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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 () = (In the limit).
• = Moment of Inertia.
• = Angular velocity
• = Angular velocity of precession.

## Overview

Key facts

For an electromagnet characterized by the area , the magnetic flux density , and the relative magnetic permeability , the magnetic pull force is:

where is the magnetic permeability of free space.

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Constants

Consider an electromagnet of area and magnetic flux density , and also imagine a displacement of as highlighted in Figure 1.

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

where is the volume, the magnetic permeability of free space, and 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 will be:

where () is the change in volume. This leads to:

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

where is the force ().

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.

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