An iron rod of volume $$10^{-3}\,\text{m}^3$$ and relative permeability 1000 is placed as core in a solenoid with 10 turns $$\text{cm}^{-1}$$. If a current of $$0.5\,\text{A}$$ is passed through the solenoid, then the magnetic moment of the rod will be:

We are asked to find the magnetic dipole moment of the iron rod after it is inserted in the solenoid and an electric current is sent through the turns. Let us introduce the symbols that will be used:

$$$V = 10^{-3}\,\text{m}^3 \qquad (\text{volume of the rod})$$$ $$$\mu_r = 1000 \qquad (\text{relative permeability of iron})$$$ $$$n = 10\,\text{turns cm}^{-1} \qquad (\text{number of turns per centimetre})$$$ $$$I = 0.5\,\text{A} \qquad (\text{current through the solenoid})$$$

First we convert the turn density into SI units (turns per metre). Since $$1\,\text{cm} = 10^{-2}\,\text{m},$$ we obtain

$$$n = 10\,\text{turns cm}^{-1} = 10 \times \frac{1}{10^{-2}}\,\text{turns m}^{-1} = 10 \times 100\,\text{turns m}^{-1} = 1000\,\text{turns m}^{-1}.$$$

Now we recall the definition of the magnetising field $$H$$ inside an ideal long solenoid. The standard formula is

$$H = n I,$$

where $$n$$ is the number of turns per metre and $$I$$ is the current in amperes. Substituting the known values we get

$$$H = (1000\,\text{turns m}^{-1})(0.5\,\text{A}) = 500\,\text{A m}^{-1}.$$$

Next we connect the magnetising field $$H$$ to the magnetisation $$M$$ of the iron core. For a linear magnetic material we use the relation

$$M = \chi_m H,$$

where $$\chi_m$$ is the magnetic susceptibility given by

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

For the present rod,

$$\chi_m = 1000 - 1 = 999.$$

Hence the magnetisation becomes

$$$M = 999 \times 500\,\text{A m}^{-1} = 499{,}500\,\text{A m}^{-1} = 4.995 \times 10^{5}\,\text{A m}^{-1}.$$$

The total magnetic dipole moment $$m$$ of the rod is obtained by multiplying the magnetisation $$M$$ by the volume $$V$$ (because magnetisation is magnetic moment per unit volume):

$$m = M V.$$

Substituting $$M = 4.995 \times 10^{5}\,\text{A m}^{-1}$$ and $$V = 10^{-3}\,\text{m}^3$$ gives

$$$m = (4.995 \times 10^{5}\,\text{A m}^{-1})(10^{-3}\,\text{m}^3) = 4.995 \times 10^{2}\,\text{A m}^2.$$$

In scientific notation this is essentially

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

This numerical value exactly matches option B among the given choices.

Hence, the correct answer is Option B.