16. Magnetic carousel

Official Problem Statement

“Several neodymium magnets are attached to a horizontal disc, with adjacent magnets oriented in opposite directions. A fixed non-magnetic plate is placed horizontally above the disc. When the disc rotates about its vertical axis and a steel ball is placed on the plate, the ball may move either in the same direction as, or opposite to, the rotation of the disc. Investigate this phenomenon and determine how the motion depends on relevant parameters.”

Rotating Magnetic Field and Periodic Potential

$n$ pairs of alternating-polarity magnets on the disc create a checkerboard magnetic field pattern that rotates with the disc. In the lab frame, the magnetic potential energy of the steel ball at angular position $\phi_{\rm ball}$ above a disc rotating at $\phi_{\rm disc} = \Omega_{\rm disc}\, t$ is:

$$U(\phi_{\rm ball} – \phi_{\rm disc}) = -U_0\cos\!\left(n(\phi_{\rm ball} – \phi_{\rm disc})\right)$$

where $U_0 > 0$ is the potential amplitude and $n$ is the number of magnet pairs. The ferromagnetic steel ball is attracted toward field maxima.

The magnetic torque on the ball at relative angle $\Delta\phi = \phi_{\rm ball} – \phi_{\rm disc}$:

$$\tau_{\rm mag} = -\frac{dU}{d\phi_{\rm ball}} = -n U_0\sin(n\Delta\phi)$$

Equation of Motion: Co-rotation vs. Counter-rotation

The ball rolls on the fixed plate. With ball mass $m$ and radius $r_{\rm ball}$, the rotational inertia of a solid sphere is $I = \frac{2}{5}mr_{\rm ball}^2$. For a ball at radius $R$ from the disc center, the equation of angular motion of the ball about the disc axis:

$$(mR^2 + I)\ddot{\phi}_{\rm ball} = \tau_{\rm mag}(\Delta\phi) – \tau_{\rm friction}$$

where $\tau_{\rm friction} = \mu_k\, mg\, R$ is the rolling friction torque.

Motion Regimes

Low-speed regime (co-rotation): When $\Omega_{\rm disc}$ is small, the magnetic torque is sufficient to keep the ball synchronized with the disc: $\dot{\phi}_{\rm ball} \approx \dot{\phi}_{\rm disc} = \Omega_{\rm disc}$. The ball moves in the same direction as the disc rotation.

High-speed regime (counter-rotation): When $\Omega_{\rm disc}$ exceeds the critical speed $\Omega_c$, the ball can no longer stay synchronized. In the disc’s rotating frame, the ball slips backward through the periodic potential. The nonlinearity of $\sin(n\Delta\phi)$ produces a net torque that, on average, drives the ball opposite to the disc’s rotation direction.

The critical angular velocity is reached when the maximum magnetic torque can no longer overcome inertia and friction at the required synchronization rate:

$$n U_0 \geq I_{\rm eff}\,\Omega_{\rm disc} / \tau_{\rm lag} + \tau_{\rm friction}$$

More simply: the transition from co-rotation to counter-rotation occurs when the ball’s lag behind the rotating field exceeds half a spatial period $\pi/(n\Omega_{\rm disc})$.

Parameters

Experimental Setup

To map ball direction and speed as a function of $\Omega_{\rm disc}$:

1. Drive the disc with a DC motor at adjustable speed. Install $n = 4$–$8$ magnet pairs symmetrically, with adjacent magnets reversed.
2. Fix a transparent (e.g., acrylic) non-magnetic plate at a well-defined height above the disc.
3. Place a steel ball on the plate and record its angular velocity and direction with an overhead camera.
4. Increase $\Omega_{\rm disc}$ from zero and note the critical speed $\Omega_c$ at which the ball reverses direction.
5. Repeat for different values of $n$, air gap $g_{\rm gap}$, ball radius, and radial position $R$.
6. To measure $U_0$ independently, use a torque sensor to record the static magnetic torque as a function of angle.

References

• Griffiths, D. J., Introduction to Electrodynamics, 4th ed., Pearson, 2013 — Magnetic dipole field and potential energy.
• Jiles, D., Introduction to Magnetism and Magnetic Materials, 3rd ed., CRC Press, 2015 — Ferromagnetism and attractive force.
• Chikazumi, S., Physics of Ferromagnetism, 2nd ed., Oxford, 1997.
• Sandri, G. et al., “Synchronization and desynchronization of rotating magnetic fields,” Am. J. Phys. 78 (2010) 301–307.