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Stopping a Falling Permanent Magnet at a Conductor

Introduction

For this animation, a cylindrical permanent magnet will fall toward a conductor. The polarization of the magnet will be along the normal to the conductor. The eddy current that is induced will slow the magnet before it touches the conductor.

Magnetic Field of a Cylindrical Permanent Magnet

If no large current sources are relevant then we may compute the field from a potential, `V`, which amounts to summing the effects of the oppsitely signed dipole charges on the north and south ends of the magnet.

`V(x,z) = int_0^(2*pi)int_0^asigma_d[1/sqrt((x-rcosphi)^2+(z-L/2)^2)- 1/sqrt((x-rcosphi)^2+(z+L/2)^2)] dphidr`

where `a` is the radius of the magnet, L is its length, and `sigma_d` is the density of magnetic dipoles on both ends. Since the magnet is symmetrical about its axis, its potential will also be symmetrical about its axis. I have chosen to define the vertical center of the magnet to be zero. The magnetic field of this potential is the `x and z` gradient

`bbvecB=bbgradV(x,z)=(delV)/(delx)bbhatx+(delV)/(delz)bbhatz`

Discussion of Eddy Current Induction

When the magnet falls toward the conductor, it induces circular electric fields in the conductor. The electric fields are of a given radius and strength depending on the magnet's rate of fall and the magnetic field strength at that depth. The reader might object that the electrons in the conductor always move in circles even if the magnetic field is not changing. This sustained circular motion due to the magnetic field can be true only if the electrons can complete a full circuit without running into the thermal motion of the core ions of the conductor. That is not possible unless the thermal motion of the core ions is zero which means that the temperature is absolute zero or the conductor is a superconductor.

However, when the conductor has circular electric fields due to the approaching magnet, significant circular currents (eddy currents) can arise and these are always of a direction such that their magnetic field repels that of th approaching magnet.

The Eddy Current Density in the Conductor due to the Magnet's `bbvecB` Field

The pertinent equation for the driver of the eddy curent density is one of Maxwell's equations.

`bbgradxxbbvecE=-(delbbvecB)/(delt)`

where `bbvecE` is the electric field and `bbvecB` is the magnetic induction (which we will call the magnetic field here). To make use of this equation we will integrate both sides and apply Stokes Theorem

`intintgradxxbbvecEdA=oint(bbvecE.dbbvecl)=del/(delt)intbbvecB*dA`

where the integral on the right is the sum of the magnetic flux going through the area, `A`, and the integral on the left is the sum of the electric field on the boundary of that area. In our case the loop will be one of the eddy current loops in the conductor. If the eddy current loop is circular of radius `a_l`, then we can say that If the voltage across an angular arc `deltaphi` of the loop is then `V_(phi)=bbE(deltaphia_l)/(2pi)`.

`oint(bbvecE.dbbvecl)=2pia_l|bbvecE|=del/(delt)intbbvecBbb*dA`

It is the permanent magnet that provides the magnetic field `bbvecB`. If the permanent magnet is moving in the `z` (axial) direction then the variation of `B_z` with time can be written:

`(dB_z)/(dt)=(dB_z)/dzdz/dt`

where `v_z=dz/dt` is the speed of the falling magnet. Then, assuming that `B_z` is constant over the loop, the equation for `E(z)` becomes

`E(z)=1/2a_l(dB_z(z))/dzdz/dt`

Then the current density at radius r and elevation `z` is

`J(r,z)=sigmaE(z)=sigma/2r(dB_z(r,z))/dzdz/dt`

where `sigma` is the conductivity of the conductor.

Magnetic field, `bbB_J` due to a current density distribution, `bbJ(x,z)`

The basic equation for `bbB_J` is the integral

`B_J(bbr)=intbbJ(bbr')xx(bbr-bbr')/(|bbr-bbr'|^3)d^3r'`

where `bbr` is the field point and `bbr'` is the source point. But we know that `J(r',z')` is axisymmetric about the axis of the permanent magnet.

`J(r',z')=sigmapir'(dz')/dtdel/(delz')B_M(r',z')`

where `bB_M` is the magnetic field due to the permanent magnet and `sigma` is the conductivity of the stopping conductor. Then the magnetic field at `z_M`, the vertical center of the magnet is

`B_J(0,z_M)=intJ(r',z')2pir'((0-bbr')bbhatr+(z_M-z')bbhatz)/(sqrt((z'-z_M)^2+r'^2)^3)dz'dr'`

Force on a PM dipole due to the magnetic field of the current distribution

In order to compute the force, `bbF_j` on a PM dipole, we need the gradient of the field, `bbB_j`, from the current density distribution.

`bbF_j=bbM*gradbbB_j`

where we have modeled the PM as a magnetic dipole of strength `bbM`. Now we need to substitute our expression for the current density, `bbJ`, into the expression for its magnetic field.

`B_j(bbr)=intbbJ(bbr')xx(bbr-bbr')/(|bbr-bbr'|^3)d^3r= v_zsigma/(rho)del/(delz)int_0^rhobbvecB_M(rho',z)_Mbbrho'drho'xx(bbr-bbr')/(|bbr-bbr'|^3)d^3r'`

In this expression the cross product is simple because `J` is perpendicular to `bbB_M`. Then we have two expressions of which we have to take the derivative with respect to `z`. First we define `bbr` and `bbr'` in terms of `(rho,z)`

`bbr'=rho'bbhatphi+zbbhatz'`
`bbr=rhobbhatphi+zbbhatz`

where `bbhatphi` is along the circumference of the present eddy loop. For the PM which we are defining as a dipole, `rho=0` and `z=z_M`. Therefore the ratio in the integrand becomes

`(bbr-bbr')/(|bbr-bbr'|)^3=(z_Mbbhatz)/(z-z_M)^3`

which has the derivative

`d/dz((bbr-bbr')/(|bbr-bbr'|^3))=-3z_Mbbhatz/(z-z_M)^4`

Putting this together with the derivative of `bbB_j` with respect to `z` we have the expression:

`(dB_j(bbr))/dz= v_zsigma/(rho)(d)/(dz)int_0^rho(dbbB_M(rho',z'))/dz(z_M)/(|z'-z_M|^3)drho'dz`

We know that `gradbbB` is a function of `z` and `rho`. However, since it is axisymmetric, it will be sufficient to model `gradbbB` as just a function of `z`. The preceding is what is done to provide an animation of the stopping magnet.

Displaying the field vectors due to the current distribution

For the PM, it was easy to show the origins of several field vectors since the origins were all dipoles that were concentrated on the North pole of the PM.

For the continuous current density distribution, the number of orgins of the field vectors is infinite. So here we integrate over all `J(x',z')` and compute the `B(x,z)` at representative values (x,z) with particular interest in the value at the center of the PM that we have assumed is a dipole.

Explanation of Viewer Controls

For this complex animation the only adjustable value is the length, 'ds', of the compute increment. If `ds` is too small the computation takes too long. If it is too large, some fine details are washed out; The viewer may also choose a radio button for one out of three quantities to view.

First is the potential,`V_B`,from which the PM magnetic field is computed by the equation `bbB_M=-gradV_B`. `V_B` is shown as a colored contour plot.

Second is the plot of `bbB_M` which is shown as "lines of force" coming out of the North pole of the PM.

Third is the eddy current density as well as the "lines of force" due the eddy current.

Explanation of the Two Canvases

A side view, Canvas 1a, shows the fields and the eddy current. A top view, Canvas 1b, shows only the direction and magnitudes of the eddy currents at the top of the conductor.