Thompson's Jumping Ring with the ring in the centre of the solenoid

[catkin]

Homework Statement

This is a personal enquiry, not a homework question. I think I understand but would like confirmation.

In a modified Thompson's Jumping Ring apparatus, the solenoid is horizontal and the ring free to move on the iron core. If the ring is either side of the central position when the current is switched on the ring is accelerated away from the centre of the solenoid. If the current is already flowing and the ring is positioned either side of the central position then, when the current is turned off the ring is accelerated toward the centre. If the ring is in the centre when the current is switched on or off it does not move.

What is happening, in terms of induced currents and magnetic fields, when the ring is in the centre and why is no net force generated?

Homework Equations

• Faraday: e.m.f. = - dΦ/dt
• Magnetic flux density in a long solenoid: B = μ₀ n I
• Magnetic flux: Φ = BA
• Lenz' law: "An induced current is always in such a direction as to oppose the motion or change causing it"
• Fleming's left hand rule

The Attempt at a Solution

With the ring at the centre, when the current is switched on the flux density in the solenoid rises. The flux passes through the ring, changing the ring's flux linkage (proportional to its cross-sectional area), the changing linkage generates an e.m.f in the ring which generates a current in the ring which generates a magnetic field. By Lenz' law this opposes the change that produced it; the direction of the ring's magnetic field is opposite to the solenoid's.

There is a magnetic field (the ring's field only opposes the solenoid's, it does not cancel it out) and there is a current so there must be force. Applying Fleming's left hand rule the current flows around the ring, the magnetic flux passes axially through it. The resultant force is radial and evenly distributed around the circumference resulting in no net force to accelerate the ring.

The implication of this reasoning is that it is only the the non-axial component of the magnetic flux that causes the ring to move. The classic Thompson's Jumping Ring apparatus has an iron-cored solenoid and I understand that an iron core is so effective at containing the flux that approximately no flux escapes; it is ~all contained within the core. But it cannot be or the ring would not move! Am I getting this right?

This line of reasoning also suggests that a Thompson's Jumping Ring apparatus could be made more effective by terminating the core at the end of the solenoid, replacing its extension beyond the solenoid with a magnetically inert guide for the ring. This should produce a flared magnetic field where gravity positions the ring thus maximising the non-axial component of the magnetic flux and increasing the force on the ring. Does that make sense?

 

[kuruman]

Your reasoning is generally correct however I think you need to pay careful attention to the force that would accelerate the ring in one direction or another.

The resultant force is radial and evenly distributed around the circumference resulting in no net force to accelerate the ring.

Why is the resultant force radial? An axial force component is needed to accelerate the ring in the axial direction. That can only come from a radial component of the field. Think about $\vec F=I~\vec L\times \vec B$ and what component of $$\vec B## provides an axial acceleration.