow \chi_2 = 750 - 1 = 749$

Thus, $m_2 \propto 749$

4. Fractional change in magnetic moment

Fractional change is $\frac{m_2 - m_1}{m_1} = \frac{749 - 499}{499} = \frac{250}{499}$

Comparing with the given form $\frac{x}{499}$ we get $x = 250$

5. Final answer

$\boxed{250}$