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Magnetic Pull Force

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

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.

746/image_em17.png
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Figure 1

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 (3) and (4), 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 \inline l_1 = 20 \; cm, \inline l_2 = 15 \; cm, and \inline l_g = 0.001 \; cm, and the area \inline A = 10 \; cm^2. Given that a current of \inline i = 1  A passes through a coil with \inline N = 200 turns and relative magnetic permeability of \inline \mu_r = 3000, 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 \inline l, cross-sectional area \inline A, and relative magnetic permeability \inline \mu_r, with an air gap of length \inline l_g, is given by:

As, in our case, \inline l=l_1+l_2=20+15 \; cm (\inline =(20+15)\cdot 10^{-2} \; m), \inline A=10\; cm^2 (\inline =10\cdot 10^{-4} \; m^2), \inline \mu_r = 3000, and \inline l_g=0.001 \; cm (\inline =0.001 \cdot 10^{-2} \; m), we obtain the total magnetic reluctance:

which gives:

The total magnetic flux is given by:

where \inline \mathcal{F} is the magnetomotive force:

As, in our case, \inline N=200, \inline i=1\; A, and \inline \mathcal{R}=10.88 \cdot 10^4 \; At/Wb (from equation 3), we obtain from (4) and (5) that the total magnetic flux is:

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

and also considering (6) and that \inline A=10\; cm^2 (\inline =10\cdot 10^{-4} \; m^2), we obtain the magnetic flux density in the air gap:

As the magnetic pull force is given by:

and also considering (8), and that \inline A=10\; cm^2 (\inline =10\cdot 10^{-4} \; m^2), and the relative magnetic permeability of air is \inline \mu_r = 1, the magnetic pull per pole becomes:

Thus, we obtain the total magnetic pull force:
Solution

Last Modified: 3 Aug 10 @ 19:47     Page Rendered: 2022-03-14 15:51:04