In case of 2 equal charged particles of charge $e$ (as the proton) at rest in a reference frame their mutual force is according to Colomb's law the well-known formula (in SI-units): $F=\frac{e^2}{4\pi \epsilon r^2}$

$r$ is the distance between both, and $\epsilon$ the dielectricity constant of the medium. If both particles move with velocity $v$ if observed from another reference frame its force (both electric and magnetic force are considered in the following) is

$F=\frac{e^2}{4\pi \epsilon r^2}\left(1-\frac{v^2}{c^2}\right)$

where $c$ is the speed of light. If $v=0$ the old Colomb's law is recovered. On the other hand in case both protons move at speed of light $c$ the force between is zero. In other words: at speed of light electric colomb force and magnetic Lorentz force cancel out each other (in this particular case). This is how electrodynamics works. Length contraction only happens in the direction of motion which is perpendicular of the direction of force, it does not play any role here. You could imagine that due to the time dilation which infinite at speed of light that the repulsion is in fact frozen, it fits with the intuitive view one might have on this effect. A derivation of this formulas can be found in the standard textbooks, Jackson for instance.

In case of 2 equal charged particles of charge $e$ (as the proton) at rest in a reference frame their mutual force is according to Colomb's law the well-known formula (in SI-units): $F=\frac{e^2}{4\pi \epsilon r^2}$

$r$ is the distance between both, and $\epsilon$ the dielectricity constant of the medium. If both particles move with velocity $v$ if observed from another reference frame its force (both electric and magnetic force are considered in the following) is

$F=\frac{e^2}{4\pi \epsilon r^2}\left(1-\frac{v^2}{c^2}\right)$ (to be multiplied by $\times\gamma$, see below)

where $c$ is the speed of light. If $v=0$ the old Colomb's law is recovered. On the other hand in case both protons move at speed of light $c$ the force between is zero. In other words: at speed of light electric colomb force and magnetic Lorentz force cancel out each other (in this particular case). This is how electrodynamics works. Length contraction only happens in the direction of motion which is perpendicular of the direction of force, it does not play any role here. You could imagine that due to the time dilation which infinite at speed of light that the repulsion is in fact frozen, it fits with the intuitive view one might have on this effect. A derivation of this formulas can be found in the standard textbooks, Jackson for instance.

Edit: Apparently the second formula has to be corrected by a factor $\gamma$. Then it turns out to be $F=\frac{e^2}{4\pi \epsilon r^2}\sqrt{1-\frac{v^2}{c^2}}$ The qualitative argumentation is not changed, however.

In case of 2 equal charged particles of charge $e$ (as the proton) at rest in a reference frame their mutual force is according to Colomb's law the well-known formula (in SI-units): $F=\frac{e^2}{4\pi \epsilon r^2}$

$r$ is the distance between both, and $\epsilon$ the dielectricity constant of the medium. If both particles move with velocity $v$ if observed from another reference frame its force (both electric and magnetic force are considered in the following) is

$F=\frac{e^2}{4\pi \epsilon r^2}\left(1-\frac{v^2}{c^2}\right)$

where $c$ is the speed of light. If $v=0$ the old Colomb's law is recovered. On the other hand in case both protons move at speed of light $c$ the force between is zero. Length contraction only happens in the direction of motion which is perpendicular of the direction of force, it does not play any role here. You could imagine that due to the time dilation which infinite at speed of light that the repulsion is in fact frozen, it fits with the intuitive view one might have on this effect. A derivation of this formulas can be found in the standard textbooks, Jackson for instance.

In case of 2 equal charged particles of charge $e$ (as the proton) at rest in a reference frame their mutual force is according to Colomb's law the well-known formula (in SI-units): $F=\frac{e^2}{4\pi \epsilon r^2}$

$r$ is the distance between both, and $\epsilon$ the dielectricity constant of the medium. If both particles move with velocity $v$ if observed from another reference frame its force (both electric and magnetic force are considered in the following) is

$F=\frac{e^2}{4\pi \epsilon r^2}\left(1-\frac{v^2}{c^2}\right)$

where $c$ is the speed of light. If $v=0$ the old Colomb's law is recovered. On the other hand in case both protons move at speed of light $c$ the force between is zero. In other words: at speed of light electric colomb force and magnetic Lorentz force cancel out each other (in this particular case). This is how electrodynamics works. Length contraction only happens in the direction of motion which is perpendicular of the direction of force, it does not play any role here. You could imagine that due to the time dilation which infinite at speed of light that the repulsion is in fact frozen, it fits with the intuitive view one might have on this effect. A derivation of this formulas can be found in the standard textbooks, Jackson for instance.