Quantifying the magnetic force on a magnet moving through a coil?

[rayjbryant]

So I'm familiar with the magnet falling through a copper tube demonstration that shows the induced magnetic fields slowing the magnet down.

I know that this experiment is also possible with a conducting coil as long as the coil forms a closed circuit. I'm trying to find a way to calculate the force acting on the magnet as a function of velocity. Does anyone have a paper they can point me toward?

Thank you,
Raymond Bryant

 

[vanhees71]

https://arxiv.org/abs/physics/0603270v2

 

[rayjbryant]

https://arxiv.org/abs/physics/0603270v2

Hello vanhees, I've looked at this paper before.

Something I tried to do was treat each wrap of the coil as an individual segment and integrate to get its contribution. I wasn't sure if my results were reasonable. I get a very small terminal velocity.

clc
clear all
close all

% magnet dimensions [m]

d = .0127; %magnet height
r = .00238; %magnet radius

%mass of magnet [kg]

m_w = .0017;

% other constants

u_0 = 1.26E-6; % permeability of free space constant T m/A
g = 9.81; % gravitational constant m/s^2

%coil properties [m]

a = .00635; %radius
w = .000635; %width of wire
N = 100; % number of turns
c = pi*a*2; %circumference
wl = c*N; %wire length
cs = pi*(w/2)^2; %cross sectional area
rho = 1.7e-8; % resistivity of copper [ohm/m]
wr = (rho*wl)/cs; % resistance in wire [ohm]
lt = 0.3048; % length of tube

%magnetic properties

sm = 72730000; % magnetic surface charge density [Mx/m^2]

qm = pi*sm*r^2; % total charge Mx

eff_dist = .003175; %effective distance of magnet [m]

%terminal velocity calculation

p = qm*d;
x = d/a;
val = scalingfunction(x);

v = (8*pi*m_w*g*rho*a^2)/(u_0^2*qm^2*w*val); %terminal velocity calculation

%for one ring

flux = [];
z = 0:.00001:eff_dist;

for i = 1:1:length(z) %z varies from zero to effective distance of magnet
    
    flux(end+1) = (u_0*qm*.5)*(((z(i)+d)/sqrt((z(i)+d)^2+a^2))-(z(i)/sqrt(z(i)^2+a^2)));
    
end

Total_flux = 2*sum(flux);

delta_t = (2*eff_dist)/v;

emf = (N*Total_flux)/delta_t;

function [val] = scalingfunction(x)
fun = @(x,y) ((1./(y.^2+1).^(3/2))-(1./((y+x).^2 + 1)).^(3/2)).^2;
val = integral(@(y) fun(x,y),-Inf,Inf);
end

 

[rayjbryant]

I also found this paper, I might try implementing the model of eddy current forces formulated there.

https://aapt.scitation.org/doi/abs/10.1119/1.4933295

 

[rude man]

You will not get much in the way of quantitative results.
Trying to analyze one coil turn at a time is hopeless. Just calculating the flux in a 1-turn coil with given current is beyond any introductory physics course.

You'd have better luck with a long solenoid in which case the two
situations (solenoid & tube) are equivalent. You'd have to know the mag moment of your magnet, for openers.

 

[Delta2]

Try to contact @kuruman , he has written a mini treatise on this problem , its not my intellectual property so I am not sure I am entitled to give it.