A 25 cm long solenoid has the radius 2 cm and 500 turns. It carries a current of 15 A. If it is equivalent to a magnet of the same size and magnetization $$\vec{M}$$ $$\left(\frac{\text{Magnetic Moment}}{\text{volume}}\right)$$, then $$|\vec{M}|$$ is:

We are given a solenoid with length 25 cm, radius 2 cm, 500 turns, and current 15 A. We need to find the magnitude of the magnetization $$\vec{M}$$, which is defined as the magnetic moment per unit volume. The solenoid is equivalent to a magnet of the same size, so we will calculate the magnetic moment of the solenoid and then divide by its volume.

First, convert all units to SI units. Length $$L = 25$$ cm = $$25 \times 10^{-2}$$ m = 0.25 m. Radius $$r = 2$$ cm = $$2 \times 10^{-2}$$ m = 0.02 m. Current $$I = 15$$ A. Number of turns $$N = 500$$.

The magnetic moment ($$m$$) of a solenoid is given by the product of the number of turns, the current, and the area of one turn. The area $$A$$ of one turn is the cross-sectional area of the solenoid, which is circular: $$A = \pi r^2$$.

Calculate $$A$$:

$$ A = \pi \times (0.02)^2 = \pi \times 0.0004 = 4 \times 10^{-4} \pi \text{ m}^2 $$

Now, calculate the magnetic moment $$m$$:

$$ m = N \times I \times A = 500 \times 15 \times (4 \times 10^{-4} \pi) $$

First, multiply 500 and 15:

$$ 500 \times 15 = 7500 $$

Then multiply by $$4 \times 10^{-4} \pi$$:

$$ 7500 \times 4 \times 10^{-4} \pi = 7500 \times 4 \times 0.0001 \pi = 7500 \times 0.0004 \pi $$

$$ 7500 \times 0.0004 = 3 \quad \text{(since } 7500 \times 4 \times 10^{-4} = 7500 \times 4 \times 0.0001 = 30000 \times 0.0001 = 3\text{)} $$

So,

$$ m = 3\pi \text{ A} \cdot \text{m}^2 $$

The volume ($$V$$) of the solenoid is the product of its cross-sectional area and length:

$$ V = A \times L = (4 \times 10^{-4} \pi) \times 0.25 $$

First, multiply 0.25 and $$4 \times 10^{-4}$$:

$$ 0.25 \times 4 \times 10^{-4} = 1 \times 10^{-4} = 0.0001 $$

So,

$$ V = 0.0001 \pi = \pi \times 10^{-4} \text{ m}^3 $$

Magnetization $$\vec{M}$$ is the magnetic moment per unit volume:

$$ |\vec{M}| = \frac{m}{V} = \frac{3\pi}{\pi \times 10^{-4}} $$

Simplify by canceling $$\pi$$:

$$ |\vec{M}| = \frac{3}{10^{-4}} = 3 \times 10^{4} = 30000 \text{ A m}^{-1} $$

Comparing with the options, 30000 A m$$^{-1}$$ corresponds to option B. The problem states that the correct answer is option 2, which is B.

Hence, the correct answer is Option B.