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# Field Strength

A description of the magnetic field strength, also considering its relation with the magnetic flux density

## Overview

Key facts

The magnetic field strength is defined as:

$\int{H&space;dl}&space;=&space;i$

For an isolated conductor of radius $\inline&space;r$, the magnetic field strength is:

$H&space;=&space;\frac{i}{2&space;\pi&space;r}$

The magnetic field strength is related to the magnetic flux density $\inline&space;B$ by the equation:

$B&space;=&space;\mu_r&space;\mu_0&space;H$

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

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Constants

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

The magnetic fields generated by electric currents are characterized by the magnetic flux density $\inline&space;B$, measured in $\inline&space;Weber&space;(Wb)&space;/&space;m^2$. But when the generated fields pass through magnetic materials which themselves contribute internal magnetic fields, ambiguities can arise concerning what part of the field comes from the external currents, and what part comes from the material itself. In order to clarify this issue, we introduce another magnetic field quantity, the magnetic fields strength $\inline&space;H$, which unambiguously designates the magnetic influence from external currents, independent of the material's magnetic contribution.

The magnetic field strength $\inline&space;H$ can be broadly defined by:

$\int{H&space;dl}&space;=&space;i$

where $\inline&space;H$ is expressed in amperes/meter, and the current $\inline&space;i$ in amperes.

For the particular case of an isolated conductor of radius $\inline&space;r$, as the one diagramed in Figure 1:

the magnetic field strength can be defined by writing that:

$\int{H&space;dl}&space;=&space;H&space;\cdot&space;2&space;\pi&space;r&space;=&space;i$

$H&space;=&space;\frac{i}{2&space;\pi&space;r}$

As was previously noted, magnetic materials, such as iron, can contribute internal magnetic fields. Indeed, iron contains within its atoms a number of tiny current loops. Normally, their random distribution produces no effect. However, when a current flows, these current loops become oriented, the magnetic effect due to the iron becomes more important, and the resulting current is much higher than it would have been without the iron. We can thus write that:

$\int{H^i&space;dl}&space;=&space;NI&space;+&space;ni^i$

where $\inline&space;I$ is the current in the coil, and $\inline&space;N$ is the number of turns, while $\inline&space;ni$ represents the equivalent effect of the atomic current loops (see Figure 2).

We redefine $\inline&space;H$ in iron to retain the magnetic circuit law in terms of the known current $\inline&space;NI$:

$\int{H&space;dl}&space;=&space;NI$

Thus, we can write that:

$\int{H^i&space;dl}&space;-&space;ni^i&space;=&space;\int{H&space;dl}$

## Magnetic Field Strength And Magnetic Flux Density

A commonly used relation between the magnetic flux density $\inline&space;B$ and the magnetic field strength $\inline&space;H$ is:

$B&space;=&space;\mu&space;H$

where $\inline&space;\mu$, the magnetic permeability, is:

$\mu&space;=&space;\mu_r&space;\mu_0$

In (8) $\inline&space;\mu_0$ is the magnetic permeability of free space, also known as the magnetic constant:

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

and $\inline&space;\mu_r$ the relative magnetic permeability of the material.

Thus, the magnetic flux density in air can be written as:

$B&space;=&space;\mu_0&space;H$

whereas the magnetic flux density in iron as:

$B&space;=&space;\mu_r^i&space;\mu_0&space;H$

where $\inline&space;\mu_r^i$ is the relative magnetic permeability of iron (for example graphs see Figures 3 and 4).

Example:
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##### Example - Magnetic flux of a toroid
Problem
Given a toroid with the mean diameter of $\inline&space;20&space;\;cm$ and the cross section of $\inline&space;2&space;\;&space;cm^2$ (see Figure E1), calculate the number of ampere-turns ($\inline&space;At$) which would give a magnetic flux of $\inline&space;2.3&space;\cdot&space;10^{-4}&space;\;&space;Wb$.

The $\inline&space;BH$ curve for iron is given in Figure E2:

Also, calculate the number of ampere-turns required to give the same magnetic flux in the hypothetical case that the toroid has an air gap of length $\inline&space;l_g&space;=&space;0.1&space;\;&space;cm$ (see Figure E3).

Workings
Given that the cross section of the toroid is $\inline&space;2&space;\;cm^2$ ($\inline&space;=&space;2&space;\cdot&space;10^{-4}&space;\;&space;m^2$), the magnetic flux density in the iron should be:

$B&space;=&space;\frac{2.3&space;\cdot&space;10^{-4}}{2&space;\cdot&space;10^{-4}}&space;=&space;1.15&space;\;&space;\frac{Wb}{m^2}$

Therefore, from Figure E2, we obtain:

$H&space;=&space;335&space;\;&space;\frac{At}{m}$

As the mean diameter of the toroid is $\inline&space;20&space;\;&space;cm$ ($\inline&space;=0.2&space;\;&space;m$), we thus obtain the required ampere-turns:

$335&space;\cdot&space;0.2&space;=&space;67&space;\;&space;At$

In the case that the toroid has an air gap, we can write the number of ampere-turns for the air gap as $\inline&space;H_g&space;l_g$, or, furthermore, as $\inline&space;\frac{B_g}{\mu_0}&space;l_g$. Taking into account that $\inline&space;\mu_0&space;=&space;4&space;\pi&space;\cdot&space;10^{-7}&space;\;&space;\frac{N}{A^2}$, $\inline&space;B_g&space;=&space;1.15&space;\;&space;\frac{Wb}{m^2}$ (from equation 1), and $\inline&space;l_g&space;=&space;0.1&space;\cdot&space;10^{-2}&space;\;&space;m$, we get the required ampere-turns for the air gap:

$0.8&space;\cdot&space;1.15&space;\cdot&space;0.1&space;\cdot&space;10^{-2}&space;\cdot&space;10^6&space;=&space;920&space;\;&space;At$

Taking into account (3) and (4), we obtain the total required ampere-turns for the toroid with an air gap:

$67&space;+&space;920&space;=&space;987&space;\;&space;At$

As a side point, it can be noted that the effect of putting an air gap into a toroid is to alter the $\inline&space;BI$ curve to an almost linear relationship, as diagramed in Figure E4.

Solution
For the toroid without an air gap, the required ampere-turns are:

$67&space;\;&space;At$

For the toroid with an air gap, the total required ampere-turns are:

$987&space;\;&space;At$