1. Given data

• Volume of iron rod:
  $V = 10^{-3}\,\text{m}^3$
• Relative permeability:
  $\mu_r = 1000$
• Turns per cm:
  $10\,\text{turns/cm} = 1000\,\text{turns/m}$ So, $n = 1000\,\text{m}^{-1}$
• Current through solenoid:
  $I = 0.5\,\text{A}$

2. Magnetic field intensity inside the solenoid

For a long solenoid, $H = nI$ So, $H = 1000 \times 0.5 = 500\,\text{A/m}$

3. Magnetization of the iron core

For a magnetic material, $M = \chi_m H$ where magnetic susceptibility $\chi_m = \mu_r - 1 = 1000 - 1 = 999$ Since $\mu_r$ is very large, we can take $\chi_m \approx 1000$ Hence, $M \approx 1000 \times 500 = 5 \times 10^5\,\text{A/m}$

4. Magnetic moment of the rod

Magnetic moment is $m = MV$ Therefore, $m = 5 \times 10^5 \times 10^{-3} = 5 \times 10^2\,\text{A m}^2$

So, $m = 500\,\text{A m}^2$

5. Match with options

$500\,\text{A m}^2 = 5 \times 10^2\,\text{A m}^2$

So the correct option is:

A: $5 \times 10^2\,\text{A m}^2$

6. Comparison with stored answer

Stored correct answer: A
Derived answer: A

They agree.