Calculating the terminal velocity of a magnet falling through a copper coil

[rayjbryant]

The magnet being used

[CODE lang="matlab" title="The main body code."]% magnet dimensions [m]

d = .0127;
r = .00238;

%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 in/s^2

%coil properties [22 gauge wire] [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; %

eff_dist = .003175;

%terminal velocity

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);
[/CODE]

[CODE lang="matlab" title="The scaling function code."]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[/CODE]

terminal velocity ends up being: 1E-4 m/s

 

[berkeman]

I'm attempting to calculate the terminal velocity of a magnet falling through a copper coil by assuming there are enough turns to treat it as a copper tube.

How is your coil terminated? Are the ends connected together? How are eddy currents going to form in a coil to generate a retarding force?

When you use a copper tube in your simulation, do you get the expected result?

 

[rayjbryant]

Yes, the ends are connected to allow current to flow and eddy currents to form. The desired effect was achieved with a physical model using a weaker magnet and 20 gauge coil but not enough damping or current flow occured. The purpose of this code is to find a combination of magnet strength, wire size, and turn number to build a model that will work to desired specifications. After which a cost analysis will be performed to see if the system is economically feasible to use in the desired commercial application.

 

[rayjbryant]

And sorry I misread the first part of your post. The equation used in the code to calculate the terminal velocity automatically assumes the apparatus is a solid tube. I've performed several copper tube and magnet experiments as well as observed quite a few, and none have a terminal velocity as low as 1E-4 m/s. If any error does occur, it should be an experimental terminal velocity higher than the computer model, since copper coils are not as effective at damping as copper tubes.