The Superconducting Ring - Motion in Magnetic Field · RMPH 2018 #2
Physics · electromagnetism
Problem
A thin ring of mass m, radius r and inductance L is maintained in a horizontal plane above a cylindrical magnetic bar which is placed vertically (see figure). The vertical axis of symmetry of the cylindrical magnetic bar is aligned with the center of the ring. The magnetic field due to the cylindrical magnet is shown in the right figure and each of its components are given by: the radial component is given by B_r = B₀βr, while the vertical component is given by B_z = B₀(1-γz). B₀, β, γ are all positive constant with appropriate dimensions, while z and r denotes the vertical and, respectively, the radial coordinate of the system. Initially, there is no electric current passing through the ring and it is kept fixed above the magnet. It is then allowed to fall due to the gravitational pull of the Earth, given by the gravitational acceleration, g. During the fall the ring will still be in a horizontal plane and will have the same vertical axis. Answer the following questions and write your results as a function of the variables specified above. 2.a. Derive the equations of motion for the ring in the reference frame specified in figure. 2.b. Find the expression of current I = I(t) which flows through the ring at time t.Intuitive explanation
The key insight is that as the ring falls through the non-uniform magnetic field, the vertical magnetic flux through it changes, inducing an EMF that drives a current—which then creates a magnetic force opposing the motion. You need to apply Newton's second law while accounting for both gravity and the magnetic force from the induced current interacting with the radial field component. The induced current depends on how fast the flux is changing, which connects the ring's velocity to the electromagnetic response through Faraday's law.
Intuitive explanation
The key insight is that as the ring falls through the non-uniform magnetic field, the vertical magnetic flux through it changes, inducing an EMF that drives a current—which then creates a magnetic force opposing the motion. You need to apply Newton's second law while accounting for both gravity and the magnetic force from the induced current interacting with the radial field component. The induced current depends on how fast the flux is changing, which connects the ring's velocity to the electromagnetic response through Faraday's law.