# Magnetic Reluctance

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

### 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. <br/>

**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:
which leads to:
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