Electromagnetic Force

The electromagnetic force is, with the force of Gravitation, the weak Interaction, and the strong Interaction, one of the four fundamental forces of the Physique. When one takes account of the quantum Mécanique the electromagnetic force must be re-studied within the framework of the quantum electrodynamic .

Power

The electromagnetic interaction is the second of the four elementary interactions in the order of the powers. With low energy, that is to say that of the chemical reactions or nuclear, it is about hundred times weaker than the strong Interaction, but exceeds the weak interactions and gravitational of a factor 10¹¹ and 10⁴² respectively.

Mathematical description

The electromagnetic Champ exerts the following force (often called the Force of Lorentz) on particles electrically charged

$$

\ vec {F} = Q \ vec {E} + Q \ vec {v} \ wedge \ vec {B} or:

$$

\ vec {F} = Q \ vec {E} + Q \ frac {\ vec {v}} {C} \ wedge \ vec {B} en unit of Gauss

Electrostatic interaction, Force of Coulomb (on presumedly motionless particles):

$F\; =\; K\; \backslash \; frac\; \{d^\; \{2\}\}$ or $\backslash \; vec\; \{F\}\; =\; -\; K\; \backslash \; frac\; \{q*q\_\; \{2\}\}\; \{d^\; \{2\}\}\; *\; \backslash \; vec\; \{U\}$; with $\backslash \; vec\; \{U\}$ the unit vector directed of Q towards $q\_\; \{2\}$

with:

• $\backslash \; vec\; \{F\}$ the force undergone by the load Q;
• Q the load on which $\backslash \; vec\; \{F\}$ is exerted;
• $q\_\; \{2\}$ the load exerting the froce $\backslash \; vec\; \{F\}$ on Q in the case of the last formula
• $\backslash \; vec\; \{E\}$ the Electric field where the load is located;
• $\backslash \; vec\; \{B\}$ the Magnetic field where the load is located;
• $\backslash \; vec\; \{v\}$ the Speed of the load;
• C the Speed of light;
• $\backslash \; wedge$ the vector Product usual.

(all the sizes are measured in the same reference frame supposed galiléen).

The first description of the force between charged particles, contrary to the Loi of Coulomb, is correct in theory of relativity, and in fact, the magnetic field is then seen as a relativistic interaction of the loads moving that the law of Coulomb alone does not express.

Bond between the force of Lorentz and Laplace

The force of Lorentz makes it possible to find the Force of Laplace, in the absence of electric field ^{$(\backslash \; vec\; \{E\}\; =\; \backslash \; vec\; \{0\})$}. Indeed, the force of Lorentz being exerted on a particle of load Q is

^{$\backslash \; vec\; \{F\}\; =q\; \backslash \; vec\; \{v\}\; \backslash \; wedge\; \backslash \; vec\; \{B\}$.}

Let us consider a whole of its particles in a volume dτ of driver, by neglecting the interactions between the particles and by noting N the particulate density, the resulting force is worth then

, but , with ^{$\backslash \; vec\; \{J\}$} the Vecteur density of current thus:

^{$d\; \backslash \; vec\; \{F\}\; =\; \backslash \; vec\; \{J\}\; D\; \backslash \; tau\; \backslash \; wedge\; \backslash \; vec\; \{B\}$}, is

^{$d\; \backslash \; vec\; \{F\}\; =\; \backslash \; vec\; \{j\_\; \{S\}\}\; dS\; \backslash \; wedge\; \backslash \; vec\; \{B\}$} if the current is surface, and

^{$d\; \backslash \; vec\; \{F\}\; =\; I\; D\; \backslash \; vec\; \{L\}\; \backslash \; wedge\; \backslash \; vec\; \{B\}$} in the model of the linear current.

If one is only in the presence of one uniform magnetic field ^{$\backslash \; vec\; \{B\}$} (the same one everywhere) and of a uniform current, this formula is integrated in

^{$\backslash \; vec\; \{F\}\; =\; \backslash \; vec\; \{I\}\; \backslash \; wedge\; \backslash \; vec\; \{B\}$}, which is the formula of Laplace.

The electric field

See also: Electric field

The electric field $\backslash \; vec\; \{E\}\; \backslash ,$ is related to the force undergone by a particle known as of test of load $q\_0\; \backslash ,$ by:

$\backslash \; vec\; \{F\}\; =\; q\_o\; \backslash \; vec\; \{E\}$

where $\backslash |\backslash \; vec\; \{F\}\; \backslash |\backslash ,$ is expressed in newtons, $\backslash |\backslash \; vec\; \{E\}\; \backslash |\backslash ,$ is in newtons by Coulomb (N/C), or in Volt S by Mètre (V/m), these units being identical.

It will be noted that what counts here it is the electric field without taking account of the electromagnetic field produced by the particle itself . It is often said that one neglects the field produced by the particle, which is not in fact not possible: the electrostatic field varies, like the gravitation, in 1/r ², therefore the clean field of the particle of test is in fact, not only considerable, but even dominating when one approaches the particle, as small as is the load.

In electrostatic, where the loads are not moving, the law of Coulomb is valid, which gives in the vacuum:

$$

\ vec {E} = \ sum_ {i=1} ^ {N} \ frac {q_i (\ vec {R} - \ vec {R} _i) } {4 \ pi \ epsilon_o \ left| \ vec {R} - \ vec {R} _i \ right|^3}

where

• N is the number of loads,
• q_i is the quantity of loads associated with the load I
• $\backslash \; vec\; \{R\}\; \_i$ is the position of the load I
• $\backslash \; vec\; \{R\}$ is the position where the electric field is given
• ε_o is a universal constant called the Permittivité of the vacuum (to be replaced by the permittivity of the medium, when one is not in the vacuum).

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