Magnetic force between moving charges Given two infinite parallel charged rods with equal charge d

poklanima5lqp3 2022-05-10 Answered
Magnetic force between moving charges
Given two infinite parallel charged rods with equal charge density λ. They are moving with same constant velocity v parallel to the rods. Find the speed v for which the magnetic attraction is equal to the electrostatic repulsion.
Well, I know how to solve this problem: we first find out the magnetic field created by one rod on the other using Biot's and Savart's law, then we use the definition of B ( d F = v d q × B ) to find the magnetic force, then equate magnetic and electrostatic forces to find v, which will be greater than or equal to c, thus conclude it is impossible for the forces to be equal.
However, one can argue as the following:
We all know that "same laws of physics apply in all inertial frames". With a constant velocity v ,the rest frame of the rods is an inertial frame. Therefore, if Biot-Savart law applies in our frame, it has to apply in the rest frame. If so, none of the rods will feel a magnetic field from the other one because their relative speed is zero, and there will be no magnetic force between the rods.
I've seen this question several times before in references, exams, exercise sheets,and in many different forms (parallel planes, beam of electrons ...),but no one ever used this argument.What is the problem in it ? Is it something related to Maxwell's equations or special relativity ? Or what else ?
I know a similar question was asked before, but the answers weren't satisfying. Please provide your answers with necessary mathematics.
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (2)

Eliezer Olson
Answered 2022-05-11 Author has 16 answers
I think about this problem the same way that you do, but I phrase it slightly differently.
First the intuitive solution. An observer in the rest frame of the two line charges would observe them accelerating away from each other due to electrostatic repulsion. Relativity demands, then, that no inertial frame exists where the two line charges accelerate towards each other due to magnetic attraction.
Now the mathematical solution. The Biot-Savart and Lorentz-force laws (which are already fully relativistic) tell you that in the limit that v c, you find that F B approaches F E . You know from your prior experience with relativity that v c is physically an unreachable limit, so just because you can plug v = c into some expression about forces doesn't mean you should get sloppy and say such things.
The reason for continuing to the mathematical solution is mainly to confirm that the Lorentz force is consistent with your special-relativistic intuition. It's possible to imagine a wrong expression for the Lorentz force which would predict that boost to a frame traveling at c / 2 would cause the direction of the total force, and thus the net acceleration, to change sign. That's the kind of killer error you have to look for when you are coming up with solutions to new problems or trying to model new phenomena.
Here's a mathematically precise way to phrase your relativistic argument if we make the minor change that the two lines are oppositely charged:
In the rest frame of the line charges, electrostatic attraction will cause them to touch after some finite time t. In reference frames where the line charges have velocity v the time for them to touch is magnified by time dilation to t / 1 v 2 / c 2 ; an observer in this reference frame would ascribe the reduced acceleration to an additional force ("magnetism"). The time before contact can made arbitrary large in the limit v c, so the magnetic force can be made arbitrarily close to, but not equal to or larger than, the electric force.
Not exactly what you’re looking for?
Ask My Question
Aedan Tyler
Answered 2022-05-12 Author has 1 answers
As far as I can tell that seems perfectly valid. As long as the net force is equal in each reference frame then that should tell you that you did it right.
When you say "No one has ever used this argument" what did you mean? I believe it's pretty fundamental to understand that you can change your reference frame so that the force from a magnetic field is zero. The force from the electric field will change however because of relativistic effects on the charge density of the rods.
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2022-05-08
Question about magnetic force
I have some confusion regarding the magnetic force. I know that the magnetic field created by a moving charge or current EXERTS a force on any moving charge or current that is present in the field. But when trying to understand the motion of a charged particle in an uniform magnetic field, the youtube video I saw explained it like this: "The magnetic force on a charged particle ALWAYS points perpendicularly with respect to the velocity and magnetic field. Whenever a force acts on an object perpendicular to its motion, the object will undergo circular motion--this creates a centripetal acceleration)"
I am confused. Is the charged particle exerting a force on itself? Or what is the force that acts on the charged particle that is moving? If the charge particle creates a force due to the magnetic field, is the force it creates itself the force that makes it undergo a circular motion?
asked 2022-05-10
Magnetic force and work
If the magnetic force does no work on a particle with electric charge, then: How can you influence the motion of the particle? Is there perhaps another example of the work force but do not have a significant effect on the motion of the particle?
asked 2022-04-30
Does protons moving parallel to each other exert magnetic force?
If two protons are moving parallel to each other in same direction with equal velocities then do they exert magnetic force on each other.? As protons are moving they should produce magnetic field and there should be some magnetic force + electric force on each of them. But if look from their refrence frame then both the protons are at rest and hence there will be no magnetic force (as magnetic field is created by a moving charge), only electric force on each of them. How can this be possible that force(magnetic force) on a particle becomes different on changing the reference frame even though the reference frames are non accelerating with respect to each other?
asked 2022-05-18
Magnetic force between two point charges
I tried to derive the magnetic force between two point-charges for iterative computation. Starting out with Lorentz force and Biot–Savart law for a point charge.
F = q 2 ( Δ v × B )
B = ( Δ v × Δ x ) ( q 1 | | Δ x | | 3 μ 0 4 π )
And got this for the answer by direct subtitution:
F = q 2 ( Δ v × ( Δ v × Δ x ) ( q 1 | | Δ x | | 3 μ 0 4 π ) )
It does not seem to be right for several reasons.
1.It seems as if electron would be attracted to the nucleus by both magnetic and electrostatic force. Considering hydrogen atom.
2.It is possible to show that between non-moving particle and a non-moving wire with current in it should exist magnetic force. (Magnetic force acts between charge carriers in the wire and the point-charge. Non-moving particles in the wire do not have influence on the magnetic force)
Where have I gone wrong and how to find the correct expression for the magnetic force between two point-charges? Does the equation hold if Δ v << c? If this equation proves to be wrong, what would be the correct approach?
asked 2022-04-07
How Can There Be Magnetic Force Without Velocity
Suppose there is a space with constant magnetic field, and a charged particle is moving in that space with a constant velocity, ofcourse it experiences a magnetic force and gets deflected.
But the particle it not necessarily moving wrt all frames. There may be some frame for which the particle doesn't move at all but still gets deflected. How is this possible, there must be some velocity for the particle to experience any deflection under influence of a magnetic field?
Also a still particle in the same space doesn't experience any deflection. True, but the same particle may be moving wrt to some other frame without experiencing a magnetic force, so here we have a velocity but no magnetic force case.
I remember I have used the relation | F | = q v | B | a lot before, but I am afraid to use it any more, these things are really confusing.
asked 2022-05-15
Direction of magnetic force
Does magnetic force act along the line joining the centres like gravitational and electric forces do?
Are the directions of magnetic field and magnetic lines of force the same? I have read that the direction of the field is tangential to the direction of the line of force.
asked 2022-05-14
Can magnetic force do work?
I have been told numerous times that magnetic force do no work at all but I have some trouble digesting this fact. Now suppose we have two straight wire with some current, they certainly can feel force which may be repulsive or attractive depending upon current direction, can magnetic force do work? We also have magnetic potential energy defined to U = μ B which suggests magnetic field can store energy and hence do some kind of work.