Forgotten Physics — A Bead, a Spinning Loop, and the Magnetic Dance of Time

What happens when you combine a charged bead, a spinning circular loop, a magnetic field, and a time-varying rotation? Most physics problems don't touch this—but this one opens a gateway to advanced motion dynamics, chaos, and control systems.

The Thought Experiment

A bead of mass m and charge q slides without friction on a circular wire loop (radius R), lying horizontally. The loop rotates around its vertical axis with angular velocity ω(t), and a uniform magnetic field B⃗ = B𝑧̂ points upward.

Question:
What is the equation of motion for the bead, observed from the rotating frame, including centrifugal, Coriolis, and Lorentz forces?

The Physics

Let θ be the bead’s angle from the top of the loop.

We use the Lagrangian approach in the rotating frame, which includes:

• Kinetic energy from motion and rotation
• Potential energy from gravity
• Magnetic vector potential via minimal coupling

Lagrangian:

L = (1/2) m R² θ̇² + (1/2) m ω(t)² R² sin²θ + (1/2) q B R² θ̇ sinθ + mgR cosθ
        

Equation of Motion:

Applying the Euler-Lagrange equation gives:

m R² θ̈ + (1/2) q B R² cosθ · θ̇ − m R² ω(t)² sinθ cosθ − (1/2) q B R² θ̇ cosθ + mgR sinθ = 0
        

Simplified:

m R² θ̈ = m R² ω(t)² sinθ cosθ − mgR sinθ
        

Why This Is Exciting

This system reveals:

• Equilibrium bifurcation points depending on ω(t)
• Interplay between magnetic torque and non-inertial forces
• A realistic, chaotic physical model useful for:
  • Teaching non-inertial frames
  • Exploring time-varying angular systems
  • Simulating charged particle motion in engineered magnetic fields

If this sparks your curiosity or you'd love to model, simulate, or teach with this system, let’s connect.

Let’s reimagine what physics can do — one forgotten equation at a time.

— Bubbles (Jeff), Developer | Educator | Streamer | Founder of FNBubbles420 Org