Infinite linear distribution of charge
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)
Electromagnetic paradox
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.
At the speed of light, Fe is exactly counterbalanced by Fm.
The absence of force between the two charge distributions traveling at the speed of light is equivalent to say that the linear charge density is 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
The value of Fm/Fe expresses the effectiveness with which the Fm component is opposed to the Fe component. A value of Fm/Fe equal to 0.1 corresponds to a reduction of the linear density of electric charge equal to 10%. With Fm/Fe equal to 0.2 the linear density of electric charge is reduced by 20% and so on, up to Fm/Fe equal to 1 which corresponds to a 100% reduction of the linear density of electric charge or to his annihilation.
Its trend as a function of speed v is represented in the image below.
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)
Electromagnetic paradox
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.
At the speed of light, Fe is exactly counterbalanced by Fm.
The absence of force between the two charge distributions traveling at the speed of light is equivalent to say that the linear charge density is 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
The value of Fm/Fe expresses the effectiveness with which the Fm component is opposed to the Fe component. A value of Fm/Fe equal to 0.1 corresponds to a reduction of the linear density of electric charge equal to 10%. With Fm/Fe equal to 0.2 the linear density of electric charge is reduced by 20% and so on, up to Fm/Fe equal to 1 which corresponds to a 100% reduction of the linear density of electric charge or to his annihilation.
Its trend as a function of speed v is represented in the image below.
The graph shows that when v/c=0.5 (speed equal to half the speed of light), Fm/Fe is 0.25, ie the linear charge density is reduced by 25%. For v/c=0.95 Fm/Fe is equal to 0.9, ie the linear charge density is decreased by 90%.
Deepening
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. The force is only electrostatic and as seen above its value depends on the speed. Looking at the equation electrostatic force Fe, a decrease in Fe implies a decrease of the linear electric charge density λ. Since the linear electric charge density λ is the quantity of electric charge in the unit of length (λ=Q/l), or the unit of length increases (ie the space expands) or the amount of electric charge decreases.
In this way, when the charges are motionless the force is equal to Fe. When the charges move at the speed of light, since Fe is completely counterbalanced by Fm and their resultant is zero, the linear charge density λ must be null. The null value of λ is obtained either because the unit of length has increased to infinity (ie the space has extended to infinity) or because the electric charge has been cancelled.
Deepening
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. The force is only electrostatic and as seen above its value depends on the speed. Looking at the equation electrostatic force Fe, a decrease in Fe implies a decrease of the linear electric charge density λ. Since the linear electric charge density λ is the quantity of electric charge in the unit of length (λ=Q/l), or the unit of length increases (ie the space expands) or the amount of electric charge decreases.
In this way, when the charges are motionless the force is equal to Fe. When the charges move at the speed of light, since Fe is completely counterbalanced by Fm and their resultant is zero, the linear charge density λ must be null. The null value of λ is obtained either because the unit of length has increased to infinity (ie the space has extended to infinity) or because the electric charge has been cancelled.
