# Charged particle moving relativistically through E field

• eck
In summary, the student is confused about the SR arguments used to derive the Lorentz force law. They are also concerned about an intuitively counterintuitive consequence of the law.
eck
Right now I'm taking an introductory E&M class that uses Purcell's book "Electricity and Magnetism." The chapter we're covering focuses on deriving the Lorentz force law for moving charges using SR arguments. This is confusing because we've never covered SR before, so I have a lot of difficulty going through the appropriate transformations and understanding what's going on. One problem that is troubling me a lot goes like this:

Consider the trajectory of a charged particle moving with a speed 0.8c in the x direction when it enters a large region in which there is a uniform electric field in the y direction. Show that the x velocity of the particle must actually decrease. What about the x component of momentum?

This is totally counterintuitive to me, and if someone could explain it for me I would really appreciate it.

In the coordinate system where the particle is moving at .8c, is the electric field stationary? If so, there should be no magnetic field, and thus no reason for the momentum of the particle in the x direction to change.

If the electric field was moving, it'd be a different story.

I'm not sure what you mean by the field being stationary. If you're asking if the electric field varies with time, then the answer is no. The observer seeing the electron moving past at 0.8c measures no magnetic field. In the reference frame of the particle, however, there is a magnetic force caused by the lorentz contraction of the field lines. In fact, the Lorentz force equation is the definition for the magnetic field. From my book: "It will turn out that a field B with [the properties of the Lorentz force equation] must exist if the forces between electric charges obey the postulates of special relativity."

I think I just figured it out. The quantity $$c^2p^2 - E^2$$ is not changed by the Lorentz transformation, so it seems like a good equation to start from. Since the electric field does work on the particle the energy term gets bigger, which means the momentum term must get smaller. Because the particle gains y-velocity, the x-velocity must decrease if the momentum is going to decrease. Does this sound reasonable?

## 1. How does a charged particle move through an electric field?

When a charged particle moves through an electric field, it experiences a force due to the interaction between its electric charge and the electric field. The direction of the force depends on the direction of the electric field and the charge of the particle. If the electric field is in the same direction as the particle's motion, it will experience a force in the same direction, causing it to accelerate. If the electric field is in the opposite direction, the particle will experience a force in the opposite direction, causing it to decelerate.

## 2. What is a relativistic particle?

A relativistic particle is a particle that moves at speeds close to the speed of light, meaning its velocity is a significant fraction of the speed of light (c). In this scenario, special relativity must be taken into account, as the particle's mass, length, and time measurements will be affected by its high velocity.

## 3. How does special relativity affect the motion of a charged particle in an electric field?

Special relativity affects the motion of a charged particle in an electric field by altering its mass, length, and time measurements. As the particle's velocity approaches the speed of light, its mass increases, and its length contracts. This results in a change in the particle's inertia, making it harder to accelerate. Additionally, time dilation occurs, meaning the particle's perception of time is different from an observer at rest, resulting in a change in the particle's acceleration.

## 4. What is the equation for calculating the relativistic motion of a charged particle in an electric field?

The equation for calculating the relativistic motion of a charged particle in an electric field is F = q(E + v x B), where F is the force experienced by the particle, q is the charge of the particle, E is the electric field, v is the velocity of the particle, and B is the magnetic field. This equation takes into account special relativity and the Lorentz force law, which describes the force on a charged particle in an electric and magnetic field.

## 5. How does the velocity of a charged particle affect its motion in an electric field?

The velocity of a charged particle has a significant impact on its motion in an electric field. As the particle's velocity increases, it experiences a greater force due to its interaction with the electric field. This results in a greater acceleration and a shorter time for the particle to travel through the electric field. As the particle's velocity approaches the speed of light, its motion will be significantly affected by special relativity, resulting in a change in its acceleration and trajectory.

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