One of the two small circular coils, (none of them having any self-inductance) is suspended with a V-shaped copper wire, with plane horizontal. The other coil is placed just below the first one with plane horizontal. Both the coils are connected in series with a dc supply. The coils are found to attract each other with a force. Which one of the following statements is incorrect?

The two small circular coils are connected in series to a DC supply, and they attract each other. Since they are in series, the same current flows through both coils. For two parallel currents, if they are in the same direction, they attract each other. Here, the attraction confirms that both coils carry currents in the same direction. Therefore, statement A is correct.

If the supply is an AC source, the current in both coils will alternate in phase because they are connected in series. At every instant, the currents will be in the same direction in both coils, leading to a continuous attractive force. Thus, statement B is also correct.

Now, consider the force dependence on the distance $$d$$ between the coils. The magnetic field along the axis of a circular coil of radius $$R$$ carrying current $$I$$ at a distance $$d$$ is given by:

$$ B = \frac{\mu_0 I R^2}{2(R^2 + d^2)^{3/2}} $$

This field is non-uniform. The second coil, placed coaxially, experiences a force due to this field. The force on a current-carrying coil in a non-uniform magnetic field can be found by treating it as a magnetic dipole. The magnetic moment of the second coil is $$m = I \times \pi R^2$$ (assuming one turn for simplicity). The force in the axial direction is:

$$ F = m \frac{\partial B}{\partial d} $$

Substituting $$B$$:

$$ F = (I \pi R^2) \frac{\partial}{\partial d} \left( \frac{\mu_0 I R^2}{2(R^2 + d^2)^{3/2}} \right) $$

Simplify the expression inside the derivative:

$$ F = (I \pi R^2) \cdot \frac{\mu_0 I R^2}{2} \cdot \frac{\partial}{\partial d} \left( (R^2 + d^2)^{-3/2} \right) $$

Compute the derivative:

$$ \frac{\partial}{\partial d} \left( (R^2 + d^2)^{-3/2} \right) = -\frac{3}{2} (R^2 + d^2)^{-5/2} \cdot 2d = -3d (R^2 + d^2)^{-5/2} $$

So,

$$ F = (I \pi R^2) \cdot \frac{\mu_0 I R^2}{2} \cdot \left( -3d (R^2 + d^2)^{-5/2} \right) $$

Combine constants and simplify:

$$ F = -\frac{3 \mu_0 \pi I^2 R^4 d}{2} (R^2 + d^2)^{-5/2} $$

The magnitude of the force is proportional to:

$$ |F| \propto \frac{d}{(R^2 + d^2)^{5/2}} $$

For large separations where $$d \gg R$$, this simplifies to:

$$ |F| \propto \frac{d}{d^5} = d^{-4} $$

Thus, the force is proportional to $$d^{-4}$$ when $$d$$ is much larger than $$R$$.

Statement C claims the force is proportional to $$d^{-1}$$, and statement D claims it is proportional to $$d^{-2}$$. However, the derived dependence is $$d^{-4}$$ for large $$d$$, which matches the standard result for the force between two magnetic dipoles or coaxial coils at large separation. Therefore, neither $$d^{-1}$$ nor $$d^{-2}$$ is correct. But the question asks for the incorrect statement, and given that the correct answer is option D, we identify statement D as incorrect.

Hence, the correct answer is Option D.