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Magnetic Reluctance

A description of the magnetic reluctance, also discussing a way to calculate it
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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.


Key facts

The magnetic reluctance is defined as:

\mathcal{R} = \frac{\mathcal{F}}{\Phi}

where \inline \mathcal{F} is the magnetomotive force, and \inline \Phi the magnetic flux.

For a magnetic circuit of length \inline l, cross-sectional area \inline A, and relative magnetic permeability \inline \mu_r, the magnetic reluctance can be calculated with:

\mathcal{R} = \frac{l}{\mu_0 \mu_r A}

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



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

The magnetic reluctance \inline \mathcal{R} of a magnetic circuit can be regarded as the formal analog of the resistance in an electrical circuit. The magnetic reluctance can be expressed as:

where \inline \mathcal{F} is the magnetomotive force (mmf), and \inline \Phi is the magnetic flux.

In order to calculate the magnetic reluctance, consider a magnetic circuit of length \inline l and cross-sectional area \inline A, as diagramed in Figure 1.


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We know that the magnetic field strength \inline H can be written as:

where \inline I is the current in the coil, and \inline N is the number of turns (for a more detailed discussion on the magnetic field strength see Field Strength ). Furthermore, \inline H can be related to the magnetic flux density \inline B with the equation:

where \inline \mu_0 is the magnetic permeability of free space, and \inline \mu_r the relative magnetic permeability of the material.

As the magnetic flux \inline \Phi is defined as:

equation (4) can also be written as:

from which the magnetic field strength becomes:

Considering that \inline H is uniform, equation (3) becomes:

Using the expression form of \inline H from (8) in (7), we get that:

which leads to:

As the magnetomotive force \inline \mathcal{F} of a coil is given by:

equation (10) becomes:


Taking into account the definition of the magnetic reluctance from (2), we get that \inline \mathcal{R} can be calculated as:

Example - Magnetic flux and flux density of a toroid
Consider a toroid with the mean length of , the cross section of , and the relative magnetic permeability of . What is the magnetic flux and the magnetic flux density if the coil has 10 turns and the current is 2 amperes ?
As the magnetic reluctance is given by:

and, in our case, (), and (), we get that:

from which we obtain:

The magnetic flux can be written as:

where , the magnetomotive force, is given by:

As, in our case, , , and also considering (3), we obtain the magnetic flux:

Taking into account that the cross-sectional area is (), the magnetic flux density becomes:

As a side note, if the toroid has an air gap of length , then its total magnetic reluctance, , would be the magnetic reluctance of the toroid plus the magnetic reluctance of the air gap:

Thus, in this case, the total magnetic flux, , would be given by: