An infinite linear distribution of charge consists of an infinite wire on which there are static electric charges uniformly distributed over the length. Its electric charge linear density λ1, constant over the whole length of the distribution, is equal to the ratio between the charge Q1 present in a portion of wire and the length l1 of this portion.
The electric field vector at a point which is outside the charge distribution at the distance r from it has direction perpendicular to the wire and direction given by the sign of the charge (exiting for the positive charge, entering for the negative charge) and its module E in that point is given by the formula
E = λ1/2·π·ε0·r
where ε0 is the dielectric constant of the vacuum.
If a second infinite linear distribution of charge appears at the distance r from the first, the electric force per unit of length is equal to
Fe = λ1·λ2/(2·π·ε0·r)
This force is repulsive for charges of the same sign, attractive for charges of opposite sign.
Currents on infinite rectilinear wires
When a current i1 and a current i2 travel on two infinite rectilinear wires parallel at the distance r, an attractive force is generated when the currents have the same verse and repulsive when they have opposite verses. Its module per unit of wire length is given by the following relation (Biot-Savart’s law)
Fm = µ0·i1·i2/(2·π·r)
where µ0 is the magnetic permeability of the vacuum.
Since the current intensity i is defined as the amount of charge Q which passes through a cross-section of the wire in the unit of time
i = Q/t
assuming that the distribution of the moving charges is homogeneous along the conducting wire, so we can speak of linear charge density λ, the current i is given by the product of the linear charge density λ for the mean speed v of charges
i = Q/t = λ·v
The force equation can therefore be rewritten as follows
Fm = µ0·i1·i2/(2·π·r) = µ0·λ1·v1·λ2·v2/(2·π·r)
Taking again the case of the two infinite linear distributions of charge, if the charges move in the same direction of their distribution instead of being static, so their speed v is different from zero, in addition to the force Fe also appears the force Fm. Taking into account the case in which the charges have the same sign and are moving in the same direction the first component is repulsive, while the second is attractive.
The resulting strength module is therefore equal to the Fe value decreased by the Fm component and therefore lower than that in which the charges were static.
As the speed increases, while Fe does not change as it is independent of speed, Fm continues to increase. When the velocity v assumes the value of the speed of light, the value of Fm is identical to that of Fe in that the equality c²=1/ε0·µ0 applies
Fm = µ0·λ1·v1·λ2·v2/(2·π·r) = µ0·λ1·c·λ2·c/(2·π·r) = µ0·c²·λ1· λ2 /(2·π·r) = λ1·λ2/(2·π·ε0·r) = Fe
This means that when the two charge distributions travel at the speed of light the resulting force to which they are subjected is null.
What has been written so far is valid in a reference system in which the motion of the charges is appreciable. When we adopt a reference system that moves together with the charges, their speed v is always null and therefore the value of Fm is also null. Instead, the value of Fe is always the same. In this reference system between a force equal to Fe should continue to manifest between the two charge distributions, which is in open contradiction with what was previously established.
In order to reconcile this prediction with the result in a reference system which does not move with the charges, it is necessary to admit that the value of the electric charge depends on its speed of displacement with respect to an absolute reference system. The charge value is maximum for zero speed, decreases with increasing speed and is canceled when the speed of light is reached.
In this way, when the charges are stopped, the force is equal to Fe. When the charges move at the speed of light together with the reference system, since the electric charge is zero, the value of the linear charge distribution is zero and therefore the value of Fe is also zero.
If until now the force of Coulomb has established what is allowed and what is not, at the speed of light everything becomes possible and the impossible becomes obvious.
Making the ratio between Fm and Fe we obtain the following equation
Fm/Fe = µ0·λ1·v1·λ2·v2/(2·π·r)/[λ1·λ2/(2·π·ε0·r)] = µ0·ε0·v1·v2 = v1·v2/c²
Setting v1=v2=v, the equation is simplified into
Fm/Fe = v²/c²
Its trend as a function of speed v is represented in the image below