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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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Overview

Key facts

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

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

where \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 A and magnetic flux density B, and also imagine a displacement of \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 V is the volume, \mu_0 the magnetic permeability of free space, and \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 \delta x will be:

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

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

where F is the force (Newtons).

Taking into account equations (3) and (4), we get that:

Instant calculator eq(6)
 

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

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

which gives:

The total magnetic flux is given by:

where \mathcal{F} is the magnetomotive force:

As, in our case, N=200, i=1\; A, and \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 B is given by:

and also considering (6) and that A=10\; cm^2 (=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 A=10\; cm^2 (=10\cdot 10^{-4} \; m^2), and the relative magnetic permeability of air is \mu_r = 1, the magnetic pull per pole becomes:

Thus, we obtain the total magnetic pull force:
Solution