Table of Contents
Magnetic flux density
| Stan Zurek, Magnetic flux density, Encyclopedia Magnetica, http://www.e-magnetica.pl/doku.php/magnetic_flux_density |
Magnetic flux density B, or simply flux density - a physical quantity used as one of the fundamental measures of the intensity of magnetic field.1) The unit of magnetic flux density2) is tesla or T. Magnetic flux density should not be confused with magnetic flux.
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Magnetic field is a vector field in space, and is a type of energy whose full quantification requires the knowledge of the vector fields of both magnetic field strength $H$ and flux density $B$ (or other values correlated with them, such as magnetisation $M$ or polarisation $J$). In vacuum, at each point the $H$ and $B$ vectors are oriented along the same direction and are directly proportional through permeability of free space, but in other media they can be misaligned (especially in highly anisotropic materials - see also the illustration in the next section).
From theoretical physics viewpoint, the very definition of magnetic field involves $B$, which is called “the magnetic field”.3)4)5)
From engineering viewpoint, magnetic field strength $H$ can be thought of as excitation, and the magnetic flux density $B$ as the response of the magnetised medium. The quantity $B$ encompasses all of the magnetic response of the medium (including $M$ or $J$, as well as any non-magnetic contribution arising from the applied $H$ due to external electric current, or even internal eddy currents).
These two viewpoints are equivalent for the concerned numerical quantities, but different emphasis might be put on “importance” of either of them. This is also reflected in the naming convention.
The name magnetic flux density and the symbol $B$ are defined by International Bureau of Weights and Measures (BIPM) as one of the coherent derived physical units.6) Therefore, strictly speaking, other names like induction or B field which can be encountered in everyday technical jargon7) are incorrect if referring to a specific value expressed in the units of T.
There are many other names which are used in the literature, all denoting the same quantity:
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Magnetic field strength H
 See separate article on: Magnetic field strength H |
| See separate article on: Confusion between B and H |
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The magnetic field strength H is a separate physical quantity, with different physical units in the SI system. The H and B are interlinked such that:24)
| $$\vec{B} = \vec{J} + μ_0 · \vec{H} = μ_0 · (\vec{H} + \vec{M}) $$ | (T) |
| where: $μ_0$ - absolute permeability of vacuum (H/m), $μ_r$ - relative permeability of material (unitless), $μ = μ_0 · μ_r$ - absolute permeability of material (H/m), $J$ - magnetic polarisation (T), $M$ - magnetisation (A/m) | |
Magnetisation M represents orientation of subatomic magnetic dipole moments per unit volume, and magnetic polarisation J is M scaled by the permeability of vacuum.
In a general case, all the three vectors B, H and J (or B, H and M) can point in different directions (as shown in the illustration for a ferromagnetic anisotropic material), but always such that the vector sum in the equation above is fulfilled.
For uniaxial magnetisation the equation can be simplified to the scalar form, which is widely used in engineering applications:25)26)
| $$B = μ_0 · μ_r · H = μ · H $$ | (T) |
| where: $μ_r$ - relative permeability (unitless), $μ = μ_r · μ_0$ - absolute permeability (H/m) | |
Relative permeability $μ_r$ is a figure of merit of soft magnetic materials and has values significantly greater than unity.
For hard magnetic materials $μ_r \approx$ 1, and it is a less important parameter, as compared to remanence and coercivity.
For non-magnetic materials also $μ_r \approx$ 1, but such that paramagnets are weakly attracted to any polarity of magnetic field ($μ_r$ slightly greater than unity), and diamagnets are always weakly repelled it ($μ_r$ slightly less than unity). Depending on the viewpoint, superconductors can be classified as ideal diamagnets for which $μ_r$ = 0, and thus they are quite strongly repelled from magnetic field, sufficiently for magnetic levitation.27)
Difficulty with definition
It is difficult to give a concise definition of such a basic quantity like magnetic field, but various authors give at least a descriptive version. The same applies to magnetic flux density, as well as the other basic quantity - magnetic field strength.
In the view of theoretical physics, the Lorentz force (described below) is used as the definition of $B$.
The table below shows some examples of definitions of $B$ given in the literature (exact quotations are shown).
| Publication | Definition of magnetic field | Definition of magnetic field strength $H$ | Definition of magnetic flux density $B$ |
|---|---|---|---|
| R. Feynman, R. Leighton, M. Sands The Feynman Lectures on Physics28) | First, we must extend, somewhat, our ideas of the electric and magnetic vectors, E and B. We have defined them in terms of the forces that are felt by a charge. We wish now to speak of electric and magnetic fields at a point even when there is no charge present. We are saying, in effect, that since there are forces “acting on” the charge, there is still “something” there when the charge is removed. | We choose to define a new vector field H by $$\mathbf{H} = \mathbf{B} − \frac{\mathbf{M}}{ε_0 c^2} $$ […] Most people who use the mks units have chosen to use a different definition of H. Calling their field H' (of course, they still call it H without the prime), it is defined by $$\mathbf{H'} = ε_0 c^2\mathbf{B} − \mathbf{M}$$ Also, they usually write $ε_0 c^2$ as a new number 1/μ0 | We can write the force F on a charge q moving with a velocity v as $$\mathbf{F} = q(\mathbf{E} + \mathbf{v} × \mathbf{B})$$ We call E the electric field and B the magnetic field at the location of the charge. |
| Richard M. Bozorth Ferromagnetism29) | A magnet will attract a piece of iron even though the two are not in contact, and this action-at-a-distance is said to be caused by the magnetic field, or field of force. | The strength of the field of force, the magnetic field strength, or magnetizing force H, may be defined in terms of magnetic poles: one centimeter from a unit pole the field strength is one oersted. | Faraday showed that some of the properties of magnetism may be likened to a flow and conceived endless lines of induction that represent the direction and, by their concentration, the flow at any point. […] The total number of lines crossing a given area at right angles is the flux in that area. The flux per unit area is the flux density, or magnetic induction, and is represented by the symbol B. |
| David C. Jiles Introduction to Magnetism and Magnetic Materials30) | One of the most fundamental ideas in magnetism is the concept of the magnetic field. When a field is generated in a volume of space it means that there is a change of energy of that volume, and furthermore that there is an energy gradient so that a force is produced which can be detected by the acceleration of an electric charge moving in the field, by the force on a current-carrying conductor, by the torque on a magnetic dipole such as a bar magnet or even by a reorientation of spins of electrons within certain types of atoms. | There are a number of ways in which the magnetic field strength H can be defined. In accordance with the ideas developed here we wish to emphasize the connection between the magnetic field H and the generating electric current. […] The simplest definition is as follows. The ampere per meter is the field strength produced by an infinitely long solenoid containing n turns per metre of coil and carrying a current of 1/n amperes. | When a magnetic field H has been generated in a medium by a current, in accordance with Ampere's law, the response of the medium is its magnetic induction B, also sometimes called the flux density. |
| Magnetic field, Encyclopaedia Britannica31) | Magnetic field, region in the neighbourhood of a magnetic, electric current, or changing electric field, in which magnetic forces are observable. | The magnetic field H might be thought of as the magnetic field produced by the flow of current in wires […]32) | […] the magnetic field B [might be thought of] as the total magnetic field including also the contribution made by the magnetic properties of the materials in the field.33) |
| E.M. Purcell, D.J. Morin, Electricity and magnetism34) | This interaction of currents and other moving charges can be described by introducing a magnetic field. […] We propose to keep on calling $\mathbf{B}$ the magnetic field. | If we now define a vector function $\mathbf{H}(x, y, z)$ at every point in space by the relation $$ \mathbf{H} \equiv \frac{\mathbf{B}}{μ_0} - \mathbf{M} $$ […] As for $\mathbf{H}$, although other names have been invented for it, we shall call it the field $\mathbf{H}$, or even the magnetic field $\mathbf{H}$. | […] any moving charged particle that finds itself in this field, experiences a force […] given by $$ \mathbf{F} = q·\mathbf{E} + q·\mathbf{v} × \mathbf{B} $$ […] We shall take the equation as the definition of $\mathbf{B}$. |
Analogy to electric circuits
Electric circuit
At the fundamental level, all the electricity is linked to the presence and movement of electric charges, so knowing their positions would be sufficient to fully quantify all electric effects, including electric field. However, in practice, it is much simpler to operate with directly measurable quantities such as current $I$ and voltage $V$.
From a macroscopic viewpoint, values of $I$ and $V$ are both required to fully quantify the effects of electricity in electric circuits. In direct current circuits the proportionality between $V$ and $I$ is dictated by electrical resistance $R$ of a given medium (according to Ohm's law), such that $V = R·I$.
The product of $V$ and $I$ is proportional to power $P$ and energy $E$ in a given electric circuit.
Magnetic circuit
By analogy both magnetic field strength $H$ and magnetic flux density $B$ (or their representations by other related variables) are required for quantifying the effects of magnetism in magnetic circuits. The proportionality between $H$ and $B$ is dictated by magnetic permeability $μ$ of a given medium.35)36)37)
All magnetic field effects are also linked to the movement and intrinsic properties of electric charges. Knowing these properties (such as spin magnetic moment) and the details of movement of the charges (taking into account relativistic effects) it would be possible to completely describe the magnetic field. However, in practice it is much simpler, especially from engineering viewpoint, to utilise the directly measurable quantities such as $H$ and $B$ to quantify power and energy in a given magnetic circuit.
Under steady state conditions, the product of $H$ and $B$ is a measure of specific energy in J/m3, stored in the magnetic field contained in the given medium. The $B·H$ product (the amount of stored energy) is used for example for classification of permanent magnets.38)
Forces
From a theoretical physics viewpoint, the very existence of magnetic field is defined by the at-a-distance force which acts on an electric charge.
The magnetic forces are large enough to be useful for practical applications, such as electric motors.
Magnetic force / Lorentz force
An electric charge moving in a magnetic field experiences a mechanical force which acts in the direction perpendicular to both the direction of the magnetic field and the direction of movement of the charge. If the charge is stationary with respect to the magnetic field, or if it moves in a direction parallel to such field then it experiences no magnetic force (but it can experience other forces, as for example cause by an electric field acting in a different direction). The direction of the magnetic force follows the right-hand rule.
This force is called magnetic force or Lorentz force, and the magnetic field which satisfies such force is the definition of magnetic field $B$.39)40)41) The magnetic force acting on a charge is always proportional to $B$, regardless if it is applied in vacuum, non-magnetic or magnetic materials.
| (with both E and B) | $$ \vec{F} = q·\vec{E} + q·\vec{v} × \vec{B} $$ | (N) |
| (with just B) | $$ \vec{F}_{\vec{E} = 0} = q·\vec{v} × \vec{B} $$ | (N) |
| where: $q$ - electric charge (C), $E$ - electric field (V/m), $v$ - velocity (m/s) of the moving charge, $B$ - magnetic field (T) | ||
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Attraction and repulsion
It is possible to apply the concept of magnetic poles to magnetised objects. Like magnetic poles repel, opposite poles attract.
Magnetic force acting on a given body can be calculated with the help of magnetic energy over the concerned volume. For simple structures, under certain assumptions (e.g. small air gap, uniform B, high permeability of the magnetised body, two faces with the same area A positioned close to each other) using the magnetic energy density in the air gap it is possible to derive a simplified equation for magnetic force between two magnetised faces:
| $$F = \frac{A·B^2}{2·μ_0}$$ | (N) |
| where: $A$ - surface area of the air gap (m2), $B$ - flux density (T) acting on both faces, $μ_0$ - permeability of vacuum (H/m) | |
However, this is true only if there are magnetic moments (related to electric charges) which react to magnetic field. There is no force acting on vacuum itself, or on matter in which the magnetic moments completely cancel each other (e.g. as in a material in which the diamagnetic and paramagnetic contributions are precisely equal). In weakly magnetic materials, the force can be calculated on the basis of the interaction of magnetic moments with the magnetic field (so information about the magnetisation M is required; if M = 0 then there is no force). Magnetic dipoles experience torque in a uniform magnetic field, and are attracted or repelled according to its gradient.42)
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Maxwell's equations
| Maxwell's equations in differential form43) | |
|---|---|
| Gauss's law for electrostatics | $$ \text{div } \mathbf{E} = \frac {\rho_{charge}}{\epsilon_0}$$ |
| Gauss's law for magnetism | $$ \text{div } \mathbf{B} = 0$$ |
| Faraday's law of electromagnetic induction | $$ \text{curl } \mathbf{E} = - \frac {\partial \mathbf{B}}{\partial t}$$ |
| Ampère's circuital law | $$ \text{curl } \mathbf{B} = \mu_0 · \mathbf{J} + \mu_0 · \epsilon_0 \frac {\partial \mathbf{E}}{\partial t}$$ |
Maxwell's equations fully describe mathematically the interrelation between electric and magnetic fields.
The equations can be mathematically written in many ways (e.g. differential or integral form) or different units (e.g. CGS or MKS). They can also be formulated on the basis of more fundamental theory of quantum electrodynamics.
The application of Faraday's law of induction and Ampere's circuit law are fundamental for operation of all electric generators, transformers, as well as many transducers and sensors.44)
The equations are typically given with respect to magnetic flux density B because in that form they are valid under more general conditions.45)
However, under certain conditions it is also possible to express them with respect to H. This approach is extensively used in numerical calculations such as finite-element modelling (FEM), where the direct link between the electric current (expressed by current density J) and H is exploited, through the Ampere's law, both for solutions and formulations of boundary conditions.46)47)48)49)50)
B in electromagnetic waves
In vacuum, in the absence of charges and currents, the Maxwell's equations simplify, and because of the linearity of vacuum (or other non-magnetic medium without free charges) they can be written either with respect to magnetic flux density B (as shown in the table below), or magnetic field strength H.51) The format with B is valid under more general conditions.52)
| Maxwell's equations in vacuum (in a differential form)53)54) | |||
|---|---|---|---|
| magnetic field represented by H | magnetic field represented by B | ||
| $$ \text{div } \mathbf{E} = 0$$ | $$ \text{div } \mathbf{H} = 0$$ | $$ \text{div } \mathbf{E} = 0$$ | $$ \text{div } \mathbf{B} = 0$$ |
| $$ \text{curl } \mathbf{E} = - \mu_0 · \frac {\partial \mathbf{H}}{\partial t}$$ | $$ \text{curl } \mathbf{H} = \epsilon_0 · \frac {\partial \mathbf{E}}{\partial t} $$ | $$ \text{curl } \mathbf{E} = - \frac {\partial \mathbf{B}}{\partial t}$$ | $$ \text{curl } \mathbf{B} = \mu_0 · \epsilon_0 · \frac {\partial \mathbf{E}}{\partial t} $$ |
In vacuum the two notations, with B or H are exactly equivalent, with the latter quite popular for analysing radiation from antennas.55) For example, using the Poynting vector which is proportional to power, as a product of electric field E in V/m and magnetic field H in A/m, the result is V·A/m2 or W/m2 (power density).
Measurement of B
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There are several physical phenomena which can be exploited for measurement of B. One of the most fundamental and widely used is the induction coil (search coil) which directly employs the Faraday's law of induction. However, there are several others, such as Hall effect, fluxgate sensor, nuclear resonance principles, etc.
At the interface between two materials (with different permeabilities $μ_1$ and $μ_2$) the normal component of B does not change, so: $B_{n1} = B_{n2}$.
This property of the boundary condition is utilised in a number of techniques for sensing B, including the use of search coils, Hall effect sensors, indirect B-coils, and several others.57)
At the same interface between two materials, it is the tangential component of magnetic field strength H which is preserved.
This principles are used especially for measuring the local values of B and H components, for example with the fieldmetric technique.58)
Electromagnetic induction - search coil
The Faraday's law of induction states that the amount of electromotive force (EMF) induced in a closed loop or winding is proportional to the number of turns and the rate of change of magnetic flux penetrating such loop.
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Flux density is linked to the flux through the area, so the equation can be written in either of the formats as shown below, as well as in a differential or integral forms as used in vector calculus notation.59)60)
| $$ EMF = - N ⋅ \frac{dΦ}{dt} $$ | (V) |
| $$ EMF = - N ⋅ A ⋅ \frac{dB_{avg}}{dt} $$ | (V) |
| where: $EMF$ - electromotive force (V), $N$ - number of turns in the coil (unitless), $Φ$ - magnetic flux (Wb), $t$ - time (s), $A$ - area of the coil (m2), $B_{avg}$ - spatial average of flux density in the coil (T) | |
The value of flux density can be learned by performing an integral (in an analogue electronic circuit or numerically) of the voltage induced in the search coil:
| $$ B_{avg} = - \frac{1}{N⋅A} ⋅ \int_0^T (EMF) dt $$ | (T) |
| $$ B_{avg} = \frac{1}{N⋅A} ⋅ \int_0^T (V) dt $$ | (T) |
| where: $V$ - voltage (V) measured on the terminals of the coil induced due to EMF, assuming very high impedance of the voltmeter such that $V = - EMF$, $T$ - time interval (s) | |
The same principle can be applied to needle probes, which can be used only to measure local B.
The electromagnetic induction can be utilised only if there is some change of magnetic field penetrating the coil, and a completely static field cannot be detected without additional measures. The relative changes can be introduced for example by moving the sensing coil with relation to the sample under test, or vice versa, hence there are devices known as vibrating coil magnetometer and vibrating sample magnetometer.61)62)
Also, for measurement of DC field it is possible to move the coil from the field to be measured to some location sufficiently far at which B = 0, and the integral of the induced voltage is proportional to the initial value of B. This technique is known as ballistic measurement.63)
Hall effect
The Hall effect is a direct demonstration of the Lorentz force. Electric charges are forced to flow through a thin plate, by applying a small current $I$. The path of the moving charges is deflected to the sides, if flux density $B$ is applied perpendicular to the plate. This results in generation of imbalance of charges on both sides which is measurable as the Hall voltage $V_H$.
As in the Lorentz force equation, the effects are proportional to all the input quantities, and the Hall voltage reverses sign if the magnetic field reverses its direction.
| $$V_H = k_H · B · I$$ | (V) |
| where: $k_H$ - proportionality constant of the Hall effect device (m2/C) | |
For a functional, commercially available sensor, there is a simple proportionality constant, for a given excitation current:64)
| $$B = k_s · V_{out}$$ | (V) |
| where: $k_s$ - proportionality constant or gain of the sensor (T/V), $V_{out}$ - output voltage of the sensor (V) | |
Typical commercial Hall-effect sensors exhibit non-negligible non-linearities, such as temperature dependence, which is why the excitation current cannot be too large to avoid self-heating.
There are other correlated fundamental effects such as anomalous Hall effect.
Nuclear resonance
Magnetic moments of atoms and subatomic particles (electrons, protons, neutrons) when exposed to magnetic field B have their spins quantised. The torque acting on the spins makes them precess. The frequency of the precession is linked to the strength of magnetic field, which can be detected as absorption of magnetic field supplied at the exact frequency at which the precession occurs.
Such phenomena can be employed for detecting extremely small changes of magnetic properties of matter, such as the distribution of various tissues in human body (which can be distinguished because the contain slightly different amount of water). Magnetic resonance imaging (MRI) relies on a very complex scheme of interactions between DC and high-frequency AC magnetic fields, including superconducting electromagnets, with the DC fields generated with a known gradient so that the response of each partial volume is space can be pinpointed with high accuracy. This allows generation of 3D images of inside of human body in a non-invasive way.65)
Energy and energy density
Energy density $E_d$ of the energy stored in magnetic field, in a given material, can be calculated as:66)
| $$E_d = \int H · dB $$ | (J/m3) |
which for a material with linear characteristics, including high-energy permanent magnets, can be simplified to:
| $$E_d = \frac{H·B}{2} $$ | (J/m3) |
It should be noted that the last equation above encompasses both the field which is applied as well as the response of the material to being magnetised (regardless which quantity is assumed to be “fundamental”, B or H).
However, in non-magnetic materials for which $μ_r$ ≈ 1 it can be written that:
| $$B = μ_0 · H $$ | (T) | and | $$\frac{B}{μ_0} = H $$ | (A/m) |
Therefore, substitution can be made such that eliminates one of the variables, making the energy density proportional to the square of either just B or just H. Depending on the publication, both forms are used,67)68) often not stating the implicit assumption of $μ_r$ ≈ 1. These two forms are equivalent, although expression with B appears to be more popular. If the assumption $μ_r$ ≈ 1 cannot be made then energy is proportional to the product of $B·H$, or the equation has to include also the relative permeability $μ_r$.69)
| for $μ_r \approx$ 1 | $$E_d = \frac{B^2}{2·μ_0} = \frac{μ_0·H^2}{2} $$ | (J/m3) |
| for $μ_r \neq 1$ | $$E_d = \frac{B^2}{2·μ_r·μ_0} = \frac{μ_r·μ_0·H^2}{2} $$ | (J/m3) |
Hysteresis loop and power loss
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Soft magnetic materials are used for energy transformation under alternating or pulsed magnetisation regimes. Energy efficiency of a magnetic circuit depends on the power lost in the given magnetic material.
For one cycle of magnetisation (for time from 0 to T), the total energy lost in the material is proportional to the area of the traced B-H loop. The numerical value of loss can be calculated as:70)
| $$P = \frac{f}{D}·\int_0^T \left(\frac{dB}{dt} · H \right) dt $$ | (W/kg) |
| where: $f$ - frequency of magnetisation (Hz), $D$ - density of material (kg/m3) | |
The specific power loss is an important figure of merit for soft magnetic materials, and for example it is the basis of categorisation of electrical steels.71)
Because of the operating conditions such B-H loops are measured under conditions of sinusoidal voltage, which also enforces sinusoidal B. The waveform of H can become severely distorted especially when material operates close to saturation. This is effect is responsible for example for the inrush current in transformers.
Remanence
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Remanence Br is defined as the point at which the B-H loop (hysteresis loop) crosses the vertical axis (when H = 0), as measured for a given material.
The value of remanence is an important figure of merit for hard magnetic materials, for which the theoretical maximum achievable working energy density of a magnet is proportional to the square of Br.73)74)
| $$BH_{max} = \frac{B_r·H_c}{4} \approx \frac{{B_r}^2}{4·μ_0} $$ | (J/m3) |
In soft magnetic materials the value of remanence is useful for defining the factor of B-H loop squareness, which is important for some pulse applications.
Saturation flux density
The magnetic response of all magnetic materials (ferromagnetic and ferrimagnetic) increases with the applied magnetic field, but eventually reaches magnetic saturation at sufficiently high amplitude.
Beyond a certain level of excitation the value of magnetisation saturation Msat (equivalent to saturation polarisation Jsat) is reached when all the internal magnetic moments are aligned (all domain walls disappear and there remains a single domain). Further increase of excitation does not cause for the M to increase any more, but the flux density B keeps increasing with the excitation H, as dictated by the equation:
| B due to H above saturation | ||
|---|---|---|
| (expressed with magnetisation M) | $B = μ_0 · (H + M_{sat})$ | (T) |
| (expressed with polarisation J | $B = μ_0 · H + J_{sat}$ | (T) |
| where: $M_{sat}$ - saturation magnetisation (A/m), $J_{sat}$ - saturation polarisation (T) | ||
Therefore, there is no limiting value of “saturation flux density” Bsat or “saturation induction” even though some professional literature refers to such concepts.75)76) Nevertheless, in some cases it useful to use the concept of technical saturation defining some other point like state of magnetic domains or value of permeability.77)
Saturation magnetisation is mostly dictated by the chemical composition and physical state of a given material.78)79) For anisotropic materials it does not depend on the direction of the applied field, but it might require larger amplitude to reach saturation.80)
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