Static electricity is everywhere : in the cars, the houses, the factories, nature, This energy is in all the proportions (of the simple spark by painting the hair, while passing by the discharges in the industrial machines, until the lightning and its dangers of electrocutions and fires). However, even in minor amount static electricity can prove to be dangerous, especially in the presence of flammable or explosive products.

The atomic theory of the matter is one of the basic principles related to the phenomena of static electricity. The nature of an atom depends on the particles which compose this one. Indeed, there are three various types of particles: neutrons and the protons which form the core of the atom and the electrons like a cloud around the core.

The whole of the phenomena of static electricity is explained in particular thanks to the electric charges of these particles. It is necessary, therefore, knowledge that the protons carry a positive load (+ 1,6..10-19 Coulomb) and the electrons a negative charge (- 1,6..10-19 C), while the neutrons do not have loads. The loads + and - are normally in equal quantities and the matter is, therefore, electrically neutral.

But, of share their position in orbit of the atom, the electrons are most likely to be transferred from an atom to another, which modifies the load of the matter which becomes positive (if it lost electrons) or negative (if it received some). Thus, one can thus say that the friction of a body on another fact of appearing negative charges on one and of the positive loads on the other and once the separated bodies, these loads remain such as they are. Electrification thus results from a contact, friction is an amplification which makes it possible to put forward the phenomenon. The more one provides energy to an electron, the more it tends to move away from the core and to separate some.

Indeed, when two objects are charged, a force applies between the two. If the two objects have of the same loads the force signs will be repulsive and if the two objects are opposed signs, the force will be gravitational. For experiment, hang two plastic materials identical to the same wire, rub them on wool and you will be able to observe the phenomenon of repulsion.

It is also important to know that the electric charges depend on a material support that it is solid, liquid or gas. There are two various types of solids: drivers and insulators. In insulators, the electrons are strongly related to the atoms and they resist same under strong external actions. There are thus very few transfers of electrons and electric charges.

There are various types of electrostatic discharges but they all are generated by an excessive accumulation of electrostatic loads. The discharge spark: It can occur, for example, between a metal container not connected to the ground which stored loads during a filling (for example), and which is close to a driver connected to the ground. Here another example which explains this phenomenon: a person isolated from the ground with her shoes can take a discharge while wanting to touch a handle of door which, it, is connected to the ground.

In industry, one can thus expect that there is a spark with each time one is in the presence of isolated metal elements. The human body has a capacity compared to the ground which varies between 100 and 300 picofarads and releasable energy maximum can reach tens of mJ what can cause explosions in the presence of flammable gas.

Brush discharge: This type of discharge resembles as its name indicates it to a brush, and occurs in the presence of an insulator. When one approaches a metal element connected to the ground of an insulating surface charged, it occurs an electric shock different from the discharge spark, because it finishes in filament and resembles a brush. To avoid this type of discharge, should be used anti-static additives.

Discharge of cone: It occurs, for example, during filling of large containers (silos). The products, which will be poured in the silo, accumulated electrostatic loads during their transport towards the silo and during the payment, the discharges are propagated along the walls of the silo towards the top of the cone consisted the versed products. Discharge of the lightning type: The flash is the luminous demonstration of the lightning (thunder being the sound demonstration). The flash is highly energy (tens of thousands of megawatts), and is generated by the appearance of cloudy masses of opposed loads, separated by potential differences about tens of million of volts. This separation of the loads takes place inside the same cloud, one will attend a flash intranuage then. It can also take place from one cloud to another, because they carry each one a distinct load. They are flashes internuages.

As you can imagine it, these discharges are génantes to see even dangerous that it is in the hearths or industries and it is possible real to decrease them. To decrease static electricity and the risks, in particular of discharges sparks, one can carry out the equipotentiality and earthing of the drivers isolated but also the use of shoes allowing dissipation from the loads. In the zones at the risks, it should be checked that the ground is sufficiently conducting for this same dissipation of the loads.

Pieces of paper attracted by CD in charge of static electricity

The lightning generating a luminous flash

Electrostatics is the branch of the physics which studies the phenomena created by static electric charges for the observer.

Since Antiquity it is known that certain materials, whose amber attract objects of small size after being rubbed.

The electron, gave its name to many scientific disciplines. Electrostatics describes in particular the forces which the electric charges between them exert: it is about the law of Coulomb. This law states that the force F creates by a load Q on another load Q is proportional to the product of these two loads and contrary with the square of the distance separating them.

Although they seem on our scale relatively weak, the forces of electrostatic origin between an electron and a proton, for example in a hydrogen atom, are of 40 orders of magnitude higher than the forces of gravitation acting between them.

The fields of study covered by electrostatics are numerous: static electricity, with the explosion of the grain silos while passing by certain technologies of photocopiers or the lightning.

The laws of electrostatics proved also useful in biophysics in the study of proteins. Its extensions to the loads moving are studied within the framework of the electromagnetism which itself is generalized by the quantum electrodynamics.

There exists a simple experiment, that everyone can make, making it possible to perceive an electrostatic force: it is enough to rub a plastic rule with a quite dry rag and to approach it paper short periods: it is electrification. Papers are stuck to the rule. The electrified bodies have electricity. The experiment is simple to realize, however interpretation is not simple since, if the rule is charged by friction, the bits of paper are not it a priori! Another experiment of the same style: a filament of water is deviated if a film of cellophane is approached.

More simply, a common experiment of the effects of electrostatics is the feeling to receive a discharge by catching a carriage in very dry weather or while going down or getting into a car. They are phenomena where there was an accumulation of loads, of static electricity.

From there, one can consider two categories of body: insulators, or dielectric, where the state of electrification is preserved locally and the drivers where this state is distributed on the surface of the driver. The electrification of the bodies could be observed thanks to the insulating properties of the dry air, which prevents the flow towards the ground of the loads created by friction.

The distinction between insulators and drivers does not have anything absolute: the resistivity is never infinite (but very large), the free electric charges, practically absent in good insulators, can be created there easily by providing to an electron normally related to an atomic building a quantity of energy sufficient for releasing some (by irradiation or heating, for example). At a temperature of 3000°C, there are more insulators, but only drivers.

It is noted as in experiments as there exist two kinds of loads which one distinguishes by their signs, and which the matter consists of particles of varied loads, all multiples of that of the electron, called elementary charge, however in electrostatics one will be satisfied to say that when an object is charged in volume, it contains a voluminal density of load

ρ (x,y,z)

This corresponds to a statistical approximation, taking into account the smallness of the elementary charge.

In the same way a small experiment makes it possible to show the importance of static electricity: it is enough to charge a plastic comb (while painting itself with dry hair) then with approaching the comb charged with a tube lamp with neon: in the darkness, by approaching the comb of the tube, this one ignites locally. The electric field produced by the comb is sufficient to excite gas inside the tube. From where importance of static electricity: if the electric field of a simple comb is sufficient to excite a gas, the discharge of static electricity in a sensitive electronic device can also destroy it.

The fundamental equation of electrostatics is the law of Coulomb, which describes the force of interaction between two concentrated loadings.

Force of 1 out of 2 = - Force of 2 out of 1:

F^{→}_{1} (2) = q_{2} * (q_{1}^{→}e_{r}) ⁄ 4πεr²_{12} = q_{2} * (q_{1}r^{→}_{12}) ⁄ 4πεr³_{12} = -q_{1} * (q_{2}r^{→}_{21}) ⁄ 4πεr³_{21} = -F^{→}_{2} (1)

Force of 1 out of 2 = - Force of 2 out of 1:

F

Here, constant the E is a constant characteristic of the medium, called the permittivity. In the case of the vacuum, one notes it e0. The permittivity of the air being of 0,5% higher than that of the vacuum, it is thus often comparable for him.

This writing translates the fact that two of the same loads sign are pushed back and that two loads of contrary signs attract each other proportionally with the product of their loads and conversely proportionally to the square of their distance, the forces are equal values and opposed directions, in accordance with the principle of the action and the reaction.

As in gravitation, the remote action is done via a field: the electric field:

Product by 1 in 2:

E^{→}_{1} (2) = (q_{1}r^{→}_{12}) ⁄ (4πε_{0}r³_{12})

product by 2 in 1:

E^{→}_{2} (1) = (q_{2}r^{→}_{21}) ⁄ (4πε_{0}r³_{21})

The field created in M by N loads IQ located in points ρ_{i} is additive (principle of superposition). In the case of a discrete charge distribution:

E^{→}_{T} = E^{→}_{1} + E^{→}_{2} + E^{→}_{3} + ... + E^{→}_{n} = E^{→}_{T} (M) = ∑^{n}_{i=1} * q_{i} ⁄ 4πε_{0} * (ρ_{i}^{→}M) ⁄ (||ρ_{i}^{→}M||³)

In the case of a continuous charge distribution in space, the field caused by a small volume charged is worth:

dE^{→} (x_{m},y_{m},z_{m}) = ρ (x_{i},y_{i},z_{i}) ⁄ 4πε0 * (r^{→}_{im}) ⁄ (r³_{im}) * dx_{i}dy_{i}dz_{i}

and while integrating on all the space where there are loads, one obtains:

E^{→} (x_{m},y_{m},z_{m}) = ∫∫∫ ρ (x_{i},y_{i},z_{i}) ⁄ 4πε_{0} * (r^{→}_{im}) ⁄ r³_{im} * dx_{i}dy_{i}dz_{i}

where is the voluminal density of load in ρ_{i}, r^{→}_{im}is the going vector of Piau not Mr. In the element of dxidyidziautour volume of point ρ_{i} there is an element of load (x_{i},y_{i},z_{i}). The integrals indicate that it is necessary to add, according to the principle of superposition, on all volumes containing of the loads.

E

product by 2 in 1:

E

The field created in M by N loads IQ located in points ρ

E

In the case of a continuous charge distribution in space, the field caused by a small volume charged is worth:

dE

and while integrating on all the space where there are loads, one obtains:

E

where is the voluminal density of load in ρ

The electric potential (of which the differences are called tensions) is a current and important concept electrostatics: it is a scalar function in the space, whose electric field is the gradient.

V (x_{m},y_{m},z_{m}) = ∫∫∫ ρ (x_{i},y_{i},z_{i}) ⁄ 4πε_{0}|r_{im}| * dx_{i}dy_{i}dz_{i}

V (x,y,z) = 1 ⁄ 4ρε_{0} ∫∫∫ ρ (x_{i},y_{i},z_{i}) * dx_{i}dy_{i}dz_{i} ⁄ √ (x - x_{i})² + (y -_{i})² + (z - z_{i})²

and by calculating the derivative partial

∂V ⁄ ∂x, ∂V ⁄ ∂y, ∂V ⁄ ∂z

E^{→} (x,y,z) = 1 ⁄ 4πε_{0} ∫∫∫ ρ (x_{i},y_{i},z_{i}) * (x - x_{i}) * e^{→}_{x} + (y - y_{i}) * e^{→}_{y} + (z - z_{i}) * e^{→}_{z} ⁄ [(x - x_{i})² + (y - y_{i})² + (z - z_{i})²]^{3}/_{2} * dx_{i}dy_{i}dz_{i}

All electrostatics in a homogeneous medium is in these last formulas, though it should be noticed that these formulas are not defined if the point of coordinates (x_{i},y_{i},z_{i}) carries a concentrated loading, which is only one approximation not-physics besides (should be infinite there).

V (x

V (x,y,z) = 1 ⁄ 4ρε

and by calculating the derivative partial

∂V ⁄ ∂x, ∂V ⁄ ∂y, ∂V ⁄ ∂z

E

All electrostatics in a homogeneous medium is in these last formulas, though it should be noticed that these formulas are not defined if the point of coordinates (x

One places the load which produces the potential out of O and one then looks at the potential produced in M and his gradient. In this paragraph, it is supposed that O and M are not confused, if not the formulas would not have any direction.

O^{→}M = r^{→} = re^{→}_{r}

However, by definition of the derivative partial:

dV = grad^{→}V * d^{→}OM = -E^{→}(M) * d^{→}OM

knowing that one can show that

r^{→} ⁄ r³ = - grad^{→} * 1 ⁄ r^{1}

one from of deduced while multiplying by

q ⁄ 4πε_{0}

that:

E^{→} (x,y,z) = q /4πε_{0} * r^{→} ⁄ r³ = - grad^{→} * q ⁄ 4πε_{0}r = - grad^{→}V (r)

with

V (r) = q ⁄ 4πε_{0}r

fields in

r^{→} ⁄ ||r^{→}||³ = 0

are such as their divergence is null:

div * r^{→} ⁄ ||r^{→}||³ = 0^{2}

However, by definition of the derivative partial:

dV = grad

knowing that one can show that

r

one from of deduced while multiplying by

q ⁄ 4πε

that:

E

with

V (r) = q ⁄ 4πε

fields in

r

are such as their divergence is null:

div * r

The theorem of flow-divergence is a theorem of analysis vectorial, usable in electrostatics to obtain a local equation of the electric field.

theorem indicates that:

[E_{x} (x + dx) - E_{x} (x)]dy dz + [E_{y} (y + dy) - E_{y} (y)] dz dx + [E_{z} (z + dz) - E_{z}(z)] dx dy= (∂E_{x} ⁄ ∂x + ∂E_{y} ⁄ ∂y + ∂E_{z} ⁄ ∂z) dx dy dz = divE^{→} dv = ∑_{i} E^{→}_{i} * dS^{→}_{i}

here FD = dx Dy dz represents an elementary volume, which one can regard as a parallelepiped and the contributions of the six faces dSireprésentent them, each one being length equal to its surface and directed perpendicular to the face, towards outside. If one divides a great volume v into elementary volumes and if one summons the electric field of all these elementary volumes, the contributions of the faces located inside volume are compensated exactly, and there remains only the contribution of external surface:

∫∫∫_{v}div E^{→} dv = ∫∫_{s}E^{℮} * dS^{℮}

for any volume. In particular, in a sphere charged in volume by a voluminal density of load, having its center out of O and of ray R sufficiently small so that one can neglect the variations of:

dS * r^{8494;} ⁄ r

is the normal vector on the surface directed towards outside, and of length equal to the element of surface dS which it represents.

∫∫_{s} r^{℮} ⁄ r³ * dS^{℮} = ∫∫_{s} r^{℮} ⁄ r³ = ∫∫_{s} e^{℮}_{r} * e^{℮}_{r} * dS ⁄ r² = ∫∫_{s} dS ⁄ r² = S ⁄ r² = 4πr² ⁄ r² = 4π

What means that the result does not depend on R. And if one multiplies by

ρv ⁄ 4πε_{0}

where v are the volume of the sphere, one obtains:

ρv ⁄ 4πε_{0} ∫∫_{s} r^{℮} ⁄ r³ * dS^{8494;} = ∫∫_{s} ρv ⁄ 4πε_{0} * r^{℮} ⁄ r³ * dS^{℮} = ρv ⁄ 4πε_{0} * 4π = q ⁄ ε_{0}

where Q is the total load v of the sphere. Maybe with the final one:

∫∫_{s} E^{℮} * dS^{8494;} = q ⁄ ε_{0} = ∫∫∫ div E^{℮} dv = ∫∫∫ ρ ⁄ ε_{0} dv

from where the theorem of Gauss under its local version:

and the integrated expression, known by the physicists under the name of theorem of Gauss:

∫∫∫_{v} div E^{℮} dv = ∫∫∫_{v} ρ ⁄ ε_{0} dv = ∫∫_{s} E^{℮} * dS^{℮}

[E

here FD = dx Dy dz represents an elementary volume, which one can regard as a parallelepiped and the contributions of the six faces dSireprésentent them, each one being length equal to its surface and directed perpendicular to the face, towards outside. If one divides a great volume v into elementary volumes and if one summons the electric field of all these elementary volumes, the contributions of the faces located inside volume are compensated exactly, and there remains only the contribution of external surface:

∫∫∫

for any volume. In particular, in a sphere charged in volume by a voluminal density of load, having its center out of O and of ray R sufficiently small so that one can neglect the variations of:

dS * r

is the normal vector on the surface directed towards outside, and of length equal to the element of surface dS which it represents.

∫∫

What means that the result does not depend on R. And if one multiplies by

ρv ⁄ 4πε

where v are the volume of the sphere, one obtains:

ρv ⁄ 4πε

where Q is the total load v of the sphere. Maybe with the final one:

∫∫

from where the theorem of Gauss under its local version:

and the integrated expression, known by the physicists under the name of theorem of Gauss:

∫∫∫

The Poisson equation combines the preceding relations to give a local relation between the charge distribution and the potential:

-div E^{℮} = ∇^{℮} * ∇^{℮} V = ∇^{℮}² V = ΔV = -ρ ⁄ ε_{0}

-div E

The fact is found that the influences of the various loads are added linearly, that is to say to know the force exerted on a load by several other loads, it is enough to calculate the force which each load would exert taken separately, and to add the results: one finds the principle of superposition well, another manner of expressing the linearity of the law of Coulomb.

The law of Coulomb is very close to the expression of the gravitational forces, but these last are (for a given particle) much weaker. However, the electrostatic forces have little effect with large scales, while the gravitation explains the movement of the stars.

That comes owing to the fact that on average, the matter contains as many positive loads than negative charges and thus, beyond the scale of the inhomogeneousness, their influences are compensated. For the gravitation, on the contrary, whose expression of the force has a sign opposed to that of electrostatics, although the masses have all the same positive sign, they attract itself all, instead of pushing back itself as do it electric charges of the same sign.

The electric fields can seldom be calculated analytically by the direct calculation of the last formula but can always be calculated numerically, especially with progress of data processing.

When there exist symmetries, one can often make calculation by applying the theorem of Gauss to the electric field:

The flow of the electric field through a closed surface S is proportional to the sum of the loads which are inside this surface.

∫∫_{s} E^{℮} * dS^{℮} = Q ⁄ ε_{0}

∫∫

let us suppose that one with the x axis charged on a segment AB with a density of constant linear load and, a point M(x_{M}, y_{M}) in the xOy plan where one wants to determine the field produced by the distributed loads on AB.

Let us consider the point P(x, 0). It is in an interval dx of AB having a load dx. These loads create in M a field. Let us pose PM = R:

E^{℮} (M) = ∫_{a}^{b} dE^{℮} (M) = λ ⁄ 4πε_{0} ∫_{a}^{b} P^{℮}M ⁄ r³ dx = λ ⁄ 4πε_{0} ∫_{a}^{b} {(x_{M} - x)i^{℮} + y_{M}j^{℮} ⁄ r³} dx = {λ ⁄ 4πε_{0} ∫_{a}^{b} (x_{M} - x) ⁄ r³} dxi^{℮} + (λ ⁄ 4πε_{0} ∫_{a}^{b} y_{M} ⁄ r³) dx j^{℮}

It remains to make the two integrals on X to obtain the components of:

E^{℮} (x_{M},y_{M}) = E_{x} (x_{M},y_{M})i^{℮} + E_{y} (x_{M},y_{M}) j^{℮}By noting that:

(x_{M} - x) ⁄ r = sinαand

yM ⁄ r = cosαone deduces:

(x_{M} - x) ⁄ y_{M} = tgα

where has is complementary to angle BPM

λ ⁄ 4πε_{0} ∫_{a}^{b} x_{M} - x ⁄ r³ dx = λ ⁄ 4πε_{0} ∫_{a}^{b} (x_{M} - x) dx ⁄ r³ = - λ ⁄ 4πε_{0}y_{M} ∫_{α1}^{α2} sin α dα

easy to integrate

One used:

dx = -y_{M} ⁄ cos²α * dα

1 ⁄ r² = cos² α ⁄ y²_{M}

and

x_{M} - x ⁄ r = sin α

E

It remains to make the two integrals on X to obtain the components of:

E

(x

yM ⁄ r = cosαone deduces:

(x

where has is complementary to angle BPM

λ ⁄ 4πε

easy to integrate

One used:

dx = -y

1 ⁄ r² = cos² α ⁄ y²

and

x

For a charge distribution having a symmetry compared to a plan, it is easy to deduce that for a point M of the symmetry plane, the resulting field E (M) has components only in the symmetry plane (the component perpendicular to the symmetry plane is cancelled: by gathering the loads per symmetrical pairs indeed, one notes this nullity).

Example: If one has a spherical distribution of burden of center O, then any plan passing by O is a symmetry plane: consequently, the resulting field in M is in all the plans containing OM and thus

E^{℮} (r,θ,Φ) = E_{r} (r,θ,Φ)e^{℮}_{r}

since

EΔ (R, Δ, F) = 0 and EFF (R, Δ, F) = 0.

E

since

EΔ (R, Δ, F) = 0 and EFF (R, Δ, F) = 0.

More generally, if, for an Euclidean transformation T, the distribution (T (M)) is identical to (M), the field in T (M) will be transformed per T of that into Mr. One says that the distribution is invariant by the transformation T.

It is the case, for a spherical distribution, by any rotation around the center and one from of deduced that the field is purely radial, and its value measured along the ray depends only on its distance to the center. In polar coordinates:

E^{℮} (r,θ,Φ) = E_{r} (r,θ,Φ) e^{℮}_{r} = E_{r} (r) e^{℮}_{r}

E

The static electrical production can be not wished even constraining within the framework of industrial productions because being able to lead to the faulty operation, the deterioration of equipment on the long run, or, in the cases at the risk, by explosions. It is to this end that are developed anti-static materials.

Magnetic field created by an electric current

Magnetic field created by a magnet

Magnetic field created by a magnet

The magnetostatic one is the study of magnetism in the situations where the magnetic field is independent of time.

More specifically, the magnetostatic one attempts to calculate the magnetic fields when the sources of these fields are known. There exist two possible sources for the magnetic fields:

- on the one hand electric currents
- in addition magnetized matter.

Magnetic field created by an assembly of magnets. In complex situations like this one, the field is calculated by solving the equations of magnetostatic using numerical methods the such finished differences or the finite elements.

The fundamental relations of magnetostatic result from the Maxwell equations in the matter by removing the derivative compared to time. When these temporal variations are removed, the equations of electricity and magnetism are uncoupled, which allows the separate study of electrostatics and the magnetostatic one. The fundamental relations of magnetostatic, written in their local form, are:

∇ * B = 0

∇ * H = j

∇ * H = j

where

- B indicates the magnetic field, called sometimes also magnetic induction or density flux magnetic
- H indicates the magnetic excitation, called sometimes also magnetic field
- J is the density of electric current
- Δ is the operator nabla, who is used here to write the divergence (Δ) and the rotational one (Δ×).

It is necessary to note the ambiguity of the expression magnetic field which can, according to the context, to indicate B or H. In the continuation of the article, we will explicitly indicate the fields by B or H with each time it is important to make the distinction.

For the relations above, it is necessary to add that which connects B and H:B = µ_{0} (H + M)

where

- M is the magnetization of the medium considered
- µ0 is a fundamental constant called magnetic permeability of the vacuum.

It is seen that the distinction between B and H is really useful only in the magnetized mediums (where M 0). Magnetization being supposed known, the relation above makes it possible to calculate very simply B according to H and reciprocally. Consequently, with each time one wants to calculate a magnetic field, one will be able to choose to calculate indifferently B or H, the other resulting some immediately. These two choices correspond to two approaches of magnetostatic calculations:

- the approach ampérienne
- Coulomb approach.

The approach ampérienne sticks to the calculation of B. It is currently privileged in teaching because it is close to electromagnetism in the vacuum. The equations to be solved are:

∇ * B = 0

∇ * B = µ_{0} (j + ∇ * M)

∇ * B = 0

∇ * B = µ

One can notice that term Δ×M in the second equation acts like an additional current, which was worth to him to be interpreted like a density of current microscopic (called dependant current) rising from the movement of the electrons in their atomic orbits. This traditional interpretation of a quantum phenomenon has its limits however: if it describes rather well magnetism rising from the orbital kinetic moment, it does not give well an account of that related to the spin of the electrons.

In practice, the approach ampérienne is privileged in the situations where there is no magnetized matter and the field is due exclusively to the current. We will place ourselves thereafter in this case where one has then Δ×B = µ0j. To find the general case (in the presence of magnetized matter) it is enough to replace J by J + Δ×M.

It often happens that one deals with systems presenting of surfaces where magnetization is discontinuous. For example, if a magnet with uniform magnetization is plunged in the vacuum, magnetization on the surface of the magnet passes in a discontinuous way of a finished value (inside) to zero (outside). In this case, the density of dependant current Δ×M can be infinite. In such a case one replaces on the surface the voluminal density of current bound by a surface density:

j_{ms} = (M_{1} - M_{2}) * n_{12}

j

where M1 and m2 are magnetizations on each side of the surface of discontinuity and n12 is the normal unit vector on this surface, directed of 1 towards 2. The effect at once of this surface current is to induce a discontinuity of b:

ΔB = µ_{0}ΔM_{|}

ΔB = µ

where

- ΔM and ΔB represent discontinuities of M and B (counted in the same direction)
- ΔM
_{|}represent the part of ΔM which is parallel to surface.

This discontinuity affects only the part of B parallel at surface. The normal part of B remains as for it continues.

Two interesting relations can be obtained by applying the theorem of Stokes to the local relations. The relation ΔB = 0 gives us:

∫_{s} B * dS = 0

∫

where the integral, which extends on a closed surface S, is the outgoing flow of B. It acts of the theorem of flow-divergence. The other relation is obtained by integrating _{Δ×B=µ0j} on an open surface S:

∫_{∂s} B * dL = µ_{0} ∫_{s} j * dS

∫

where the integral of left is the circulation of B on the contour of S. This relation is known under the name of theorem of Amp. The member of right-hand side is interpreted simply like the current crossing surface.

These integral relations often make it possible to simply calculate B in the situations of high symmetry.

That is to say to calculate the field creates by an infinite rectilinear driver. Considerations of symmetry give the orientation of the field: this one turns in plans perpendicular to the fine driver. Its module can be calculated by applying the theorem of Amp to surface S delimited by a line of field of ray a:

∫_{∂s} B * dL = 2παB = µ_{0}I

where I is the current one transported by the wire. One from of deduced the module from b:

B = µ_{0}I ⁄ 2πα

It is seen that the field decrease in opposite proportion of the distance to the wire.

∫

where I is the current one transported by the wire. One from of deduced the module from b:

B = µ

It is seen that the field decrease in opposite proportion of the distance to the wire.

The divergence of B being null, one can make derive B from a potential vector a:

B = ∇ * A

To ensure the unicity of has, one it in general constrained to respect the gauge of Coulomb:

∇ * A = 0

B = ∇ * A

To ensure the unicity of has, one it in general constrained to respect the gauge of Coulomb:

∇ * A = 0

In consideration of which, has is solution of the Poisson equation:

∇² * A + µ_{0}j = 0

∇² * A + µ

One can show that has is given by the integral

A = µ_{0} ⁄ 4π ∫ j /r dv

where the integral extends to all space (or at least at the zones where jΔ0) and

A = µ

where the integral extends to all space (or at least at the zones where jΔ0) and

- R indicates the distance between the point running of the integral and that where has is calculated
- FD is the element of volume.

Same manner, B is given by

B = µ_{0} ⁄ 4π ∫ j * r ⁄ r³ dv

B = µ

where

- R is the active vector of the current point of the integral to that where B is calculated
- R is the module of R.

This last relation is known under the name of law of Biot and Savart.

If there is magnetized matter, it is of course necessary to take account of the dependant currents by replacing J by j+Δ×M. In the presence of dependant currents of surface, it is necessary to add to the integrals of volume of the integrals of surface which result from the preceding ones by substitution

∇ * Mdv → j_{ms} dS

∇ * Mdv → j

A situation usually met is that where the current circulates in a thread-like circuit, and where the section of the wire is neglected. In this case, the voluminal integrals for has and B are replaced by linear integrals along the wire with the help of substitution

jdv → i dl

jdv → i dl

where I is the current one in the wire and DLL the element length, directed according to I.

Magnetic fields B and H created by a uniformly magnetized bar. Magnetization is in blue. In top: in the approach ampérienne, dependant currents Δ×M (mauve) create a field B (in red) similar to the field created by a reel. In bottom: in the Coulomb approach, the magnetic loads - Δ·M (C. - with-D. the magnetic poles, in cyan) create a field H (in green) similar to the electric field in a plane condenser. The fields B and H are identical outside the magnet but differ inside.

In the Coulomb approach one sticks to the calculation of H. This approach finds his roots in work of Coulomb on the forces generated by the magnetic poles. It is still usually employed by the magneticians. It is a question of solving the equations for H:

∇ * H = ρ_{m}

∇ * H = j

where one defined

ρ_{m} = -∇ * M

∇ * H = ρ

∇ * H = j

where one defined

ρ

By analogy with electrostatics, Δm is called density of load magnetic. It should be noticed that with the difference in the electric charges, it magnetic loads cannot be isolated. The theorem of flow-divergence shows indeed that the total magnetic load of a matter sample is null. A magnet thus has always as much positive load (north pole) that negative (south pole).

In practice, the magnetic load is often in the form of density of surface charge localized on surfaces of the magnet. This density of surface charge rises from discontinuities of the normal component of M on the surface, where ΔM is locally infinite. Surfaces thus charged are called magnetic poles. Positively charged surface is the north pole, that negatively charged is the south pole. On these surfaces, one replaces the voluminal density of load by a surface density:

σ_{m} = (M_{1} - M_{2}) * n_{12}

This density of surface charge causes to induce a discontinuity of H:

ΔH = -ΔM_{⊥}

σ

This density of surface charge causes to induce a discontinuity of H:

ΔH = -ΔM

where ΔMι is the part of ΔM which is normal on the surface. This discontinuity assigns only the normal part of H to surface. The parallel part of H remains as for it continues.

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