An insulated wire is wound so that it forms a flat coil with $$N = 200$$ turns. The radius of the innermost turn is $$r_1 = 3$$ cm, and of the outermost turn $$r_2 = 6$$ cm. If 20 mA current flows in it then the magnetic moment will be $$\alpha \times 10^{-2}$$ A·m². The value of $$\alpha$$ is :

The magnetic moment of a planar coil equals the algebraic sum of magnetic moments of its individual turns.
For one circular turn of radius $$r$$ carrying current $$I$$, magnetic moment $$m = I \, A = I \, (\pi r^{2})$$, directed perpendicular to the plane.

The coil here is a tightly wound flat spiral: the radius grows uniformly from $$r_1 = 3 \text{ cm}$$ to $$r_2 = 6 \text{ cm}$$ in $$N = 200$$ turns. If the turns are uniformly spaced in the radial direction, the number of turns per unit radius is$$ \frac{dn}{dr} = \frac{N}{\,r_2 - r_1\,}. $$

Total magnetic moment $$M$$ is obtained by integrating the contribution $$I \pi r^{2}\, dn$$ from $$r_1$$ to $$r_2$$:$$ M = \int_{r_1}^{r_2} I \pi r^{2}\, dn = I \pi \int_{r_1}^{r_2} r^{2}\left(\frac{N}{r_2 - r_1}\right) dr. $$

Carry out the integral:$$ M = \frac{I \pi N}{r_2 - r_1}\left[ \frac{r^{3}}{3} \right]_{r_1}^{r_2} = \frac{I \pi N}{3}\,\frac{r_2^{3} - r_1^{3}}{r_2 - r_1}. $$

Factor the difference of cubes: $$r_2^{3} - r_1^{3} = (r_2 - r_1)(r_2^{2} + r_1 r_2 + r_1^{2})$$. The factor $$r_2 - r_1$$ cancels, giving the compact formula

$$ M = \frac{I \pi N}{3}\left(r_2^{2} + r_1 r_2 + r_1^{2}\right). \quad -(1) $$

Insert the numerical data (convert cm to m):
$$r_1 = 0.03 \text{ m}, \; r_2 = 0.06 \text{ m}, \; N = 200,\; I = 20 \text{ mA} = 0.02 \text{ A}.$$ Compute the bracket in $$(1)$$:
$$ r_2^{2} = 0.06^{2} = 0.0036, \quad r_1 r_2 = 0.03 \times 0.06 = 0.0018, \quad r_1^{2} = 0.03^{2} = 0.0009. $$ Sum $$ = 0.0036 + 0.0018 + 0.0009 = 0.0063.$$ Dividing by 3: $$\frac{0.0063}{3} = 0.0021.$$ Now evaluate $$M$$:
$$ M = I \pi N \times 0.0021 = \bigl(0.02 \times 200\bigr)\pi \times 0.0021 = 4 \times \pi \times 0.0021 \approx 4 \times 3.1416 \times 0.0021 \approx 0.0264 \text{ A·m}^{2}. $$

Expressing $$M$$ as $$\alpha \times 10^{-2} \text{ A·m}^{2}$$ gives $$\alpha = 2.64.$$ Hence

Option B which is: $$2.64$$