Paricles in crossed electric and magnetic field

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Discussion Overview

The discussion revolves around the behavior of an electron and a positron moving in crossed electric and magnetic fields, specifically addressing the forces acting on them and the implications for their separation. The participants explore the conditions under which these particles can maintain their trajectories without separation, focusing on the Lorentz force and the configuration of the fields.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant calculates the forces on the electron and positron and finds them unequal, suggesting a potential separation, which contradicts the problem statement.
  • Another participant questions the sign of the electric field in the force equations for the positron compared to the electron, raising concerns about the treatment of the forces.
  • A participant asserts that the forces on an electron and positron should be in opposite directions for a given electric field, challenging the initial calculations.
  • Further clarification is provided regarding the definitions of the electric and magnetic fields, leading to a reevaluation of the forces acting on the particles.
  • One participant concludes that for the particles to maintain a straight path, the relationship between the electric and magnetic fields must satisfy a specific condition, suggesting that the magnetic field direction may need to be reversed.

Areas of Agreement / Disagreement

Participants express differing views on the signs of the forces acting on the electron and positron, and whether the attraction between them should be considered. The discussion remains unresolved regarding the implications of these forces on the separation of the particles.

Contextual Notes

There are unresolved assumptions regarding the treatment of the electric and magnetic fields, particularly concerning their signs and the implications for the forces acting on the particles. The discussion also touches on the potential relevance to a cathode ray tube (CRT) experiment, though this connection is not fully explored.

iAlexN
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An electron and a positron are moving in the +x-direction with the same velocity in a crossed electric and magnetic field (the fields are perpendicular). The question states it's impossible to separate them using this configuration.

The electric field is pointing in the -y-direction and the magnetic field, out of the page (+z-direction). Taking the +y-direction as positive for the electric field and vB, I get this:

F_e = -e(E-vB) = e(-E+vB) (force on electron)
F_p = e(-E-vB) (force on positron)

The cross product between the velocity (in the +x-direction) and the magnetic field (+z-direction) points the same for the electron and the positron (I think), -y-direction.

As you can see F_e and F_p are not equal, so the particles would be separated, which is not supposed to be possible.

But I just cannot find where I go wrong.

Thank you!
 
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Why is there a minus sign for the positron E field magnitude and not for the electron E field?
How did you account for the attractive force between the positron and the electron?
 
iAlexN said:
Fe=−e(E−vB)=e(−E+vB)F_e = -e(E-vB) = e(-E+vB) (force on electron)
Fp=e(−E−vB)F_p = e(-E-vB) (force on positron)


Why do you have -E in the force on the positron?

For a given E field, the forces on an electron and a positron should be in opposite directions.
 
jtbell said:
Why do you have -E in the force on the positron?

For a given E field, the forces on an electron and a positron should be in opposite directions.

I agree, the E-field should have the same sign, in this situation I defined the minus y-direction as negative. I get:

Fe=−e(-E−vB) = e(E+vB)
Fp=e(-E−vB)

Supposedly (according to the question) you need not consider the attraction between the electron and the positron, just the Lorentz's Force to show this.

However, this gives that the magnetic force on the electron and positron are in the same direction as the electric force, which means they would separate? But they are not supposed to.
 
Oh now I get you!
Lets see - electric field is ##\vec E = -E\hat\jmath##, the magnetic field is ##\vec B = B\hat k##, and the velocity is ##\vec v = v\hat\imath##

Check my reasoning...
For an arbitrary charge q:
##\vec F = q\big[-E\hat\jmath + vB(\hat\imath \times \hat k = -\hat j)\big] = q(-E - vB )\hat\jmath##

For the particle to maintain a straight line path, ##E+ vB=0## ... if you fix the electric field and adjust the magnet, you see that it has to be ##B=-E/v < 0##
i.e. the magnetic field needs to point the other way; well done.

Is this in the context of a CRT experiment?
 

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