Coulomb's force vs the Lorentz force

[Electrodude]

Beams of electrons and protons move parallel to each other in the same direction. They ______.
a. attract each other.
b. repel each other.
c. neither attract nor repel.
d. the force of attraction or repulsion depends upon the speed of the beams.

This is a previous-year-question of CBSE Board 2023.
The answer key marks (b) as the right option.
I want to know why we are ignoring Coulomb's force?

 

[PeroK]

Beams of electrons and protons move parallel to each other in the same direction. They ______.
a. attract each other.
b. repel each other.
c. neither attract nor repel.
d. the force of attraction or repulsion depends upon the speed of the beams.

This is a previous-year-question of CBSE Board 2023.
The answer key marks (b) as the right option.
I want to know why we are ignoring Coulomb's force?

What happens if the speed of both beams is zero, or approximately zero?

 

[Dale]

This is a poorly worded question. Are they talking about one pair of beams one containing protons and the other electrons? Or one pair of beams each containing both protons and electrons? Or multiple pairs of beams one pair entirely of protons and another pair entirely of electrons?

 

[Electrodude]

This is a poorly worded question. Are they talking about one pair of beams one containing protons and the other electrons? Or one pair of beams each containing both protons and electrons? Or multiple pairs of beams one pair entirely of protons and another pair entirely of electrons?

This is the exact question. This question probably was prompted to around a million students two years back.
Anyways, with reference to the complexity level of our syllabus, it would be safe to assume that there are two separate beams of electrons and protons, some unspecified distance apart and travelling at some unspecified speed.

 

[Electrodude]

What happens if the speed of both beams is zero, or approximately zero?

Is it to assume static conditions to apply electrostatics?

 

[PeroK]

This is the exact question. This question probably was prompted to around a million students two years back.
Anyways, with reference to the complexity level of our syllabus, it would be safe to assume that there are two separate beams of electrons and protons, some unspecified distance apart and travelling at some unspecified speed.

Speed is frame dependent. Take the case where the speeds are equal and consider the frame where that speed is zero. Now we have two rows of protons and electrons at rest that repel each other?

 

[haruspex]

I'd forgotten this puzzled me in the past: whether there is a magnetic field from a current must depend on the frame of reference. Turns out it takes SR to explain it.
Consider two parallel beams of electrons at the same velocity. The relative motion to the observer increases the electrostatic repulsion, and this exactly cancels the attraction created by the observed magnetic fields.
Full details at https://physics.stackexchange.com/q...forces-on-each-other-depending-on-frame-of-re.

So the answer to the question in this thread is a.

 

[PeroK]

This question can be understood without relativity by considering a reference frame in which the beams move at the same speed in opposite directions. That's equivalent to two equal currents in the same direction, which produces an attractive force. Additionally, if we have beams of particles rather than neutral current carrying wires, then there is an additional attractive electric force.

 

[haruspex]

This question can be understood without relativity by considering a reference frame in which the beams move at the same speed in opposite directions.

I do not understand. If they move with the same velocity in one frame they must move at the same velocity in all frames.
Besides, the conundrum is why two different inertial frames appear to give conflicting results. Considering a third frame cannot resolve that.

Btw, it remains true, of course, that two parallel wires carrying (vectorially) the same current attract, but a wire carrying a current is not the same as a beam of ions: it has no net charge. That said, I have not figured out why an observer moving with the same velocity as the electrons still sees attraction. Maybe it looks like positive charges moving the other way?

 

[kuruman]

To muddy the waters some more, here is one of the many sites that feature this question
https://www.shaalaa.com/question-ba...ther-in-the-same-direction-they-_______356772
In it, the "correct" answer is as reported by the OP,

and farther down is the explanation of sorts

So I looked at the AI overview and I found concurrence that there be repulsion

Just to make sure, I "dove deeper" into AI and guess what?

Yay, there is attraction now !!!

I don't think that the physical situation of charged beams moving in vacuum can be be made equivalent to electrons confined to move within an overall neutral conductor.

 

[PeroK]

I do not understand. If they move with the same velocity in one frame they must move at the same velocity in all frames.

If they move with the same velocity in the "lab" frame, then the frame in which they move with the same velocity in opposite directions is, degenerately, the rest frame. And, in that frame, there is simply electrostatic attraction.

It did not specify in the question that the speeds of the beams were the same. I was considering the more general case.

Besides, the conundrum is why two different inertial frames appear to give conflicting results. Considering a third frame cannot resolve that.
Btw, it remains true, of course, that two parallel wires carrying (vectorially) the same current attract, but a wire carrying a current is not the same as a beam of ions: it has no net charge. That said, I have not figured out why an observer moving with the same velocity as the electrons still sees attraction. Maybe it looks like positive charges moving the other way?

As these are beams of particles, not neutral current-carrying wires, we can't ignore the electric force. In the case where the velocities are the same, we have attraction in the rest frame. In a frame where the beams are moving, we have a magnetic repulsion. But, considering length contraction in this frame makes the beams more intense and increases the electric attraction.

In any case, the electric force must dominate in all frames.

 

[haruspex]

If they move with the same velocity in the "lab" frame, then the frame in which they move with the same velocity in opposite directions is, degenerately, the rest frame. And, in that frame, there is simply electrostatic attraction.

I must be misinterpreting what you are saying. As I read the above, you are saying that if in one frame they are both moving at velocity $\vec v$ then there is some other frame, moving at $\vec u$ wrt the first frame, say, in which the velocities (both $\vec v-\vec u$) are now equal and opposite. Seems to me that implies $\vec v=\vec u$.

It did not specify in the question that the speeds of the beams were the same. I was considering the more general case.

No, you specified "same velocity".
I think what you are intending to say is that if they are moving at parallel but different velocities then there is a frame in which they are moving with equal and opposite velocities. Well, of course, but that only gives a way of solving the general problem by choice of a particular frame (as you already did for the equal velocity case in post #6) ; it doesn’t give a way of understanding the difficulty that whether there is a magnetic (Lorentz) force in the equal velocity case seems to depend on the choice of frame. That requires SR.

As these are beams of particles, not neutral current-carrying wires,

As I also noted in post #9.

In a frame where the beams are moving, we have a magnetic repulsion. But, considering length contraction in this frame makes the beams more intense and increases the electric attraction.

Exactly.