Motion Equation for a magnet on a spring

In summary, the conversation discusses the process of finding the force produced by Eddy currents induced by a moving magnet. The formula for the magnetic flux through a hole plate is calculated using the magnet's magnetic field. To find the induced EMF, the resistance of the plate needs to be determined. Suggestions are welcome for finding the induced current and magnetic force on the magnet, and there is a discussion about computing the flux and the z component of the magnetic field.
  • #1
Gonzalo Lopez
1
0
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
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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!
 
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  • #2
Current density ## J=\sigma E ##. They give you the conductivity ## \sigma ##. To compute ## E=E_{induced} ## you need to compute the flux ## \Phi ## out to a radius ## r ##. Then by symmetry, ## E_{induced}(r)=\mathcal{E} (r)/(2 \pi r ) ##.
It looks like computing the flux is no easy task, because you need to take the dot product of ## B ## and ## dS ##, and integrate it the area from ##0 ## to ## r ##. Perhaps it is ok, because you just need the z component of ## B ##.
 
Last edited:

Related to Motion Equation for a magnet on a spring

1. What is the motion equation for a magnet on a spring?

The motion equation for a magnet on a spring is given by F = -kx - μBm, where F is the force exerted on the magnet, k is the spring constant, x is the displacement of the magnet, μ is the magnetic moment, and B is the magnetic field.

2. How does the spring constant affect the motion of the magnet?

The spring constant, k, determines the stiffness of the spring and therefore affects the force exerted on the magnet. A higher spring constant results in a stronger force, causing the magnet to oscillate at a faster rate.

3. What is the significance of the magnetic moment in this equation?

The magnetic moment, μ, represents the strength of the magnet and its orientation in the magnetic field. It affects the direction and magnitude of the force on the magnet, and therefore plays a crucial role in the motion equation.

4. Can this equation be used to calculate the period of oscillation for the magnet?

Yes, this equation can be used to calculate the period of oscillation for the magnet. The period is given by T = 2π√(m/k), where m is the mass of the magnet and k is the spring constant.

5. Are there any other factors that can affect the motion of the magnet on a spring?

Yes, there are other factors that can affect the motion of the magnet on a spring. These include the strength and orientation of the magnetic field, the mass and shape of the magnet, and any external forces acting on the system.

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