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

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?

• Possibly helpful. It is helpful to think of a dipole aligned with the local field as a “strong-field seeker,” and a dipole anti-aligned with the local field as a “weak-field seeker.”
– rob
Nov 25, 2021 at 15:46