# Magnetic Pull Force

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

### 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. <br/>

**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:

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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