Kétamine

The electrostatic treats stable electric charges and force S which they exert between them, i.e. their interactions. By extension, the milked electrostatics of a succession of stable conditions taking into account the transit of intermediate loads.

High potentials (kilovolts with megavolts) can be obtained starting from electrostatic phenomena, which are with weak energy (microjoules with millijoules).

It is difficult to make sciences Physique S without knowing and using the mathematical tools of the formulas of electrostatic.

The Law of Coulomb and the Principe of superposition make it possible mathematically to deduce experimental measurements from motionless loads. This state is naturally not very current but constitutes a stage necessary in the comprehension of the laws of electromagnetism and its extension: relativistic quantum electrodynamics.

Electrostatics appeared particularly powerful in Biophysique in the study of the Protéine S.

General information

There exists a simple experiment, that everyone can make, making it possible to perceive a 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, everyone received 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 electricity, static electricity.

From there, one can consider two categories of body: the Insulator S, or dielectric , where the state of electrification is preserved locally and the conducting 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). With 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 Densité of load ρ (X, there, Z) . This corresponds to a statistical approximation, taking into account the smallness of the elementary charge.

Basic formulas of electrostatics

The fundamental equation of electrostatics is the law of Coulomb, which describes the force of interaction between two concentrated loadings. In a homogeneous medium , le only cas that we will consider in this article, the vacuum for example, it is written:

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

Here, the constant ε is a constant characteristic of the medium, called the “  Permittivity   ”. In the case of the vacuum, one notes it ε ₀ . To note that the permittivity of the air is of 0,5  ‰ higher than that of the vacuum, and is thus often comparable for him.

Note 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  ; also note that the forces are of equal value and opposite direction (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: $\backslash \; overrightarrow\; \{E\_1\}\; (2)\; =\; \backslash \; frac\; \{q\_1\; \backslash \; overrightarrow\; \{r\_\; \{12\}\}\}\; \{4\; \backslash \; pi\; \backslash \; varepsilon\_0\; r\_\; \{12\}\; ^3\}$ produced by 2 into 1: $\backslash \; overrightarrow\; \{E\_2\}\; (1)\; =\; \backslash \; frac\; \{q\_2\; \backslash \; overrightarrow\; \{r\_\; \{21\}\}\}\; \{4\; \backslash \; pi\; \backslash \; varepsilon\_0\; r\_\; \{21\}\; ^3\}$

The field created in M by N loads q_i located in points P_i is additive (Principe of superposition). In the case of a charge distribution discrète :

$\backslash \; overrightarrow\; \{E\_T\}\; =\; \backslash \; overrightarrow\; \{E\_1\}\; +\; \backslash \; overrightarrow\; \{E\_2\}\; +\; \backslash \; overrightarrow\; \{E\_3\}\; +\; \backslash \; ldots\; +\; \backslash \; overrightarrow\; \{E\_n\}\; =\; \backslash \; overrightarrow\; \{E\_T\}\; (M)\; =\; \backslash \; sum\_\; \{i=1\}\; ^n\; \backslash \; frac\; \{q\_i\}\; \{4\; \backslash \; pi\; \backslash \; varepsilon\_0\}\; \backslash \; frac\; \{\backslash \; overrightarrow\; \{P\_iM\}\}\; \{\backslash |\backslash overrightarrow\{P\_iM\}\backslash |^3\}$

In the case of a distribution ρ of loads continues in space, the field caused by a small volume charged vaut :

$D\; \backslash \; vec\; \{E\}\; (x\_m,\; y\_m,\; z\_m)\; =\; \backslash \; frac\; \{\backslash \; rho\; (x\_i,\; y\_i,\; z\_i)\}\{4\; \backslash \; pi\; \backslash \; varepsilon\_0\}\; \backslash \; frac\; \{\backslash \; overrightarrow\; \{r\_\; \{im\}\}\}\; \{r\_\; \{im\}\; ^3\}\; \backslash ,\; dx\_idy\_idz\_i$

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

$\backslash \; vec\; \{E\}\; (x\_m,\; y\_m,\; z\_m)\; =\; \backslash \; int\; \backslash !\; \backslash !\; \backslash !\; \backslash !\; \backslash \; int\; \backslash !\; \backslash !\; \backslash !\; \backslash !\; \backslash \; int\; \backslash \; frac\; \{\backslash \; rho\; (x\_i,\; y\_i,\; z\_i)\}\{4\; \backslash \; pi\; \backslash \; varepsilon\_0\}\; \backslash \; frac\; \{\backslash \; overrightarrow\; \{r\_\; \{im\}\}\}\; \{r\_\; \{im\}\; ^3\}\; \backslash ,\; dx\_idy\_idz\_i$

where ρ is the voluminal Densité of load in P_i , ^{$\backslash \; overrightarrow\; \{r\_\; \{im\}\}$} is the going vector of P_i at the point M . In the element of volume dx_i  dy_i  dz_i around the point P_i there is an element of load ρ (x_i, y_i, z_i) dx_i  dy_i  dz_i . The integrals indicate that it is necessary to add, according to the principle of superposition, on all volumes containing of the loads.

The electric Potentiel (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.

$V\; (x\_m,\; y\_m,\; z\_m)\; =\; \backslash \; int\; \backslash !\; \backslash !\; \backslash !\; \backslash !\; \backslash \; int\; \backslash !\; \backslash !\; \backslash !\; \backslash !\; \backslash \; int\; \backslash \; frac\; \{\backslash \; rho\; (x\_i,\; y\_i,\; z\_i)\}\{4\; \backslash \; pi\; \backslash \; varepsilon\_0|r\_\; \{im\}|\}\; \backslash ,\; dx\_idy\_idz\_i$

and by calculating the derivative partial

$\backslash \; frac\; \{\backslash \; partial\; V\}\; \{\backslash \; partial\; X\},\; ~\; \backslash \; frac\; \{\backslash \; partial\; V\}\; \{\backslash \; partial\; there\},\; ~\; \backslash \; frac\; \{\backslash \; partial\; V\}\; \{\backslash \; partial\; Z\}$

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 ( ρ should be infinite there).

Potential in 1/ R and field with null divergence

One places the load which produces the potential in O and one looks in the potential produced in M and his gradient. All this paragraph supposes that O and M does not coincide   ; if not the formulas would not have any direction. Posons :
$\backslash \; overrightarrow\; \{OM\}\; =\; \backslash \; vec\; \{R\}\; =\; R\; \backslash \; vec\; \{e\_r\}$
Let us recall that, by definition of derived the partielles :
$dV=\; \backslash \; overrightarrow\; \{\backslash \; mbox\; \{grad\}\}\; \backslash ,\; V\; \backslash \; cdot\; D\; \backslash \; overrightarrow\; \{OM\}\; =\; -\; \backslash \; vec\; \{E\}\; (M)\; \backslash \; cdot\; D\; \backslash \; overrightarrow\; \{OM\}$

• knowing that one can show that $\backslash \; frac\; \{\backslash \; vec\; \{R\}\}\; \{r^3\}\; =\; \backslash \; overrightarrow\; \{\backslash \; mbox\; \{grad\}\}\; \backslash ,\; \backslash \; frac\; \{1\}\; \{R\}$ ¹, one from of deduced while multiplying by $\backslash \; frac\; \{Q\}\; \{4\; \backslash \; pi\; \backslash \; varepsilon\_0\}$ que :

$\backslash \; vec\; \{E\}\; (X,\; there,\; Z)\; =\; \backslash \; frac\; \{Q\}\; \{4\; \backslash \; pi\; \backslash \; varepsilon\_0\}\; \backslash \; frac\; \{\backslash \; vec\; \{R\}\}\; \{r^3\}\; =\; -\; \backslash \; overrightarrow\; \{\backslash \; mbox\; \{grad\}\}\; \backslash \; frac\; \{Q\}\; \{4\; \backslash \; pi\; \backslash \; varepsilon\_0\; R\}\; =\; -\; \backslash \; overrightarrow\; \{\backslash \; mbox\; \{grad\}\}\; V\; (R)$

with   $V\; (R)\; =\; \backslash \; frac\; \{Q\}\; \{4\; \backslash \; pi\; \backslash \; varepsilon\_0\; R\}$

• fields in $\backslash \; frac\; \{\backslash \; vec\; \{R\}\}\; \{\backslash |\backslash \; vec\; \{R\}\; \backslash |^3\}$ is such as their divergence is nulle : $\backslash \; mbox\; \{div\}\; \backslash \; frac\; \{\backslash \; vec\; \{R\}\}\; \{\backslash |\backslash \; vec\; \{R\}\; \backslash |^3\}\; =\; 0$ ²

Theorem of Gauss

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

This theorem indicates that (summary demonstration)  :

$dy\; \backslash ,\; dz+dz\; \backslash ,\; dx+dx\; \backslash ,\; Dy\; =$

$=\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; E\_x\}\; \{\backslash \; partial\; X\}\; +\; \backslash \; frac\; \{\backslash \; partial\; partial\; E\_y\}\; \{\backslash \; there\}\; +\; \backslash \; frac\; \{\backslash \; partial\; E\_z\}\; \{\backslash \; partial\; Z\}\; \backslash \; right)\; dx\; \backslash ,\; Dy\; \backslash ,\; dz\; =\; \backslash \; mbox\; \{div\}\; \backslash \; vec\; \{E\}\; \backslash \; FD\; =\; \backslash \; sum\_i\; \backslash \; vec\; \{E\}\; \_i\; \backslash \; cdot\; D\; \backslash \; vec\; \{S\}\; \_i$

Here FD = dx  dy  dz represents an elementary volume, which one can regard as a parallelepiped and the dS_i represent the contributions of the 6 faces, 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 on all these elementary volumes, the contributions of the faces located inside volume are compensated exactly, and there remains only the contribution of surface extérieure :

for any volume. In particular, let us consider a sphere charged in volume by a voluminal density of load ρ , having its center in O and of ray R sufficiently small so that one can neglect the variations of ρ  :
$dS\; \backslash \; frac\; \{\backslash \; vec\; \{R\}\}\; \{R\}$ is the normal vector on the surface directed towards outside, and of length equal to the element of surface dS that it represents.

$\backslash int\backslash !\backslash !\; \backslash !\; \backslash !\; \backslash \; int\_\; \{S\}\; \backslash \; frac\; \{\backslash \; vec\; \{R\}\}\; \{r^3\}\; \backslash \; cdot\; D\; \backslash \; vec\; S\; =\; \backslash \; int\; \backslash !\; \backslash !\; \backslash !\; \backslash !\; \backslash \; int\_\; \{S\}\; \backslash \; frac\; \{\backslash \; vec\; \{R\}\}\; \{r^3\}\; \backslash \; cdot\; dS\; \backslash \; frac\; \{\backslash \; vec\; \{R\}\}\; \{R\}\; =\; \backslash \; int\; \backslash !\; \backslash !\; \backslash !\; \backslash !\; \backslash \; int\_\; \{S\}\; \backslash \; vec\; \{e\_r\}\; \backslash \; cdot\; \backslash \; vec\; \{e\_r\}\; \backslash ,\; \backslash \; frac\; \{dS\}\; \{r^2\}\; =\; \backslash \; int\; \backslash !\; \backslash !\; \backslash !\; \backslash !\; \backslash \; int\_\; \{S\}\; \backslash \; frac\; \{dS\}\; \{r^2\}\; =\; \backslash \; frac\; \{S\}\; \{r^2\}\; =\; \backslash \; frac\; \{4\; \backslash \; pi\; r^2\}\; \{r^2\}\; =\; 4\; \backslash \; pi$

What means that the result does not depend on R   ! and if one multiplies by $\backslash \; frac\; \{\backslash \; rho\; v\}\; \{4\; \backslash \; pi\; \backslash \; varepsilon\_0\}$ where v is the volume of the sphere, one obtains:

$\backslash \; frac\; \{\backslash \; rho\; v\}\; \{4\; \backslash \; pi\; \backslash \; varepsilon\_0\}\; \backslash \; int\; \backslash !\; \backslash !\; \backslash !\; \backslash !\; \backslash \; int\_\; \{S\}\; \backslash \; frac\; \{\backslash \; vec\; \{R\}\}\; \{r^3\}\; \backslash \; cdot\; D\; \backslash \; vec\; S\; =\; \backslash \; int\; \backslash !\; \backslash !\; \backslash !\; \backslash !\; \backslash \; int\_\; \{S\}\; \backslash \; frac\; \{\backslash \; rho\; v\}\; \{4\; \backslash \; pi\; \backslash \; varepsilon\_0\}\; \backslash \; frac\; \{\backslash \; vec\; \{R\}\}\; \{r^3\}\; \backslash \; cdot\; D\; \backslash \; vec\; S\; =\; \backslash \; int\; \backslash !\; \backslash !\; \backslash !\; \backslash !\; \backslash \; int\_\; \{S\}\; \backslash \; vec\; \{E\}\; \backslash \; cdot\; D\; \backslash \; vec\; S\; =\; \backslash \; frac\; \{\backslash \; rho\; v\}\; \{4\; \backslash \; pi\; \backslash \; varepsilon\_0\}\; \backslash ,\; 4\; \backslash \; pi\; =\; \backslash \; frac\; \{Q\}\; \{\backslash \; varepsilon\_0\}$ where Q is the total load ρv of the sphere Maybe with the final : $\backslash int\backslash !\backslash !\; \backslash !\; \backslash !\; \backslash \; int\_\; \{S\}\; \backslash \; vec\; \{E\}\; \backslash \; cdot\; D\; \backslash \; vec\; S\; =\; \backslash \; frac\; \{Q\}\; \{\backslash \; varepsilon\_0\}\; =\; \backslash \; int\; \backslash !\; \backslash !\; \backslash !\; \backslash !\; \backslash \; int\; \backslash !\; \backslash !\; \backslash !\; \backslash !\; \backslash \; int\; \backslash \; mbox\; \{div\}\; \backslash \; vec\; E\; \backslash \; FD\; =\; \backslash \; int\; \backslash !\; \backslash !\; \backslash !\; \backslash !\; \backslash \; int\; \backslash !\; \backslash !\; \backslash !\; \backslash !\; \backslash \; int\; \backslash \; frac\; \{\backslash \; rho\}\; \{\backslash \; varepsilon\_0\}\; \backslash ,\; dv$ From where (Theorem of Gauss under its local version)  :

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

The Poisson's equation

combine the preceding relations to give a local relation between the charge distribution and the potential:

See the article Nabla for the significance of the very useful symbol ^{$\backslash \; vec\; \{\backslash \; nabla\}$}

The fact is found that the influences of the various loads are added linearly, i.e. 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 résultats : one finds well the Principe of superposition , another manner of expressing the Linéarité law of Coulomb.

The law of Coulomb is very close to the expression of the nelles forces Gravitation; 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.

Electric field created by some charge distributions

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 .

$\backslash int\backslash !\backslash !\; \backslash !\; \backslash !\; \backslash \; int\_S\; \backslash \; vec\; \{E\}\; \backslash \; cdot\; D\; \backslash \; vec\; \{S\}\; =\; \backslash \; frac\; \{Q\}\; \{\backslash \; varepsilon\_0\}$

Here some examples of computation results for symmetrical charge distributions.

• infinite rectilinear Wire, taken along the axis linear OZ of Density of load λ , remotely R of the wire:

For a point M , the plan passing by M containing the axis OZ is a symmetry plane, like that passing by M and perpendicular to the axis OZ   ; one from of deduced that the resulting field has of component only suivant :

$\backslash vec\{e\_r\}\; :\; \backslash \; vec\; \{E\}\; (R,\; \backslash \; theta,\; Z)\; =\; E\_r\; (R,\; \backslash \; theta,\; Z)\; \backslash ,\; \backslash \; vec\; \{e\_r\}$

Invariances by translation according to OZ and by rotation according to θ make it possible to deduce that E_r should not depend on the variables Z and θ and donc :
$\backslash \; vec\; \{E\}\; (R)\; =\; E\_r\; (R)\; \backslash ,\; \backslash \; vec\; \{e\_r\}$

So to apply the theorem of Gauss, one chooses a cylinder passing by M , of axis OZ , of ray R and elementary thickness dz  :

$\backslash int\backslash !\backslash !\; \backslash !\; \backslash !\; \backslash \; int\_S\; \backslash \; vec\; \{E\}\; \backslash \; cdot\; D\; \backslash \; vec\; \{S\}\; =\; \backslash \; int\; \backslash !\; \backslash !\; \backslash !\; \backslash !\; \backslash \; int\_S.E.\; \_r\; (R)\; \backslash \; vec\; \{e\_r\}\; \backslash \; cdot\; \backslash \; vec\; \{e\_r\}\; \backslash ,\; dS\; =E\_r\; (R)\; \backslash \; int\; \backslash !\; \backslash !\; \backslash !\; \backslash !\; \backslash \; int\_S\; dS\; =\; E\_r\; (R)\; (2\; \backslash \; pi\; R\; \backslash ,\; dz)\; =\; \backslash \; frac\; \{Q\}\; \{\backslash \; varepsilon\_0\}\; =\; \backslash \; frac\; \{\backslash \; lambda\; dz\}\; \{\backslash \; varepsilon\_0\}$

and one obtains finalement : $E\_r\; (R)\; =\; \backslash \; frac\; \{\backslash \; lambda\}\; \{2\; \backslash \; pi\; \backslash \; varepsilon\_0\}\; \backslash \; frac\; \{1\}\; \{R\}$

• Plane infinite, uniformly charged on the surface, of surface Density of load σ , remotely R of the plan. As the system is invariant by translation parallel with the plan, the field can be only perpendicular to the plan. In addition, the fields are directly opposed in two symmetrical points compared to the plan. If M is at the distance R of the plan, consider a symmetrical elementary prism compared to the plan and whose base, from surface dS , passes by M  :

$\backslash int\backslash !\backslash !\; \backslash !\; \backslash !\; \backslash \; int\_S\; \backslash \; vec\; \{E\}\; \backslash \; cdot\; D\; \backslash \; vec\; \{S\}\; =2E\; (R)\; \backslash ,\; dS=\; \backslash \; frac\; \{\backslash \; sigma\; \backslash ,\; dS\}\; \{\backslash \; varepsilon\_0\}$   from where   $E\; (R)\; =\; \backslash \; frac\; \{\backslash \; sigma\}\; \{2\; \backslash \; varepsilon\_0\}$

the absolute value of the field is constant in all space. Its direction changes between the two sides of the plan  ; it is thus discontinuous on the level of the plan.

• Sphère digs diameter R uniformly, charged on the surface, of surface density of load σ , remotely R of the center:

• inside (R < R) : $E\; (R)\; =\; 0\; \backslash \; quad$
• just outside surface (R = R+0) : $E\; (R)\; =\; \backslash \; frac\; \{\backslash \; sigma\}\; \{\backslash \; varepsilon\_0\}$. Again, the field is discontinuous on the level of a surface charged.
• outside (R > R) : $E\; (R)\; =\; \backslash \; frac\; \{\backslash \; sigma\}\; \{\backslash \; varepsilon\_0\}\; \backslash \; frac\; \{R^2\}\; \{r^2\}$

• Sphere full with diameter R , uniformly charged in volume, of voluminal density of load ρ , remotely R of the center:

• inside (R < R) : $E\; (R)\; =\; \backslash \; frac\; \{\backslash \; rho\}\; \{3\; \backslash \; varepsilon\_0\}\; R$
• on the surface (R = R) : $E\; (R)\; =\; \backslash \; frac\; \{\backslash \; rho\}\; \{3\; \backslash \; varepsilon\_0\}\; R$
• outside (R > R) : $E\; (R)\; =\; \backslash \; frac\; \{\backslash \; rho\}\; \{3\; \backslash \; varepsilon\_0\}\; \backslash \; frac\; \{R^3\}\; \{r^2\}$

Consequence of the theorem of Gauss, we find in both cases outside the sphere a field equal to that of a specific load Q placed at the center of the sphère :

$E\; (R)\; =\; 4\; \backslash \; pi\; \backslash \; sigma\; R^2\; \backslash \; frac\; \{1\}\; \{4\; \backslash \; pi\; \backslash \; varepsilon\_0\; r^2\}\; =\; \backslash \; frac\; \{Q\}\; \{4\; \backslash \; pi\; \backslash \; varepsilon\_0\; r^2\}$

respectivement :

$E\; (R)\; =\; \backslash \; frac\; \{4\; \backslash \; pi\; \backslash \; rho\; R^3\}\; \{3\}\; \backslash \; frac\; \{1\}\; \{4\; \backslash \; pi\; \backslash \; varepsilon\_0\; r^2\}\; =\; \backslash \; frac\; \{Q\}\; \{4\; \backslash \; pi\; \backslash \; varepsilon\_0\; r^2\}$

Examples of potentials

• Potential of a wire finished (- has, a) in B in its prolongement :

$V\; (b)=\; \backslash \; int\_\; \{-\; has\}\; ^a\; \backslash ,\; \backslash \; frac\; \{\backslash \; lambda\}\; \{4\; \backslash \; pi\; \backslash \; varepsilon\_0\}\; \backslash \; frac\; \{dx\}\; \{b-x\}\; =\; \backslash \; frac\; \{\backslash \; lambda\}\; \{4\; \backslash \; pi\; \backslash \; varepsilon\_0\}\; \backslash ,\; \backslash \; ln\; \backslash ,\; \backslash \; frac\; \{b+a\}\; \{Ba\}$

• Potentiel of a disc charged with ray R at a distance Z of its center along its axe :

$V\; (Z)\; =\; \backslash \; frac\; \{\backslash \; sigma\}\; \{4\; \backslash \; pi\; \backslash \; varepsilon\_0\}\; \backslash \; int\_0^R\; \backslash \; frac\; \{2\; \backslash \; pi\; R\; \backslash ,\; Dr.\}\; \{\backslash \; sqrt\; \{r^2+z^2\}\}\; =\; \backslash \; frac\; \{\backslash \; sigma\}\; \{2\; \backslash \; varepsilon\_0\}\; \backslash \; left\; (\backslash \; sqrt\; \{R^2+z^2\}\; -\; Z\; \backslash \; right)$

A finished wire: direct calculation of the produced field

Let us suppose that one with the axis of the X charged on a segment AB with a density of constant linear load λ and, a point M  (x_M, y_M) in the plan xOy 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 λdx . These loads create in M a field. Posing PM = R  :

$\backslash \; vec\; \{E\}\; (M)\; =\; \backslash \; int\_a^b\; \backslash !\; \backslash !\; D\; \backslash \; vec\; \{E\}\; (M)\; =\; \backslash \; frac\; \{\backslash \; lambda\}\; \{4\; \backslash \; pi\; \backslash \; varepsilon\_0\}\; \backslash \; int\_a^b\; \backslash !\; \backslash !\; \backslash \; frac\; \{\backslash \; overrightarrow\; \{PM\}\}\; \{r^3\}\; dx\; =\; \backslash \; frac\; \{\backslash \; lambda\}\; \{4\; \backslash \; pi\; \backslash \; varepsilon\_0\}\; \backslash \; int\_a^b\; \backslash !\; \backslash !\; \backslash \; left\; (\backslash \; frac\; \{(x\_M-x)\; \backslash \; vec\; \{I\}\; +y\_M\; \backslash \; vec\; \{J\}\}\; \{r^3\}\; \backslash \; right)\; dx$

$=\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; lambda\}\; \{4\; \backslash \; pi\; \backslash \; varepsilon\_0\}\; \backslash \; int\_a^b\; \backslash \; frac\; \{(x\_M-x)\}\{r^3\}\; \backslash \; right)\; dx\; \backslash ,\; \backslash \; vec\; \{I\}\; +\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; lambda\}\; \{4\; \backslash \; pi\; \backslash \; varepsilon\_0\}\; \backslash \; int\_a^b\; \backslash \; frac\; \{y\_M\}\; \{r^3\}\; \backslash \; right)\; dx\; \backslash ,\; \backslash \; vec\; \{J\}$

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

$\backslash \; vec\; \{E\}\; (x\_M,\; y\_M)\; =\; E\_x\; (x\_M,\; y\_M)\; \backslash ,\; \backslash \; vec\; \{I\}\; +\; E\_y\; (x\_M,\; y\_M)\; \backslash ,\; \backslash \; vec\; \{J\}$

By noting que :

$\backslash \; frac\; \{(x\_M-x)\}\{R\}\; =\; \backslash \; sin\; \backslash \; alpha$   and   $\backslash \; frac\; \{y\_M\}\; \{R\}\; =\; \backslash \; cos\; \backslash \; alpha$   one déduit :   $\backslash \; frac\; \{(x\_M-x)\}\{y\_M\}\; =\; \backslash \; mbox\; \{tg\}\; \backslash ,\; \backslash \; alpha$ where α is complementary to angle BPM ,

$\backslash \; frac\; \{\backslash \; lambda\}\; \{4\; \backslash \; pi\; \backslash \; varepsilon\_0\}\; \backslash \; int\_a^b\; \backslash \; frac\; \{x\_M-x\}\; \{r^3\}\; \backslash ,\; dx\; =\; \backslash \; frac\; \{\backslash \; lambda\}\; \{4\; \backslash \; pi\; \backslash \; varepsilon\_0\}\; \backslash \; int\_a^b\; \backslash \; frac\; \{(x\_M-x)\; dx\}\; \{r^\; \{3\}\}\; =\; \backslash \; frac\; \{\backslash \; lambda\}\; \{4\; \backslash \; pi\; \backslash \; varepsilon\_0\}\; \backslash \; int\_\; \{\backslash \; alpha\_1\}\; ^\; \{\backslash \; alpha\_2\}\; \backslash !\; \backslash !\; \backslash \; sin\; \backslash \; alpha\; \backslash ,\; D\; \backslash \; alpha$   easy to integrate

There is utilisé :
$dx=\; \backslash \; frac\; \{there\; \_M\}\; \{\backslash \; cos^2\; \backslash \; alpha\}\; \backslash ,\; D\; \backslash \; alpha$  ,   $\backslash \; frac\; \{1\}\; \{r^2\}\; =\; \backslash \; frac\; \{\backslash \; cos^2\; \backslash \; alpha\}\; \{y\_M^2\}$   and   $\backslash \; frac\; \{x\_M-x\}\; \{R\}\; =\; \backslash \; sin\; \backslash \; alpha$

Distributions having symmetries and invariances

For charge distributions 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 annule : by gathering the loads per symmetrical pairs, one notes this nullity).

Example: If one has a spherical distribution of burden of center O , then any plan passing by O is a plan of symétrie : consequently, the resulting field in M is in all the plans containing OM and thus ^{$\backslash \; vec\; E\; (R,\; \backslash \; theta,\; \backslash \; phi)\; =\; E\_r\; (R,\; \backslash \; theta,\; \backslash \; phi)\; \backslash \; vec\; e\_r$} since E_θ (R, θ, φ) = 0 and E_φ (R, θ, φ) = 0 .

More generally, if, for an Euclidean transformation T , the distribution ρ (T (M)) is identical to ρ (M) , the field in T (M) will be transformed by T of that into M . It is said 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 coordinates polaires :

$\backslash \; vec\; E\; (R,\; \backslash \; theta,\; \backslash \; phi)\; =\; E\_r\; (R,\; \backslash \; theta,\; \backslash \; phi)\; \backslash \; vec\; e\_r\; =E\_r\; (R)\; \backslash \; vec\; e\_r$

This result simplifies much calculations.

Other exemple : case of a cylindrical symmetry, with invariance of ρ by symmetry compared to any plan containing OZ , or perpendicular to OZ , one obtient :

$\backslash \; vec\; E\; (R,\; \backslash \; theta,\; Z)\; =\; E\_r\; (R,\; \backslash \; theta,\; Z)\; \backslash \; vec\; e\_r\; =E\_r\; (R)\; \backslash \; vec\; e\_r$

See too

Related articles

• anti-static Agent

• vector Calculus
• Density of load
• Equation of electromagnetic Laplace
• Force
• electrostatic Machine
• Magnetostatic
• Potential electric
• Replenisher
• Theorem of Green

External bonds

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