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Magnetic Pull Force

An analysis of the magnetic pull force which arises between the poles of an electromagnet
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Key Facts

Gyroscopic Couple: The rate of change of angular momentum (\inline \tau) = \inline I\omega\Omega (In the limit).
  • \inline I = Moment of Inertia.
  • \inline \omega = Angular velocity
  • \inline \Omega = Angular velocity of precession.

Overview

Key facts

For an electromagnet characterized by the area \inline A, the magnetic flux density \inline B, and the relative magnetic permeability \inline \mu_r, the magnetic pull force is:

F = A \frac{B^2}{2 \mu_0 \mu_r}

where \inline \mu_0 is the magnetic permeability of free space.

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Constants

\mu_0 = 4 \pi \cdot 10^{-7} \; \frac{N}{A^2}

Consider an electromagnet of area \inline A and magnetic flux density \inline B, and also imagine a displacement of \inline \delta x as highlighted in Figure 1.

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We know that the energy stored in a magnetic field of no magnetic saturation is given by:

where \inline V is the volume, \inline \mu_0 the magnetic permeability of free space, and \inline \mu_r the relative magnetic permeability (for a more detailed discussion on the energy stored in a magnetic field see Stored Energy ).

Thus, the change in energy stored following the displacement \inline \delta x will be:

where \inline \delta V (\inline =A \delta x) is the change in volume. This leads to:

where \inline W refers to the work done. However, we also know that work can also be defined as:

where \inline F is the force (\inline Newtons).

Taking into account equations (4) and (5), we get that:

from which the magnetic pull force becomes:

Example:
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Example - Magnetic pull force of an electromagnet
Problem
Consider the electromagnet diagramed in Figure E1, characterised by the lengths , , and , and the area . Given that a current of passes through a coil with turns and relative magnetic permeability of , find the total magnetic pull force.

746/img_em18.png
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Figure E1
Workings
We know that the total magnetic reluctance of a magnetic circuit of length , cross-sectional area , and relative magnetic permeability , with an air gap of length , is given by:

As, in our case, (), (), , and (), we obtain the total magnetic reluctance:

which gives:

The total magnetic flux is given by:

where is the magnetomotive force:

As, in our case, , , and (from equation 3), we obtain from (4) and (5) that the total magnetic flux is:

Taking into account that the magnetic flux density is given by:

and also considering (6) and that (), we obtain the magnetic flux density in the air gap:

As the magnetic pull force is given by:

and also considering (8), and that (), and the relative magnetic permeability of air is , the magnetic pull per pole becomes:

Thus, we obtain the total magnetic pull force:
Solution