magnetic_field_strength
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magnetic_field_strength [2023/09/04 14:55] – stan_zurek | magnetic_field_strength [2023/09/29 20:54] (current) – [Difficulty with definition] stan_zurek | ||
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+ | ====== Magnetic field strength ====== | ||
+ | |< 100% >| | ||
+ | | // | ||
+ | |||
+ | **Magnetic field strength //H//** - a physical quantity used as one of the basic measures of the intensity of [[magnetic field]].[(Mansfield> | ||
+ | |||
+ | <box 30% left #f0f0f0> | ||
+ | [[Electric current]] $I$ generates **magnetic field strength** $H$, whose magnitude is independent on the type of the uniform isotropic surrounding medium ([[magnetic]] or [[non-magnetic]]) | ||
+ | [[file/ | ||
+ | {{page> | ||
+ | </ | ||
+ | |||
+ | From the engineering viewpoint, **magnetic field strength** $H$ can be thought of as excitation and the **[[magnetic flux density]]** $B$ as the response of the medium.[(Zurek> | ||
+ | |||
+ | From theoretical physics viewpoint, the field $H$ is defined as the vectorial difference between [[flux density]] $B$ and [[magnetisation]] $M$. The //H// field is sometimes referred to as " | ||
+ | |||
+ | These two approaches are identical in the sense of the physical quantities in question (with the same physical units of A/m), but are referred to by different names, and different emphasis put on their meaning and use in derivation of some equations. | ||
+ | |||
+ | Magnetic field is a [[vector field]] in space, and is a kind 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 [[uniform material|non-uniform]] or [[anisotropy|anisotropic]] materials). | ||
+ | |||
+ | The requirement of two quantities is analogous for example to [[electricity]]. Both [[electric voltage]] $V$ and [[electric current]] $I$ are required to fully quantify the effects of electricity, | ||
+ | |||
+ | The name // | ||
+ | |||
+ | There are many other names which are used in the literature, all denoting the same quantity: | ||
+ | * //magnetic field intensity H// [(Britannica> | ||
+ | * //magnetic field H// [(Britannica)][(Purcell)][([[http:// | ||
+ | * //field H// [(Purcell)] | ||
+ | * //field H'// [(Feynman)] | ||
+ | * //H-field strength// [(Poljak> | ||
+ | * //H-field// [([[https:// | ||
+ | * // | ||
+ | * // | ||
+ | * // | ||
+ | * //magnetic force H// [(Thompson_1890)] | ||
+ | * //intensity of magnetic force H// [(Thompson_1890)] | ||
+ | * //auxiliary field H// [([[https:// | ||
+ | * and probably several others. | ||
+ | |||
+ | {{page> | ||
+ | |||
+ | ===== Magnetic flux density B ===== | ||
+ | |||
+ | |< 100% >| | ||
+ | | {{/ | ||
+ | |||
+ | |< 100% >| | ||
+ | | {{/ | ||
+ | |||
+ | <box 45% right #f0f0f0> | ||
+ | Illustration of [[magnetic field strength]] //H//, [[flux density]] //B//, [[magnetisation]] //M//, and [[magnetic polarisation|polarisation]] //J// in a **[[ferromagnet]]** | ||
+ | [[file/ | ||
+ | {{page> | ||
+ | </ | ||
+ | |||
+ | The [[magnetic flux density]] //B// is a separate physical quantity, with different physical units in the [[SI system]]. The //H// and //B// are interlinked such that: | ||
+ | |||
+ | |||
+ | | $$\vec{B} = \vec{J} + μ_0 · \vec{H} = μ_0 · (\vec{H} + \vec{M}) $$ | (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 moment|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 an [[anisotropic material]]), | ||
+ | |||
+ | For uniaxial magnetisation the equation can be simplified to the scalar form, which is used widely in engineering applications: | ||
+ | |||
+ | | $$B = μ_0 · μ_r · H = μ · H $$ | (T) | | ||
+ | |||
+ | [[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 much less important parameter. | ||
+ | |||
+ | For non-magnetic materials also $μ_r \approx$ 1, but such that [[paramagnet|paramagnets]] are weakly attracted to any polarity of magnetic field ($μ_r$ slightly greater than unity), and [[diamagnet|diamagnets]] are always weakly repelled it ($μ_r$ slightly less than unity). Depending on the viewpoint, [[superconductor|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]].[(Tumanski)] | ||
+ | ===== 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 field strength**, as well as the other basic quantity - **[[magnetic flux density]]**. | ||
+ | |||
+ | The table below shows some examples of definitions of $H$ given in the literature (exact quotations are shown). | ||
+ | |||
+ | <WRAP clear></ | ||
+ | |||
+ | <WRAP 100% lo> | ||
+ | |< 100% 10% 30% 30% 30%> | ||
+ | ^ Publication ^ Definition of // | ||
+ | | R. Feynman, R. Leighton, M. Sands \\ **The Feynman Lectures on Physics**[(Feynman)] | ||
+ | | Richard M. Bozorth \\ **Ferromagnetism**[([[http:// | ||
+ | | David C. Jiles \\ **Introduction to Magnetism and Magnetic Materials**[(Jiles> | ||
+ | | **Magnetic field**, **Encyclopaedia Britannica**[(Britannica_Field> | ||
+ | | E.M. Purcell, D.J. Morin, **Electricity and magnetism**[(Purcell)] | //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, | ||
+ | </ | ||
+ | |||
+ | ===== 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' | ||
+ | |||
+ | 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 circuit|magnetic circuits]]. The proportionality between $H$ and $B$ is dictated by [[magnetic permeability]] $μ$ of a given medium.[(White)] | ||
+ | |||
+ | 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/ | ||
+ | |||
+ | |||
+ | ===== H due to electric current | ||
+ | |||
+ | <WRAP 30% right> | ||
+ | <box 100% right #f0f0f0> | ||
+ | Amplitude of **magnetic field strength** $H$ reduces with the distance from a conductor with electric current $I$ | ||
+ | [[file/ | ||
+ | {{page> | ||
+ | </ | ||
+ | |||
+ | <box 100% right #f0f0f0> | ||
+ | Orientation of **magnetic field strength** $H$ vector with respect to the current $I$ follows the [[right-hand rule]] | ||
+ | [[file/ | ||
+ | {{page> | ||
+ | </ | ||
+ | </ | ||
+ | |||
+ | From a [[macroscopic]] viewpoint, the fields can be treated as averaged over some volume of material, and their magnitudes can be linked to measurable signals such as [[current]] or [[voltage]]. Therefore, this approach is used extensively in engineering.[(Tumanski)][(Zurek)][(Jiles)] | ||
+ | |||
+ | $H$ is always generated around [[electrical current]] $I$, which can be a solid [[conductor]] with current or just a moving [[electric charge]] (also in [[free space]]). The direction of the $H$ vector is perpendicular to the direction of the current $I$ generating it, and the senses of the vectors are assumed to follow the [[right-hand rule]].[(Jiles)] It can be said that //H// " | ||
+ | |||
+ | Without other sources of magnetic field and in a [[uniformity|uniform]] and [[isotropic]] medium the generated magnetic field strength $H$ depends only on the magnitude and direction of the electric current $I$ and the physical sizes involved (e.g. length and diameter of the conductor, etc.) so according to the [[Ampere' | ||
+ | |||
+ | | $$ \int_C \vec{H} · d \vec{l} | ||
+ | | where: //C// - closed path over which the integral is calculated, $dl$ - [[infinitesimal]] fragment of [[magnetic path length]] (m), $I$ - current (A) || | ||
+ | |||
+ | |||
+ | |||
+ | In a linear isotropic medium the values from various sources combine and can be calculated from the superposition of the sources. For simple geometrical cases the value of $H$ can be calculated analytically, | ||
+ | |||
+ | The relationship between $H$ and $I$ is often shown by employing the [[Biot-Savart law|Biot-Savart' | ||
+ | |||
+ | In many examples given in the literature there is an implicit assumption (typically not stated) that the derivation is carried out for vacuum and not for an arbitrary medium with a different permeability[(MIT)]. When the $μ_0$ permeability is reduced in the equations on both sides then $H$ is proportional only to $I$ and this is true for any uniform isotropic medium with any permeability, | ||
+ | |||
+ | The situation is slightly different for [[anisotropic]] or discontinuous medium. They can give rise to additional sources of magnetic field because new [[magnetic pole|magnetic poles]] can be generated by the excited medium, and these poles must be taken into account in order to accurately describe distribution of $H$. For instance, pole pieces in an [[electromagnet]] affect $H$, whose distribution is no longer dictated by just the coils with electric current. | ||
+ | |||
+ | ===== H due to M and B ===== | ||
+ | |||
+ | <box 45% right #f0f0f0> | ||
+ | //H//, //B//, //M//, and //J// in a **[[paramagnet]]** | ||
+ | [[file/ | ||
+ | {{page> | ||
+ | </ | ||
+ | |||
+ | The [[microscopic]] viewpoint is used often in theoretical physics.[(Purcell)][(Griffiths> | ||
+ | |||
+ | Each atom responds to the externally applied magnetic field //B// with some [[magnetisation]] //M//, which is defined as the vector sum of magnetic moments per given volume. The " | ||
+ | |||
+ | | $$\vec{H} = \frac{\vec{B}}{μ_0} - \vec{M}$$ | ||
+ | | where: $μ_0$ - absolute [[permeability of vacuum]] (H/m) || | ||
+ | |||
+ | For DC excitation, in non-magnetic or magnetic but isotropic materials //B// and //H// vectors are parallel. For [[ferromagnetic]] (and other ordered structures) the [[crystal anisotropy|crystal]] or [[shape anisotropy]] can introduce significant angle between the two vectors. | ||
+ | |||
+ | ===== Maxwell' | ||
+ | |||
+ | [[Maxwell' | ||
+ | |||
+ | However, under certain conditions it is also possible to express them with respect to //H//. This approach is extensively used in numerical calculations such as [[finite-element modelling]] (FEM), where the direct link between the electric current (expressed by [[current density]] //J//) and //H// is exploited, through the Ampere' | ||
+ | |||
+ | ^ Example of notation used in FEM documentation (after reference [(COMSOL_Cyclopedia)] ) ^^ | ||
+ | | $$ \nabla · \mathbf{D} = ρ$$ | $$ \nabla · \mathbf{B} = 0$$ | | ||
+ | | $$ \nabla \times \mathbf{E} = - \frac {\partial \mathbf{B}}{\partial t}$$ | $$ \nabla \times \mathbf{H} = \mathbf{J} + \frac {\partial \mathbf{D}}{\partial t} $$ | | ||
+ | |||
+ | ==== H in electromagnetic waves ==== | ||
+ | |||
+ | In vacuum, in the absence of charges and currents, the Maxwell' | ||
+ | |||
+ | |< 100% >| | ||
+ | ^ Maxwell' | ||
+ | ^ magnetic field represented by //H// ^^ magnetic field represented by //B// ^^ | ||
+ | | $$ \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} $$ | | ||
+ | |||
+ | |||
+ | In vacuum the two notations, with //B// or //H// are exactly equivalent, with the latter quite popular for analysing radiation from [[antenna|antennas]].[(Suriano> | ||
+ | |||
+ | ===== Defining H with force ===== | ||
+ | |||
+ | It is shown in the literature that magnetic field strength at a given point in space can be defined as the mechanical [[force]] acting on [[unit pole]] at the given point.[(Mansfield)] However, calculation of force requires $B$, which depends on the properties of medium. Indeed, the original experiment performed by Biot and Savart involved physical forces acting on wires.[(Biot> | ||
+ | |||
+ | The forces acting on two magnetised bodies will be different if they are placed in oxygen (which is [[paramagnetic]]) or in water (which is [[diamagnetic]]). This difference will be directly proportional to the relative permeabilities of the involved media. However, the $H$ produced around the wire will be the same (as long as the medium is uniform and isotropic). | ||
+ | |||
+ | The magnitude of [[magnetic force]] (Lorentz force) is always proportional flux density $B$.[(Purcell)] | ||
+ | |||
+ | |||
+ | |||
+ | ===== Generation of H ===== | ||
+ | |||
+ | Known values of //H// are generated by utilising the Ampere or Biot-Savart laws mentioned above. If relativistic effects can be ignored, then the proportionality is exactly direct such that instantaneous values of magnetic field strength $H$ correspond to instantaneous values of the applied current $I$: | ||
+ | |||
+ | | $$ H(t) = c · I(t) $$ | (A/m) | | ||
+ | | where: $c$ - proportionality constant of a given circuit (1/m) || | ||
+ | |||
+ | Under certain conditions the generated magnetic field can be calculated so precisely that it can be used for calibration of other sensors or definition of values, as recommended by [[BIPM]].[(SI_Appendix2> | ||
+ | |||
+ | Two typical devices which can be used for generating known values of //H// are the [[solenoid]] and the [[Helmholtz coil]].[(SI_Appendix2)][(Tumanski)] They can be even used in a combined setup, in which the external Helmholtz coils compensate for [[Earth' | ||
+ | |||
+ | ==== Solenoid ==== | ||
+ | |||
+ | <box 30% right #f0f0f0> | ||
+ | [[Solenoid]] is often used a source of known value of magnetic field strength //H//, which can be calculated for its geometrical centre (black dot) | ||
+ | [[file/ | ||
+ | {{page> | ||
+ | </ | ||
+ | |||
+ | |||
+ | |< 100% >| | ||
+ | | {{/ | ||
+ | |||
+ | In an infinitely long uniform **[[solenoid]]**, | ||
+ | |||
+ | For a solenoid with a finite length, the magnetic field at its geometrical centre can be calculated as in the equation below. For " | ||
+ | |||
+ | | $$ H_{centre} = \frac{N·I}{\sqrt{l^2 + d^2}} \approx \frac{N·I}{l} $$ | (A/m) | | ||
+ | | where: $N$ - total [[number of turns]] in the [[solenoid]] (unitless), $I$ - current (A), $l$ - solenoid length (m), $d$ - solenoid diameter (m) || | ||
+ | |||
+ | If the thickness of the wire in the solenoid is significant, | ||
+ | |||
+ | |||
+ | ==== Helmholtz coil ==== | ||
+ | |||
+ | |< 100% >| | ||
+ | | {{/ | ||
+ | |||
+ | Another widely used source of //H// is the **[[Helmholtz coil]]**. The device comprises two identical coils resembling circular [[current loop|current loops]], positioned parallel on the same axis, and separated precisely by the radius of the circle. | ||
+ | |||
+ | For two coils, each with radius //r// and each comprising number of turns // | ||
+ | |||
+ | | (turns per coil) | $$ H_{centre} = \frac{N_{each}·I·\sqrt{0.8^3}}{r} \approx \frac{0.71554·N_{each}·I}{r} $$ | (A/m) | | ||
+ | | (turns total) | ||
+ | | where: $N_{each}$ - [[number of turns]] of each [[coil]] (unitless), $N_{total}$ - total number of turns of both coils (unitless), such that $N_{total}=2·N_{each}$, | ||
+ | |||
+ | Shapes other than circular are also used (e.g. square) but at the expense of uniformity of the obtained field distribution. | ||
+ | |||
+ | <box 30% left #f0f0f0> | ||
+ | [[Helmholtz coil]] is a precise source of magnetic field at its geometrical centre (black dot) | ||
+ | [[file/ | ||
+ | {{page> | ||
+ | </ | ||
+ | |||
+ | <box 30% left #f0f0f0> | ||
+ | Path of moving [[electron|electrons]] bent into a circle by [[magnetic field]] generated by external [[Helmholtz coil]] (red circular shape in the background) | ||
+ | [[file/ | ||
+ | //< | ||
+ | </ | ||
+ | |||
+ | <box 25% left #f0f0f0> | ||
+ | Set of three large orthogonal Helmholtz coils used for compensation of [[Earth magnetic field|Earth' | ||
+ | [[file/ | ||
+ | {{page> | ||
+ | </ | ||
+ | |||
+ | |||
+ | ==== Magnetic circuit with a small gap ==== | ||
+ | |||
+ | The Ampere' | ||
+ | |||
+ | This relationship is used extensively in engineering by employing the concept of [[magnetomotive force]] (product of current and turns of the coil, expressed in [[ampere-turn|ampere-turns]]). For a simple magnetic circuit with one air gap it can be written that: | ||
+ | |||
+ | | $$ N·I = H_{core}·l_{core} + H_{gap}·l_{gap} $$ | (A-turns) ≡ (A) | | ||
+ | | where: $N$ - [[number of turns]] of the [[winding]] (unitless), $I$ - current (A), $H_{core}$ - //H// in the core (A/m), $l_{core}$ - length of the core (m), $H_{gap}$ - //H// in the [[air gap]] (A/m), $l_{gap}$ - length of the air gap (m) || | ||
+ | |||
+ | In a magnetic circuit with a relatively small air gap the value of [[magnetic flux density]] is such that $B_{gap} \approx B_{core}$. However, the value of //H// required to support some value of //B// is scaled by the inverse of [[relative permeability]]. Hence, for a magnetic material with large permeability, | ||
+ | |||
+ | |||
+ | | $$ H_{gap} \approx \frac{N·I}{l_{gap}} $$ | (A/m) | | ||
+ | | where: $N$ - [[number of turns]] of the [[winding]] (unitless), $I$ - current (A), $l_{gap}$ - length of the [[air gap]] (m) || | ||
+ | |||
+ | However, for more complex magnetic circuits, effects such as [[flux fringing]] or [[magnetic energy]] stored in the material must be taken into account, and this can be done by numerical methods such as [[finite-element modelling]].[(Kozlowski)] | ||
+ | |||
+ | Addition of air gap allows storing energy in it. The B-H loop is " | ||
+ | |||
+ | <box 40% left #f0f0f0> | ||
+ | [[Magnetic circuit]] with [[magnetic path]] and [[air gap]] | ||
+ | [[file/ | ||
+ | {{page> | ||
+ | </ | ||
+ | |||
+ | <box 20% left #f0f0f0> | ||
+ | Large [[electromagnet]] with an [[air gap]]; designed with the simplified equation[(Kozlowski)] | ||
+ | [[file/ | ||
+ | {{page> | ||
+ | </ | ||
+ | |||
+ | <box 30% left #f0f0f0> | ||
+ | Addition of [[air gap]] cases [[B-H loop shearing]], linearising the loop and extending the saturation to higher //H// | ||
+ | [[file/ | ||
+ | {{page> | ||
+ | </ | ||
+ | |||
+ | ==== Demagnetising field ==== | ||
+ | |||
+ | <box 30% right #f0f0f0> | ||
+ | Simplified illustration of [[demagnetising field]] $H_d$ in a body magnetised with [[magnetisation]] $M$ | ||
+ | [[/ | ||
+ | {{page> | ||
+ | </ | ||
+ | |||
+ | Magnetic field is generated not only by the electric currents, but also by the magnetic moments which can store magnetic energy, for example by alignment due to previously applied [[process of magnetisation]]. The collections of such intrinsic magnetic moments amounts to [[magnetisation]] $M$ and becomes a source of magnetic field, as it is the case in [[permanent magnet|permanent magnets]]. If magnetic poles are created then [[magnetic field lines]] (of $H$) are by convention assumed to point from the [[north pole|N]] to the [[south pole|S pole]]. | ||
+ | |||
+ | The [[magnetic field lines]] will close through the medium surrounding the magnet, but also through the magnet itself, in the direction opposite to the magnetisation $M$, thus lowering the effective magnetisation of the body, which is the reason why this effect is called the [[demagnetising field]] $H_d$. | ||
+ | |||
+ | The effect can be quantified with a unitless [[demagnetising factor]] $N_d$, which is proportional to the magnetisation and it is a function of dimensions of the body, such that for for very long structures or for magnetically closed circuits $N_d=0$ and for thin flat structures of infinite dimensions magnetised perpendicularly to the surface $N_d=1$. | ||
+ | |||
+ | The value of demagnetising factor can be calculated analytically for ellipsoids and other very simple geometric shapes, and for a sphere $N_d=1/ | ||
+ | |||
+ | | $$H_d = - N_d·M $$ | (A/m) | | ||
+ | | where: $N_d$ - demagnetising factor (unitless), $M$ - magnetisation (A/m) || | ||
+ | |||
+ | <box 70% left #f0f0f0> | ||
+ | [[Demagnetising field]] $H_d$ in a body magnetised with [[uniform]] magnetisation $M$[(Fiorillo)] | ||
+ | [[/ | ||
+ | {{page> | ||
+ | </ | ||
+ | |||
+ | The illustration shows a [[permanent magnet]] magnetised with uniform //M// (or //J//), surrounded by [[vacuum]]. The magnetic field lines are shown separately for each field: //M//, //H// and //B//, which is the vector sum of //M// and // | ||
+ | |||
+ | The demagnetising field // | ||
+ | |||
+ | As a result, inside the body //B// is also non-uniform and in terms of magnitude //B// < // | ||
+ | |||
+ | Outside the body, the field lines of //B// and //H// have the same shape, because in vacuum the two [[vector]] quantities differ only by a [[scalar]] constant of the [[vacuum permeability]] μ< | ||
+ | |||
+ | This illustration also shows that at the boundary between the two media with different [[permeability]] values, for //H// the [[tangential component]] // | ||
+ | |||
+ | ===== Measurement of H ===== | ||
+ | |||
+ | The value of //H// cannot be measured directly, but it is derived by other means. | ||
+ | |||
+ | In some [[magnetic measurement system|magnetic measurement systems]] the proportionality to current is used explicitly, as for example in such devices as [[Epstein frame]], [[single-sheet tester]] or [[toroidal sample]]. The measured quantity is current (e.g. by means of a [[shunt resistor]]), | ||
+ | |||
+ | In some other applications, | ||
+ | |||
+ | <box 40% left #f0f0f0> | ||
+ | At the interface between two materials (with different permeabilities $μ_1$ and $μ_2$) the [[tangential component]] of //H// does not change, | ||
+ | [[file/ | ||
+ | {{page> | ||
+ | </ | ||
+ | |||
+ | <box 38% left #f0f0f0> | ||
+ | Flat [[H-coil]] made with [[PCB]] tracks[(Zurek)] | ||
+ | [[file/ | ||
+ | {{page> | ||
+ | </ | ||
+ | |||
+ | <box 12% left #f0f0f0> | ||
+ | Simple wire-wound H-coil[(Zurek)] | ||
+ | [[file/ | ||
+ | {{page> | ||
+ | </ | ||
+ | |||
+ | ===== Energy and energy density ===== | ||
+ | |||
+ | |||
+ | [[Energy density]] $E_d$ of the energy stored in magnetic field, in a given material, can be calculated as: | ||
+ | |||
+ | | $$E_d = \int H · dB $$ | (J/ | ||
+ | |||
+ | which for a material with linear characteristics, | ||
+ | |||
+ | | $$E_d = \frac{H·B}{2} $$ | (J/ | ||
+ | |||
+ | It should be noted that the last equation above encompasses both the field which is applied as well as the response of the material to being magnetised (regardless which quantity is assumed to be " | ||
+ | |||
+ | |||
+ | However, in non-magnetic materials for which $μ_r$ ≈ 1 it can be written that: | ||
+ | |||
+ | | $$B = μ_0 · H $$ | (T) | and | $$\frac{B}{μ_0} = H $$ | (A/m) | | ||
+ | |||
+ | Therefore, substitution can be made such that eliminates one of the variables, making the energy density proportional to the square of either just //B// or just //H//. Depending on the publication, | ||
+ | |||
+ | | for $μ_r \approx$ 1 | $$E_d = \frac{B^2}{2·μ_0} = \frac{μ_0·H^2}{2} $$ | (J/ | ||
+ | | for $μ_r \neq 1$ | $$E_d = \frac{B^2}{2·μ_r·μ_0} = \frac{μ_r·μ_0·H^2}{2} $$ | (J/ | ||
+ | |||
+ | |||
+ | |||
+ | ===== Hysteresis loop and power loss ===== | ||
+ | |||
+ | <box 30% right #f0f0f0> | ||
+ | In [[ferromagnet|ferromagnets]] the power or energy loss is proportional to the area of the [[B-H loop]] | ||
+ | [[file/M4 B-H loop 50Hz_png|{{M4 B-H loop 50Hz.png}}]] | ||
+ | {{page> | ||
+ | </ | ||
+ | |||
+ | [[Soft magnetic materials]] are used for energy transformation under alternating or pulsed magnetisation regimes. Energy efficiency of a [[magnetic circuit]] depends on the power lost in the given magnetic material. | ||
+ | |||
+ | For one cycle of magnetisation (for time from 0 to //T//), the total energy lost in the material is proportional to the area of the traced [[B-H loop]] (hysteresis loop). The numerical value of loss can be calculated as: | ||
+ | |||
+ | | $$P = \frac{f}{D}·\int_0^T \left(\frac{dB}{dt} · H \right) dt | ||
+ | | where: $f$ - frequency of magnetisation (Hz), $D$ - density of material (kg/ | ||
+ | |||
+ | 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 steel|electrical steels]].[(Tumanski)] | ||
+ | |||
+ | 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 [[transformer|transformers]]. | ||
+ | |||
+ | ==== Models of B-H loop ==== | ||
+ | There are several analytical, statistical and numerical models which are used for mathematical description of the B-H loop trajectories, | ||
+ | |||
+ | [[Hysteresis model|Models]] such as [[Jiles-Atherton model|Jiles-Atherton]] or [[Preisach model|Preisach]] use //H// as the independent variable representing the applied excitation, as dictated by the [[Ampere' | ||
+ | |||
+ | ===== Coercivity ===== | ||
+ | |||
+ | <box 30% right #f0f0f0> | ||
+ | In high-energy [[magnet|magnets]] there are two values of coercivity: //< | ||
+ | [[file/ | ||
+ | {{page> | ||
+ | </ | ||
+ | |||
+ | Coercivity // | ||
+ | |||
+ | The of coercivity is linked to the amount of energy which is required to magnetise (and demagnetise) a given magnetic material. Soft magnetic materials have narrow B-H loop, they are easy to magnetise and therefore they have low values of coercivity. | ||
+ | |||
+ | In high-energy permanent magnets the coercivity values are very high (wide hysteresis loop), and because of the significant differences between the values of [[magnetic flux density]] //B// and [[magnetic polarisation]] //J// two values of coercivity can be distinguished: | ||
+ | |||
+ | In soft magnetic materials under normal operating conditions (significantly below [[saturation]]) $B \approx J$ and therefore just single value of coercivity // | ||
+ | |||
+ | |||
+ | ===== See also ===== | ||
+ | *[[Magnetic field]] | ||
+ | *[[Magnetic flux density]] //B// | ||
+ | *[[Confusion between B and H]] | ||
+ | |||
+ | ===== References ===== | ||
+ | ~~REFNOTES~~ | ||
+ | |||
+ | {{tag> Magnetic_field Counter}} |
magnetic_field_strength.txt · Last modified: 2023/09/29 20:54 by stan_zurek