### Table of Contents

# Field

Stan Zurek, Field, Encyclopedia Magnetica, http://e-magnetica.pl/doku.php/field |

**Field** - a concept used in physics to describe and explain interactions of energy or matter, such that every point in space can take a different value of a given physical quantity, in the form of a scalar, vector, or tensor, making it a **scalar field**, **vector field**, or **tensor field**, accordingly.^{1)}

In particular, electromagnetic field contains energy and has momentum associated with it, so the concept of a **field** is useful for describing and explaining action-at-distance, especially for electric field, magnetic field and gravitational field. No information can travel faster than the speed of light, and therefore any changes in the field (such as electromagnetic pulse) cannot travel faster than that.^{2)}

Appropriate mathematical notation of a **field** allows mathematical analysis of the involved quantities, so that the solutions of equations can be provided for known parameters of involved excitations, geometries, and material properties. This concept is fundamental for such numerical techniques as finite element method or finite difference element method, or similar.

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## Action at a distance

**David C. Jiles,**:

*Introduction to Magnetism and Magnetic Materials*^{3)}

*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.*

The concept of field is useful when describing action-at-a-distance, for example there can be forces between two bodies which do not touch and are suspended in vacuum. There is no matter-like medium which could transfer the force, and therefore it is the field (magnetic, electric, gravitational, etc.) surrounding the bodies which can act over the separating distance.

Nature always tends to minimise the energy of a given system, and therefore the forces will act in the direction which reduces the energy. For some fields such as electrostatic, gravitational, or thermal the forces will act along the local gradient of the field, and therefore calculation of a gradient is important.

Magnetic forces are more complex, because they are interrelated with electric currents, which in turn are caused by electric fields. However, electric current generates a magnetic field which is perpendicular to it and “circulates” around the current, so the magnetic forces follow more complex paths. Nevertheless, a gradient of magnetic field is also an important quantity, as it governs the simpler case of magnetic force between magnetic poles.

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## Scalar field

In a **scalar field** each point has some scalar value associated with it. Scalar value represents just the magnitude of the value, without any additional attributes such as direction.

The intensity of a scalar field can be shown as intensity of colours in a false-colour image, as used for example in thermal cameras.

An intuitive example of a scalar field is the temperature distribution. Within some volume of solid, liquid, or gas, there is a certain value of temperature present at every point within that volume, and therefore it can be treated mathematically as a kind of field. The exact distribution of temperature is dictated by the heat sources, geometry, material properties, and cooling conditions.

The temperature can be used as an example of calculations which can be performed on a field. The heat flux out of any closed volume is equal to the heat generation within that volume^{4)}. Therefore, the divergence of the field represents the heat source:

$$∇·\vec{F} = q$$ | (W/m^{3}) |

where: $∇·\vec{F}$ - divergence (W/m^{3}) of the vector heat flux $\vec{F}$ (W/m^{2}), q - scalar volume heat generation (W/m^{3}) |

Heat flux is also related to the temperature gradient, through the thermal conductivity of the given material:

$$\vec{F} = k·\vec{G}$$ | (W/m^{2}) |

where: $\vec{F}$ - vector heat flux (W/m^{2}), k - scalar thermal conductivity of the material (W/(m·K)), $\vec{G}$ - vector temperature gradient (K/m) |

and temperature gradient is related to the temperature distribution:

$$\vec{G} = -∇T$$ | (K/m) |

where: $\vec{G}$ - vector temperature gradient (K/m), T - scalar temperature (K) |

Therefore, combination of all these equations results in a second-order partial differential equation, which directly relates the heat sources with material properties, and with the temperature distribution, which is generally of interest in such calculations, and thus can be solved numerically by using the following formulation:^{5)}

$$-∇·(k·∇T) = q$$ | (W/m^{3}) |

where: k - scalar thermal conductivity of the material (W/(m·K)), T - scalar temperature (K), q - scalar volume heat generation (W/m^{3}) |

## Vector field

In a **vector field** each point of space has some vector value associated with it. Vectors contain information about magnitude as well as direction.

An intuitive example of a vector field is a wind in the air, because each point in the given volume has some vector of velocity associated with it (and if the velocity is zero then the vector has zero length).

It is quite difficult to illustrate all aspects of a vector field on a 2D image, and there are several synonymous approaches, such as: small vectors distributed around the picture, field lines, or false-colour map of just the amplitude of the vectors (and these approaches can be combined).^{6)} The usefulness of a given illustration depends on the preferences of the author or the exact details that need to be emphasized.

Similar computations as those shown above for the temperature scalar field can be carried out for a vector field. In fact, a gradient of a scalar field is a vector field and therefore the same rules of vector calculus apply throughout.^{7)}

On the other hand, divergence of a vector field over a given volume is a scalar value (not a scalar field).

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*H*of a circular current loop, with the magnetic field lines superimposed on it. The amplitude of the field is related to the density of field lines.

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An example of a vector field is the magnetic field, which can be described by the following Maxwell's equations:^{8)}^{9)}

$$∇×\vec{H} = \vec{J}$$ | (A/m^{2}) |

$$∇·\vec{B} = 0$$ | (T/m) |

$$\vec{B} = µ·\vec{H}$$ | (T) |

where: $\vec{H}$ - vector of magnetic field strength (A/m), $\vec{J}$ - vector of electric current density (A/m^{2}), $\vec{B}$ - vector of magnetic flux density (T), $µ$ - absolute magnetic permeability (H/m) (which can be expressed as scalar, vector or tensor) |

These equations can be combined by using the magnetic vector potential *A* (which is also a vector field), and by using the Coulomb gauge the equations can be solved to obtain the value of magnetic flux density *B* as generated due to electric currents (expressed by the current density *J*), for example by the formulation as below. Such solutions are of great practical importance in engineering and design of electromagnetic devices. The false-colour map of magnetic field around a current loop shown above was calculated by using such method.^{10)}

$$ ∇× \left( \frac{1}{μ(B)}· ∇×\vec{A} \right) = \vec{J}$$ | (A/m^{2}) |

where: $µ(B)$ - magnetic permeability as a function of flux density B in the given material, $\vec{A}$ - vector of magnetic vector potential (T·m), $\vec{J}$ - vector of electric current density (A/m^{2}) |

## Tensor field

A scalar is tensor rank 0, a vector is a tensor rank 1, and the quantity commonly called “tensor” is a tensor rank 2.^{11)} Therefore, the physical quantities which have very complex interactions might require mathematical description by higher-order tensors.

A tensor rank 2 has multiple independent magnitudes in multiple directions.

It is difficult to visualise tensor field, because for rank 2 tensor there are 9 values associated with each point in space, as schematically illustrated with the image of an infinitesimal cube included below. However, there can be also higher order tensors, such as rank 3 and rank 4, with the latter assigning 81 coefficients to each point in space,^{12)} and even though some of these numbers are repeated (due to symmetry) the representation remains very complex. However, for a simplified case the tensors and tensor field can be represented by three-dimensional ellipsoids with independent radii for each principles axis.^{13)}^{14)}

Sometimes in equations the tensors are represented with a double arrow, to distinguish them from ordinary vectors (which have a single arrow).

The rules of calculations are similar to those performed on three-dimensional vectors, as described above, so that vector calculus is still applicable.

**tensor**for an infinitesimal cube, with a set of 3 orthogonal vectors interacting with the three orthogonal surfaces

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^{15)}

- a) sphere ($k_1=k_2=k_3$)
- b) oblate ellipsoid ($k_1 < k_2 = k_3$)
- c) prolate ellipsoid ($k_1 > k_2 = k_3$)
- d) triaxial ellipsoid ($k_1 \neq k_2 \neq k_3$)

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## Vector calculus

The mathematical rules of vector calculus and tensor calculus allow performing analytical and numerical calculations on scalars, vector, and tensor fields.^{16)}^{17)}

There are several quantities which are of great importance, with some examples as listed below.

### Superposition (sum)

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In a linear system the excitation from multiple field sources can be added in a linear way, or in other words they can be superposed. Therefore, the superposition rule allows adding and subtracting the fields produced by multiple sources. The calculations can be also performed in a reversed way, because knowing the values of a field (distribution of field) it is possible to derive the value of sources (by performing inverse calculations).

However, in non-linear systems the situation is more complex because inverse calculations cannot be performed directly. In numerical methods such computations are solved by iterative techniques, in which the solution is found by successive approximation of the solution, which is declared when the calculation error is below a certain threshold.

operator | input quantity | mathematical representation | outcome |
---|---|---|---|

$+$ or $-$ | scalar, vector, or tensor | $F_1 + F_2 = F_3$ $F_4 - F_5 = F_6$ | sum or difference of the input quantities |

### Gradient (change)

A gradient can be calculated for a scalar field, and quantifies the amount and direction of its change, and therefore the gradient of a scalar field is a vector field.

Vector calculus operators in a Cartesian system^{18)} |
|||
---|---|---|---|

operator | input function | mathematical representation | outcome |

gradient | scalar $F$ | $$\text{grad} \, F ≡ ∇F ≡ \hat{\imath} \frac{∂F}{∂x} + \hat{\jmath} \frac{∂F}{∂y} + \hat{k} \frac{∂F}{∂z}$$ | vector |

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### Flux of field (flow)

Vector field has a direction at a given point as if there was a movement of a substance (e.g. a flow of wind in the air), and the **flux of field** quantifies the net amount of directional “penetration” of the vector field through a given closed surface (as illustrated in the picture with the normal vectors).

For this purpose, in vector calculus it is useful to use the concept of the normal vector, because performing vector dot product of the unit normal vector to the given flat surface with the vector penetrating that surface extracts the flux of field through that surface. If the field is directed tangentially to the surface then there is no penetration, and hence flux of field through tangential surface is zero.

The Gauss's divergence theorem links the divergence of the field (the amount generated by the source) to the flux flowing outward from a closed surface.^{20)} The intuitive illustration of that relationship is shown in the image with the infinitesimal cubes and surface patches.

$$ \int \int \int_\mathbf{V} (\mathbf{∇}·\mathbf{F})dV = \oint \oint_S ( \mathbf{F} · \mathbf{\hat{a}} ) dS $$ |

where: $V$ - volume, $\mathbf{F}$ - analysed vector field, $S$ - surface surrounding the volume $V$, $\hat{a}$ - unit vector normal to $S$ |

### Divergence (source)

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Performing a divergence calculation over a given volume of space extracts the information about the net amount of source-like excitation contained in that space. If there are positive and negative sources (source and “sink”) which balance each other out, then the net amount could be zero. For example, for a thermal field divergence quantifies the heat source, and for electric field it is the net amount of electric charge. Magnetic field in the form of magnetic flux density *B* is always solenoidal, which means that the divergence of *B* is always zero, because there are no magnetic monopoles.^{22)}

Divergence of B is always zero |

$$∇·\vec{B} = 0$$ |

An example of using divergence in calculations was shown above in the **Scalar field** section, for computing the heat source.

### Curl (circulation)

Circulation (curl) of a field is especially important when considering electromagnetic field.

Dynamic magnetic field and electric field are correlated by “circulating” around each other, and therefore the curl operator $∇× ...$ appears both in the Faraday's law of induction as well as in the Ampère's circuital law.

Varying magnetic field *B* induces electromotive force *E* around it. And a varying electric field (or varying or static electric current represented by current density *J*) generates a magnetic field around it.

Faraday's law of electromagnetic induction | $$ ∇× \mathbf{E} = - \frac {\partial \mathbf{B}}{\partial t}$$ |

Ampère's circuital law | $$ ∇× \mathbf{B} = \mu_0 · \mathbf{J} + \mu_0 · \epsilon_0 · \frac {\partial \mathbf{E}}{\partial t}$$ |

Stoke's theorem links the circulation of a vector around a given boundary and the net curl over the whole surface of the patch limited by that boundary, which can be mathematically written as in the equation below.^{23)}

Also, the intuitive illustration shows how the infinitesimal vectors can be summed over the boundary of the surface, as well as over the all the content of the surface.

Curl is represented by a vector which is perpendicular to the surface around which the vector field “rotates” (in a similar manner in which the pseudovector can represent physical rotation in mathematical analysis).

$$ \int \int_S (\mathbf{∇} \times \mathbf{F}) · d\mathbf{a} = \oint_C \mathbf{F} · d\mathbf{l} $$ |

where:$S$ - analysed surface, $\mathbf{F}$ - analysed vector field, $\mathbf{a}$ - vector normal to surface $S$, $C$ - closed curve enclosing the surface $S$, $\mathbf{l}$ - vector tangential to curve $C$ |

*C*is also defined with the right hand in the right-hand system;

**normal vector**$\mathbf{\hat{n}}$ denotes vector normal (perpendicular) to the surface $a$

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## See also

## References

^{1), 2), 6), 12), 16)}Richard Feynman, Robert Leighton, Matthew Sands, Ferromagnetism, The Feynman Lectures on Physics, Vol. II, Basic Books, ISBN: 9780465079988

^{4), 5), 9), 10)}David Meeker, Finite Element Method Magnetics: Documentation, {accessed 2021-06-25}

^{8)}Electromagnetics, Magnetostatics, Theory, Multiphysics Cyclopedia, COMSOL, 2019, {accessed 2022-09-24}

^{17)}W. Schlotter, Tensors For vector calculus, Stanford University, https://web.stanford.edu, 2003, {accessed 2022-09-26}

^{20), 21), 22), 23), 24)}David J. Griffiths, Introduction to electrodynamics, 4th ed., Pearson, Boston, 2013, ISBN 0321856562