- #1

Gonzalo Lopez

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- Homework Statement
- A magnet of mass M and magnetic moment m is suspended from a spring (where k is its spring constant). At its equilibrium heigh, there is a infinitely large plate with a thickness d and conductivity σ. However, the plate has a circular hole of radius a directly below the magnet/spring system. Find the motion equation for the magnet if at t=0, z=Zo.

- Relevant Equations
- Magnetic flux, resistance, Newtons 2nd law

Apart from the trivial elements of the motion equation (m z'' = -kz -mg), I am required to find the force produced by the Eddy currents induced by the moving magnet. To do so, I calculated the magnetic flux through the hole plate:

For a magnet:

Bz=μo m 4π. 2z^2−r^2/(z^2+r^2)^5/2

so

Φ = a→ +∞ ∫μo . m . (2z^2 - r^2).r /2(z^2+r^2)^5/2 dr = μo m a^2/(2(a^2+z^2)^3/2).

In order to find the induced EMF: -dΦ/dt = μo m 3a^2z/2(a^2+z^2)^4 . z'.

However, I can't find an expression for the resistance of the plate in order to obtain the induced current and thus the magnetic force on the magnet.

(z' means the first derivative of z(t) and z'' the second derivative)

Any suggestions are welcome, thanks for your time!