What direction does magnetic dipole moment feel force in parallel magnetic field?

Question

Suppose I have a magnetic dipole oriented vertically in the $\hat{z}$ direction so

$$ \vec{m} = m\hat{z}$$

And the position vector to it is $\vec{r} = z\hat{z}$. It's entirely on the z-axis.

We're using cylindrical coordinates and I have a ring-like current loop with radius $a$ at the origin and the current ($I$) is going in the $\hat{\phi}$ and obviously the magnetic dipole is right above the center of it.

I calculated the magnetic field of the current loop using Bio-Savart's Law and got

$$\vec{B}_{loop}(\vec{r} = z\hat{z}) = \frac{\mu_o Ia^2}{2\pi(a^2 + z^2)^{3/2}}$$

Thus the force that the dipole feels should be $$\vec{F}_{dipole} = \nabla(\vec{m} \cdot \vec{B}_{loop})$$

But then I get that $\vec{F}_{dipole} < 0$ so it's pointing downwards towards the loop. Since the dipole and magnetic field are parallel to each other and aligned, shouldn't the force be upwards? Hence, a positive value?

Answer 1

You can view magnetic dipole just as yet another current loop which is parallel to the existing one, with the currents being in the same direction. Thus you basically have two parallel conductors and you know (hopefully) that there is always attractive force between them when currents are parallel.