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I'm learning the basics of electromagnetism, and something that seems trivial as an exercise is proving confusing. Here is an excerpt:

$1a)$ An electron moves initially vertically into a region of electric field which is horizontal to the right with a magnitude of $100 \ N/C$. What is the force on the electron? What is the magnitude and direction of the acceleration of the electron? Comment on your result.

Here is my thought process:

  • It doesn't matter whether the velocity of the electron is such that it comes from above or below - as long as it's vertical. In fact, it shouldn't even matter whether it's moving or not (could be wrong but I don't see anything immediately wrong with this statement). All that matters is whether it experienced a force or not.

  • The electron did in fact feel a force, from an electric field with a strength of $100 \ N/C$. The force will be parallel to the electric field vector.

  • The force can be found by the relationship $\vec F = \vec E q$, where $q$ is the elementary charge in this case: $1.6 * 10^{-19}$ coulombs. This implies the force is $1.6 * 10^{-17}$ Newtons.

Okay, my problem is with the next part. The acceleration must be found next. Application of Newton's Second Law should be fine:

$$\vec F = m \vec a$$ $$ 1.6 * 10^{-17} = m a$$ $$m = 9.11 * 10^{-31} \ kg$$ $$\implies a = 1.75 * 10^{13}\ m/s^2$$

Not feeling good about this. This is superluminal. And considering the fact the force will be acting on the particle for more than $1$ second I would think, as it is a field, this would be problematic. I take it I made a mistake somewhere, but I can't seem to spot anything.

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    $\begingroup$ In a realistic setup, the field could only act for a short time. For example if the plates were 1 meter apart with a 100 volt potential difference. $\endgroup$
    – user137289
    Commented Feb 6, 2018 at 13:57
  • $\begingroup$ "This is superluminal." Can you compare a velocity ($c = 3.0 \times 10^8\,\mathrm{m/s}$) with an acceleration ($1.75\times 10^{13}\,\mathrm{m/s^2}$)? Why or why not? $\endgroup$
    – dmckee --- ex-moderator kitten
    Commented Feb 6, 2018 at 18:59

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What you've done is fine,except for a couple of quibbles: you've forgotten (bullet 2) that the electron's charge is negative, and (third bullet) you're giving a force in coulomb! The main issue: acceleration is a different thing from velocity; even the units are different. So it makes no sense to compare their numerical values, as you tried to do when you called the electron's acceleration 'superluminal'. On the other hand if the field extends far enough the electron will reach a speed of the same order of magnitude as the speed of light in a very short time. [To find what happens subsequently you would need to use relativistic formulae.]

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  • $\begingroup$ Right, I know a faster than light acceleration isn’t impossible per se, but definitely weird, since it would reach relativistic speeds in practically no time at all. Also, ack! My mistake about expressing the force in coulombs. If the charge is negative however, wouldn’t that cause the force vector to not point parallel to the electric field vector? $\endgroup$
    – sangstar
    Commented Feb 6, 2018 at 13:59
  • $\begingroup$ " a faster than light acceleration" doesn't even make sense! This isn't a mere nit-pick. See answer above. Force on a negative charge is in the opposite direction to the electric field strength vector, $\endgroup$
    – Philip Wood
    Commented Feb 6, 2018 at 14:08
  • $\begingroup$ So it's going to move opposite to the electric field vectors? Is that due to the convention that positive test charges move in the direction of electric field lines? $\endgroup$
    – sangstar
    Commented Feb 6, 2018 at 14:15
  • $\begingroup$ Yes, though you needn't bring $lines$ into it. $\endgroup$
    – Philip Wood
    Commented Feb 6, 2018 at 14:48
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The force unit is not Coulomb but Newton, but you calculated it correctly. Also the acceleration seems to be right. How do you see anything "superluminal" in the acceleration? If you assume that the force acts for 1s, using Newton's law you formally would get a superluminal velocity. But you cannot use Newton's law for velocities that are not much smaller than the light velocity$c$. Thus, in this case, for such a long acceleration time you would have to calculate the velocity according to the laws of Special Relativity.

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