- #1

FranzDiCoccio

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- TL;DR Summary
- a textbook example considers two protons at a given distance having the same constant velocity. The analysis of their interaction in two different reference frames results in an (apparent) paradox.

This is used to highlight the inconsistency of Newtonian Mechanics and Maxwell's Electrodynamics in an introductory discussion of Special Relativity.

I'm wondering whether there's a simple way of "solving" the paradox.

Before introducing Special Relativity, a textbook highlights the inconsistency of Maxwell's Electrodynamics and Newtonian Mechanics through the standard discussion about the velocity of light in different frames of reference.

A further inconsistency discussed.

In some inertial frame of reference two protons are initially at distance ##d## and move with velocity ##v## perpendicular to the segment joining their positions.

In this frame of reference each proton generates both an electric and a magnetic field. The fields generated by each of the protons affect the other particle. Specifically, the electric and magnetic field produce an attractive and a repulsive force, respectively.

The same situation is then analyzed in a frame of reference moving at the same velocity as the (initial) velocity of the protons. There, no magnetic field is present, so only the repulsive interaction exists.

From a "classical" (non-relativistic) point of view this is a problem, because the two inertial frames of reference should be equivalent in the Galilean sense. In this case the two protons would move with different accelerations in two different inertial frames of reference.

But according to Galilean invariance their accelerations should be the same in the two frames of reference.

After discussing this, the book introduces the postulates of Special Relativity, and never goes back to this example.

In a way, this is understandable, because its discussion does not go as far as the transformations of the fields.

I was wondering whether a sufficiently simple argument exists that "solves" the paradox. I think I have something, but I am not sure I am interpreting it correctly.

The magnetic field that each proton generates is

[tex]\vec B = \frac{\mu_0}{4\pi} e \frac{\vec v \times \vec r}{r^3}[/tex]

Using ##\vec v = v \hat x## and ##\vec r = d \hat y## I get

[tex]\vec B = \frac{\mu_0}{4\pi}\frac{e v}{d^2} \hat z[/tex]

The force resulting from the combined electric and magnetic field should have a magnitude

[tex]F=\frac{1}{4\pi \varepsilon_0} \frac{e^2}{d^2} (1-\frac{v^2}{c^2})[/tex]

Classically, in the frame of reference where the protons are at rest ##v=0## and the magnitude is

[tex]F'=\frac{1}{4\pi \varepsilon_0} \frac{e^2}{d^2}[/tex]

Using the transformation of the fields between the two inertial frames I get

[tex]E' =\gamma \frac{1}{4\pi \varepsilon_0} \frac{e}{d^2}(1-\frac{v^2}{c^2}),\qquad B'=0 [/tex]

so that

[tex]F'=\gamma F[/tex]

If I get this other discussion right, this is the way the force is actually expected to transform according to Special Relativity, which should prove that there is no real paradox here.

Does this make sense?

A further inconsistency discussed.

In some inertial frame of reference two protons are initially at distance ##d## and move with velocity ##v## perpendicular to the segment joining their positions.

In this frame of reference each proton generates both an electric and a magnetic field. The fields generated by each of the protons affect the other particle. Specifically, the electric and magnetic field produce an attractive and a repulsive force, respectively.

The same situation is then analyzed in a frame of reference moving at the same velocity as the (initial) velocity of the protons. There, no magnetic field is present, so only the repulsive interaction exists.

From a "classical" (non-relativistic) point of view this is a problem, because the two inertial frames of reference should be equivalent in the Galilean sense. In this case the two protons would move with different accelerations in two different inertial frames of reference.

But according to Galilean invariance their accelerations should be the same in the two frames of reference.

After discussing this, the book introduces the postulates of Special Relativity, and never goes back to this example.

In a way, this is understandable, because its discussion does not go as far as the transformations of the fields.

I was wondering whether a sufficiently simple argument exists that "solves" the paradox. I think I have something, but I am not sure I am interpreting it correctly.

The magnetic field that each proton generates is

[tex]\vec B = \frac{\mu_0}{4\pi} e \frac{\vec v \times \vec r}{r^3}[/tex]

Using ##\vec v = v \hat x## and ##\vec r = d \hat y## I get

[tex]\vec B = \frac{\mu_0}{4\pi}\frac{e v}{d^2} \hat z[/tex]

The force resulting from the combined electric and magnetic field should have a magnitude

[tex]F=\frac{1}{4\pi \varepsilon_0} \frac{e^2}{d^2} (1-\frac{v^2}{c^2})[/tex]

Classically, in the frame of reference where the protons are at rest ##v=0## and the magnitude is

[tex]F'=\frac{1}{4\pi \varepsilon_0} \frac{e^2}{d^2}[/tex]

Using the transformation of the fields between the two inertial frames I get

[tex]E' =\gamma \frac{1}{4\pi \varepsilon_0} \frac{e}{d^2}(1-\frac{v^2}{c^2}),\qquad B'=0 [/tex]

so that

[tex]F'=\gamma F[/tex]

If I get this other discussion right, this is the way the force is actually expected to transform according to Special Relativity, which should prove that there is no real paradox here.

Does this make sense?