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

## Overview

Key facts

The magnetic reluctance is defined as:

where is the magnetomotive force, and the magnetic flux.

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

where is the magnetic permeability of free space.

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Constants

The magnetic reluctance 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 is the magnetomotive force (mmf), and is the magnetic flux.

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

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

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

where is the magnetic permeability of free space, and the relative magnetic permeability of the material.

As the magnetic flux is defined as:

equation (4) can also be written as:

from which the magnetic field strength becomes:

Considering that is uniform, equation (3) becomes:

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

As the magnetomotive force of a coil is given by:

equation (10) becomes:

or:

Taking into account the definition of the magnetic reluctance from (2), we get that can be calculated as:

Example:
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##### Example - Magnetic flux and flux density of a toroid
Problem
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 ?
Workings
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:


Solution